Proof. The proof is by now standard. One covers the domain \(S_L\) with balls of finite overlap, such that on each ball all possible functions \(\eta\) can be written as a graph. On these sets one may apply Theorem 6. Hence the partition of unity argument introduced in [20] allows to construct the desired operator. ◻

Proof. At first we consider the solution to the integral equation (which is nothing but a simple ordinary differential equation with unique local solution), \[g(t)=g(0)+\int_0^t h(s)\,\mathrm{d}s+c_1\int_0^tg^p(s)\,\mathrm{d}s\text{ with }g(0)=c_0.\] We find that \(g\) is a continuous solution to an ODE for short times. Moreover by [27] that there exists a \(\tilde{T}>0\) depending on \(h, c_0, c_1\) and \(p\), such that \[g(t)\leq 2g(0)+2\int_0^th(s)\,\mathrm{d}s\quad\text{for a.a. }t\in[0,\tilde{T}].\] Now we find that by our assumption on \(f\) that \[f(t)-g(t)\leq c_1\int_0^t(f^p-g^p)\,\mathrm{d}s\quad\text{for a.a. }t\in[0,\tilde{T}].\] It follows that \[f(t)-g(t)\leq c_1\int_0^t\int_0^1(\theta \abs{f(s)}+(1-\theta)\abs{g(s)})^{p-1}\, \mathrm d\theta(f(s)-g(s))\,\mathrm{d}s\quad\text{for a.a. }t\in[0,\tilde{T}].\] This implies for \(a(t):=(f(t)-g(t))_{+},\) \[a(t)\leq c_1\int_0^t\bigg(\int_0^1(\theta \abs{f(s)}+(1-\theta)\abs{g(s)})^{p-1}\, \mathrm d\theta\bigg) a(s)\,\mathrm{d}s\quad\text{and}\quad a(0)=0.\] We obtain \(a(t)\equiv0\) by the standard Gronwall lemma as \(\int_0^1(\theta \abs{f}+(1-\theta)\abs{g})^{p-1}\, d\theta \in L^1(0,\tilde{T})\). Hence both claims follows as \(f(t)\leq g(t)\) by the above and the continuity of \(g\) at \(0\). ◻

Proof. Based on the analysis around 30 , the system 29 is smooth with respect to the time variable. Thereby we differentiate 29 in time, multiply the resulting expression by \({\alpha_n^k}'(t)\) and sum with respect to \(k\) (this corresponds to a test by \((\partial_t^2\eta_n,\partial_t\overline{\mathbf{v}}_n)\) in the time-differentiated system). Note that we have the following identities \[\begin{align} \sum_{k=1}^n{\alpha_n^k}'(t)\bfX_k&=\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)=\mathbf{B}_{\zeta_\varepsilon}^\intercal\partial_t\overline{\mathbf{v}}_n+\partial_t\mathbf{B}_{\zeta_{\varepsilon}}^\intercal\overline{\mathbf{v}}_n, \\ \sum_{k=1}^n{\alpha_n^k}'(t)X_k&=\partial_t(J_{\zeta_{\varepsilon}}\circ\bfvarphi\,\partial_t\eta_n)=\partial_t J_{\varepsilon}\circ\bfvarphi\,\partial_t\eta_n+J_{\varepsilon}\circ\bfvarphi\,\partial_t^2\eta_n, \\ \sum_{k=1}^n{\alpha_n^k}'(t)\nabla(\mathbf{B}_{\zeta_{\varepsilon}}^{-\intercal}\bfX_k)&=\nabla(\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)), \\ \sum_{k=1}^n{\alpha_n^k}'(t)\nabla(\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\bfX_k)&=\nabla(\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)), \\ \sum_{k=1}^n{\alpha_n^k}'(t)\Delta(J_{\zeta_{\varepsilon}}^{-1}\circ\bfvarphi\, X_k)&=\Delta(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\varepsilon}\circ\bfvarphi\,\partial_t\eta_n)), \\ \sum_{k=1}^n{\alpha_n^k}'(t)\Delta(\partial_t J_{\varepsilon}^{-1}\circ\bfvarphi\, X_k)&=\Delta(\partial_t J_{\varepsilon}^{-1}\circ\bfvarphi\, \partial_t(J_{\varepsilon}\circ\bfvarphi\,\partial_t\eta_n)), \\ \sum_{k=1}^n{\alpha_n^k}'(t)\nabla(J_{\varepsilon}^{-1}\circ\bfvarphi\, X_k)&=\nabla(J_{\varepsilon}^{-1}\circ\bfvarphi\,\partial_t(J_{\varepsilon}\circ\bfvarphi\partial_t\eta_n)), \\ \sum_{k=1}^n{\alpha_n^k}'(t)\nabla(\partial_tJ_{\varepsilon}^{-1}\circ\bfvarphi\, X_k)&=\nabla(\partial_t J_{\varepsilon}^{-1}\circ\bfvarphi\,\partial_t(J_{\varepsilon}\circ\bfvarphi\,\partial_t\eta_n)). \end{align}\] Further note that \[\|\nabla\overline{\mathbf{v}}_n(t)\|_{L^2_\mathbf{x}}^2\lesssim \int_0^t\|\partial_t\nabla\overline{\mathbf{v}}_n\|^2_{L^2_\mathbf{x}}\,\mathrm{d}s+ \|\nabla\overline{\mathbf{v}}_n(0)\|_{L^2_\mathbf{x}}^2.\] Hence we obtain after integrating over time that \[\begin{align} \mathcal{E}_n(t)+\int_0^t\|\partial_t\nabla\overline{\mathbf{v}}_n\|^2_{L^2_\mathbf{x}}\,\mathrm{d}s+\varepsilon\int_0^t\|\partial_t^2\eta_n\|_{W^{1,2}_\mathbf{y}}^2\,\mathrm{d}s=\mathcal{E}_n(0)+({\tt I})_n+\cdots+ ({\tt XIII})_n, \end{align}\] where the terms \(({\tt I})_n,\dots, ({\tt XIII})_n\) are defined and estimated in the following. Since \(\mathbf{B}_{\zeta_\varepsilon}\), \(J_{\zeta_\varepsilon}\) and \(\boldsymbol{\Psi}_{\zeta_\varepsilon}\) are all smooth in time and space, it suffices to estimate the critical terms in the remaining part.

We first have by integration by parts that \[\begin{align} ({\tt I})_n:&=\int_0^t\int_\Omega\partial_t^2\overline{\mathbf{v}}_n\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal} \mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n J_{\zeta_\varepsilon}\, \mathrm{d} \mathbf{x}\mathrm ds \\ &=\int_\Omega\partial_t\overline{\mathbf{v}}_n\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n J_{\zeta_\varepsilon}\, \mathrm{d} \mathbf{x}\big|_0^t-\int_0^t\int_\Omega\partial_t\overline{\mathbf{v}}_n\partial_t(\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n J_{\zeta_\varepsilon})\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\leq \kappa\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2+c(\kappa) C_0(1+t)+\int_0^t\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds. \end{align}\] Then we also see that \[\begin{align} ({\tt II})_n:&=\int_0^t\int_\Omega\partial_t\overline{\mathbf{v}}_n\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n) J_{\zeta_\varepsilon}\, \mathrm{d} \mathbf{x}\mathrm ds+\int_0^t\int_\Omega\partial_t\overline{\mathbf{v}}_n\mathbf{B}_{\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\partial_t J_{\zeta_\varepsilon}\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\lesssim \int_0^t\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}(\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}+\|\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}})\mathrm ds\\ &\leq \int_0^t\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds+C_0 t. \end{align}\] \[\begin{align} ({\tt III})_n:&=\int_0^t\int_\Omega\partial_t\nabla\overline{\mathbf{v}}_n\partial_t{\boldsymbol{\Psi}}_{\zeta_\varepsilon}^{-1}\circ{\boldsymbol{\Psi}}_{\zeta_\varepsilon} \mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal \overline{\mathbf{v}}_n) J_{\zeta_\varepsilon}\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\leq \kappa\int_0^t\|\partial_t\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds+c(\kappa)\int_0^t\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds+C_0 t. \end{align}\] \[\begin{align} ({\tt IV})_n:&=\int_0^t\int_\Omega\nabla\overline{\mathbf{v}}_n\partial_t\left(\partial_t{\boldsymbol{\Psi}}_{\zeta_\varepsilon}^{-1}\circ{\boldsymbol{\Psi}}_{\zeta_\varepsilon}\,\mathbf{B}_{\zeta_\varepsilon}^{-\intercal} J_{\zeta_\varepsilon}\right)\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\lesssim \int_0^t\|\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds+\int_0^t(\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2+\|\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2)\mathrm ds. \end{align}\] \[\begin{align} ({\tt V})_n:&=\frac{1}{2}\int_0^t\int_\omega\partial_t^2\zeta_{\varepsilon}\partial_t\eta_n\,\partial_t(J_{\zeta_{\varepsilon}}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds+\frac{1}{2}\int_0^t\int_\omega\partial_t\zeta_{\varepsilon}\partial_t^2\eta_n\,\partial_t(J_{\zeta_{\varepsilon}}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\lesssim \int_0^t\|\partial_t^2\eta_n\|_{L^2_\mathbf{y}}^2\mathrm ds+\int_0^t\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2\mathrm ds. \end{align}\] \[\begin{align} ({\tt VI})_n:&=\frac{1}{2}\int_0^t\int_\Omega\partial_t\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds+\frac{1}{2}\int_0^t\int_\Omega\overline{\mathbf{v}}_n\partial_t\mathbf{B}_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\quad +\frac{1}{2}\int_0^t\int_\Omega\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_n\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds+\frac{1}{2}\int_0^t\int_\Omega\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}\partial_t\nabla\overline{\mathbf{v}}_n\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\lesssim \int_0^t(\|\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}+\|\partial_t\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}})(\|\overline{\mathbf{v}}_n\|_{L^4_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}_n\|_{L^4_\mathbf{x}}+\|\overline{\mathbf{v}}_n\|_{L^4_\mathbf{x}}^2)\mathrm ds+\int_0^t\|\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}_n\|_{L^4_\mathbf{x}}^2\mathrm ds\\ &\lesssim \kappa\int_0^t\|\partial_t\overline{\mathbf{v}}_n\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d}s+c(\kappa) \int_0^t\Big(\|\overline{\mathbf{v}}_n\|^{8}_{W^{1,2}_\mathbf{x}}+\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^{8}+1\Big)\,\mathrm{d}s. \end{align}\] The integral from the contribution of \[\frac{1}{2}\int_0^t\int_\Omega(\overline{\mathbf{v}}_n\cdot\mathbf{B}_{\zeta_\varepsilon}\nabla)(\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\bfX_k)\cdot\overline{\mathbf{v}}_n\,\mathrm d\mathbf{x}\,\mathrm{d}s,\] denoted by \(({\tt VII})_n\), can be treated similarly as \(({\tt VI})_n\).

We continue the estimate: \[\begin{align} ({\tt VIII})_n:&=\int_0^t\int_\Omega\partial_t\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_n\,\nabla(\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n))\, \mathrm{d} \mathbf{x}\mathrm ds+\int_0^t\int_\Omega\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_n\,\nabla(\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n))\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\quad +\int_0^t\int_\Omega\bfA_{\zeta_\varepsilon}\partial_t\nabla\overline{\mathbf{v}}_n\left(\nabla(\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t(\mathbf{B}_{\zeta_\varepsilon}\overline{\mathbf{v}}_n))-\partial_t\nabla\overline{\mathbf{v}}_n\right)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\lesssim \int_0^t\int_\Omega (\partial_t\nabla\overline{\mathbf{v}}_n+\nabla\overline{\mathbf{v}}_n)(\overline{\mathbf{v}}_n+\partial_t\overline{\mathbf{v}}_n+\nabla\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\leq \kappa\int_0^t\|\partial_t\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2\mathrm ds+c(\kappa)\int_0^t(\|\nabla\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2+\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2+\|\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2)\mathrm ds. \end{align}\] \[\begin{align} ({\tt IX})_n:&=\int_0^t\int_\Omega\partial_t(\bff\circ{\boldsymbol{\Psi}}_{\zeta_\varepsilon}\,\mathbf{B}_{\zeta_\varepsilon}^{-\intercal} J_{\zeta_\varepsilon})\partial_t(\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\lesssim \int_0^t\int_\Omega(\partial_t\overline{\bff}+\partial_t\nabla\overline{\bff})(\partial_t\overline{\mathbf{v}}_n+\overline{\mathbf{v}}_n)\, \mathrm{d} \mathbf{x}\mathrm ds\\ &\leq \int_0^tH(s)\mathrm ds+\int_0^t(\|\partial_t\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2+\|\overline{\mathbf{v}}_n\|_{L^2_\mathbf{x}}^2)\mathrm ds. \end{align}\] \[\begin{align} ({\tt X})_n:&=\int_0^t\int_\omega\partial_t(g\, J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi)\partial_t( J_{\zeta_\varepsilon}\circ\bfvarphi \partial_t\eta_n)\,\mathrm{d} t\mathrm ds\\ &\lesssim \int_0^t\int_\omega(g+\partial_tg)(\partial_t\eta_n+\partial_t^2\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\leq \int_0^t H(s)\mathrm ds+\int_0^t(\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t^2\eta_n\|_{L^2_\mathbf{y}}^2)\mathrm ds. \end{align}\] Now we take again the integration by parts with respect to time and obtain that \[\begin{align} ({\tt XI})_n:&=\int_0^t\int_\omega\partial_t^3\eta_n J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n\, \mathrm{d}\mathbf{y}\mathrm ds+\int_0^t\int_\omega\partial_t^2\eta_n\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &=\int_\omega\partial_t^2\eta_n J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n\, \mathrm{d}\mathbf{y}\big|_0^t -\int_0^t\partial_t^2\eta_n\partial_t(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad +\int_0^t\int_\omega\partial_t^2\eta_n\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\leq\kappa\|\partial_t^2\eta_n\|_{L^2_\mathbf{y}}^2+c(\kappa)\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2+\int_0^t(\|\partial_t^2\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2)\mathrm ds. \end{align}\] We also see that \[\begin{align} ({\tt XII})_n:&=\int_0^t\int_\omega\partial_t\Delta\eta_n\Delta(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad +\int_0^t\int_\omega\Delta\eta_n\Delta(\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n))\, \mathrm{d}\mathbf{y}\mathrm ds\\ &=\int_0^t\int_\omega\partial_t\Delta\eta_n\Delta(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds+\int_0^t\Delta\eta_n\partial_t\Delta\eta_n\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad +\int_\omega\Delta\eta_n\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n\, \mathrm{d}\mathbf{y}\big|_0^t -\int_0^t\int_\omega\partial_t\Delta\eta_n\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad -\int_0^t\int_\omega\Delta\eta_n\partial_t(\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi J_{\zeta_\varepsilon}\circ\bfvarphi)\partial_t\eta_n\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\lesssim \|\Delta\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2+\int_0^t(\|\partial_t\Delta\eta_n\|_{L^2_\mathbf{y}}^2+\|\Delta\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2)\mathrm ds. \end{align}\]

Finally by taking an integration by parts in space we deal with the integral with \(\varepsilon\) as below: \[\begin{align} ({\tt XIII})_n:&=\varepsilon\int_0^t\int_\omega\partial_t^2\nabla\eta_n\,\nabla(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad +\varepsilon\int_0^t\int_\omega\partial_t\nabla\eta_n\,\nabla(\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n))\, \mathrm{d}\mathbf{y}\mathrm ds\\ &=-\varepsilon\int_0^t\int_\omega\partial_t^2\eta_n\Delta(J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\quad -\varepsilon\int_0^t\int_\omega\partial_t\Delta\eta_n\,\partial_t J_{\zeta_\varepsilon}^{-1}\circ\bfvarphi\partial_t(J_{\zeta_\varepsilon}\circ\bfvarphi\partial_t\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\lesssim \int_0^t\int_\omega\partial_t^2\eta_n(\partial_t\eta_n+\partial_t\nabla\eta_n+\partial_t\Delta\eta_n)\, \mathrm{d}\mathbf{y}\mathrm ds+\int_0^t\int_\omega\partial_t\Delta\eta_n\partial_t\eta_n\, \mathrm{d}\mathbf{y}\mathrm ds\\ &\leq \int_0^t\|\partial_t^2\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t\eta_n\|_{L^2_\mathbf{y}}^2+\|\partial_t\Delta\eta_n\|_{L^2_\mathbf{y}}^2\mathrm ds. \end{align}\] Collecting all estimates and choosing \(\kappa\) small enough yields the claim. ◻

Proof. In order to control the pressure we write similarly to 14 \[\begin{align} \overline{\pi}_\varepsilon=\overline{\pi}_{\varepsilon,0}+c_{\overline{\pi}_\varepsilon}, \end{align}\] where \((\overline{\pi}_{\varepsilon,0})_\Omega:=\int_\Omega\overline{\pi}_{\varepsilon,0}\, \mathrm{d} \mathbf{x}=0\) and \(c_{\overline{\pi}_\varepsilon}\) is a function of time only. The latter satisfies \[\begin{align} c_{\overline{\pi}_\varepsilon}(t)\int_{\omega}\bfn\mathbf{B}_{{\zeta_\varepsilon}}\circ\bfvarphi\bfn\, \mathrm{d}\mathbf{y}=\int_{\omega}\bfn\big(\bfA_{{\zeta_\varepsilon}}\nabla\overline{\bfv}_\varepsilon-\mathbf{B}_{{\zeta_\varepsilon}}\overline{\pi}_{\varepsilon,0}\big)\circ\bfvarphi\bfn\, \mathrm{d}\mathbf{y}+\int_\omega\partial_t^2\eta_\varepsilon\, \mathrm{d}\mathbf{y}-\int_\omega g\, \mathrm{d}\mathbf{y}, \end{align}\] due to the structure equation. Noticing that \(\mathbf{B}_{{\zeta_\varepsilon}}\) is uniformly elliptic we infer from Poincaré’s inequality \[\begin{align} \int_{I}\|\overline{\pi}_\varepsilon\|^2_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t&\lesssim \int_{I}\|\nabla\overline{\pi}_\varepsilon\|^2_{L^{2}_\mathbf{x}}\,\mathrm{d} t+\int_{I}|c_{\overline{\pi}_\varepsilon}|^2\,\mathrm{d} t\\ &\lesssim \int_{I}\|\nabla\overline{\pi}_\varepsilon\|^2_{L^{2}_\mathbf{x}}\,\mathrm{d} t+\int_{I}\int_\omega|\partial_t^2\eta_\varepsilon|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+\int_{I}\int_\omega|g|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad +\int_{I}\|\overline{\pi}_{\varepsilon,0}\|_{L^{2}(\partial\Omega)}^2\,\mathrm{d} t+\int_{I}\|\nabla\overline{\bfv}_\varepsilon\|_{L^{2}(\partial\Omega)}^2\,\mathrm{d} t. \end{align}\] By the trace theorem and \((\overline{\pi}_{\varepsilon,0})_\Omega=0\) we obtain for \(I=[0,T]\) that \[\begin{align} \int_{I}\|\overline{\pi}_{\varepsilon,0}\|_{L^{2}(\partial\Omega)}^2\,\mathrm{d} t &\lesssim\int_{I}\|\overline{\pi}_{\varepsilon,0}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t\lesssim \int_{I}\|\nabla\overline{\pi}_{\varepsilon,0}\|_{L^{2}_\mathbf{x}}^{2}\,\mathrm{d} t=\int_{I}\|\nabla\overline{\pi}_\varepsilon\|_{L^{2}_\mathbf{x}}^{2}\,\mathrm{d} t. \end{align}\] We can use estimates for the steady Stokes system (see Theorem 4 and Remark 5) with \(s=2\), \(p=2\) and \(n=3\) to get\[\begin{align} &\int_I\int_\Omega|\nabla^2\overline{\bfv}_\varepsilon|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega|\nabla\overline{\pi}_\varepsilon|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\int_\Omega\big(|\mathbf{B}_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon\,\overline{\mathbf{v}}_\varepsilon|^{2}+|J_{\zeta_\varepsilon}\partial_t\overline{\bfv}_\varepsilon|^{2}+|J_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon\partial_t\bfPsi_{\zeta_\varepsilon}^{-1}\circ\bfPsi_{\zeta_\varepsilon}|^2+|J_{\zeta_\varepsilon}\bff \circ \bfPsi_{\zeta_\varepsilon}|^{2}\big)\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad +\int_I\|\partial_t\eta_\varepsilon\|^{2}_{W^{\frac{3}{2}, 2}(\omega)}\,\mathrm{d} t. \end{align}\] Except for the first term, all terms on the right-hand side can be controlled by \(\int_I\mathcal{E}_\eta\,\mathrm{d} t\). As for the convective term we have \[\begin{align} & \int_I\int_\Omega|\mathbf{B}_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon\,\overline{\mathbf{v}}_\varepsilon|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t \lesssim \int_{I }\|\overline{\mathbf{v}}_\varepsilon\|^2_{L^4_\mathbf{x}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^{4}_\mathbf{x}}^2\,\mathrm{d} t\\ &\leq\,c\int_{I }\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}^2\|\nabla \overline{\mathbf{v}}_\varepsilon\|^{\frac{1}{2}}_{L^{2}_\mathbf{x}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{3}{2}}\,\mathrm{d} t\\ &\leq\,c(\kappa)\int_{I }\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}^{8}\|\nabla \overline{\mathbf{v}}_\varepsilon\|^{2}_{L^{2}_\mathbf{x}}\,\mathrm{d} t+\kappa\int_{I }\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t\\ &\leq \,c(\kappa)\int_{I }\|\overline{\mathbf{v}}_\varepsilon\|^{10}_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t+\kappa\int_{I }\|\overline{\mathbf{v}}_\varepsilon\|_{W^{2,2}_\mathbf{x}}^2\, dt. \end{align}\] Choosing \(\kappa\) small enough we obtain the result. ◻

Proof. For that we take the finite difference of 26 in time and test by \[\begin{align} (\phi_\varepsilon,\overline{\boldsymbol{\varphi}}_\varepsilon)=\left(\Delta_{-h}^s\Delta_h^s \Delta_\tau\eta_\varepsilon-\mathscr K_{{\zeta_\varepsilon}}(\Delta_{-h}^s\Delta_h^s \Delta_\tau\eta_\varepsilon, \,\,\mathscr F_{\zeta_\varepsilon}^{\mathrm{div}}(\Delta_{-h}^s\Delta_h^s \Delta_\tau\eta_\varepsilon-\mathscr K_{{\zeta_\varepsilon}}(\Delta_{-h}^s\Delta_h^s \Delta_\tau\eta_\varepsilon))\circ\bfPsi_{\zeta_\varepsilon})\right). \end{align}\] Here for \(0<s\leq 1/4\), \(\Delta_h^sv(\mathbf{y})=h^{-s}(v(\mathbf{y}+h\boldsymbol{e}_\alpha)-v(\mathbf{y}))\) is the fractional difference quotient for space variable in direction \(\boldsymbol{e}_\alpha\) with \(\alpha\in\{1,2\}\). \(\mathscr F_{\zeta_\varepsilon}^{\mathrm{div}}\) is the solenoidal extension, please see for instance Proposition 1. It follows that \[\begin{align} \nonumber &\int_I \int_{ \omega }|\Delta_h^s\Delta_{\mathbf{y}}\Delta_\tau\eta_\varepsilon|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+\varepsilon\sup_I\int_\omega|\Delta_h^s\nabla_{\mathbf{y}}\Delta_\tau\eta_\varepsilon|^2\, \mathrm{d}\mathbf{y}\\ \nonumber& =\varepsilon \int_\omega|\Delta_h^s\nabla_{\mathbf{y}}\Delta_\tau\eta_\varepsilon(0)|^2\, \mathrm{d}\mathbf{y}-\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot\partial_t\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\quad +\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\nabla\overline{\boldsymbol{\varphi}}_\varepsilon \, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\nabla\Delta_\tau\overline{\mathbf{v}}_\varepsilon:\nabla\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\quad +\int_I\frac{\mathrm d}{\,\mathrm{d} t}\left(\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_\tau\overline{\mathbf{v}}\cdot\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}+\int_\omega\partial_t\Delta_\tau\eta_\varepsilon\cdot\phi_\varepsilon\, \mathrm{d}\mathbf{y}\right)\,\mathrm{d} t-\int_I\int_\omega(\partial_t\Delta_\tau\eta_\varepsilon \partial_t\phi_\varepsilon+\Delta_\tau g \,\phi_\varepsilon)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\nonumber\\ &\quad -\int_I\int_\Omega\pi_\varepsilon^\tau\mathop{\mathrm{div}}(\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal \overline{\boldsymbol{\varphi}}_\varepsilon)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega\Delta_\tau\bfh_{\zeta_\varepsilon}\cdot \overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \nonumber\\ &\nonumber\quad +\int_I\int_\Omega\left(-\Delta_\tau J_{\zeta_\varepsilon} \overline{\mathbf{v}}_\varepsilon\cdot \partial_t\overline{\boldsymbol{\varphi}}_\varepsilon+\frac{\mathrm d}{\,\mathrm{d} t}\left(\Delta_\tau J_{\zeta_\varepsilon}\overline{\mathbf{v}}_\varepsilon\cdot \overline{\boldsymbol{\varphi}}_\varepsilon\right)-\Delta_\tau(\partial_t J_{\zeta_\varepsilon}\overline{\mathbf{v}}_\varepsilon)\cdot \overline{\boldsymbol{\varphi}}_\varepsilon\right)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &=:(O)^\varepsilon+(I)^\varepsilon+...+(VII)^\varepsilon+(VIII)^\varepsilon.\label{mom:reg} \end{align}\tag{35}\] By Lemma 1, \({\zeta_\varepsilon}\in X_{\tt shell}(0,T)\), 8 and 10 \[\begin{align} (I)^\varepsilon&\lesssim \int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}(\| \partial_t\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{L^{2}_\mathbf{y}}+\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{L^2_\mathbf{y}}+\| \nabla_{\mathbf{y}}\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{L^{2}_\mathbf{y}})\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}(\| \partial_t\Delta_\tau\eta_\varepsilon\|_{W^{1/2,2}_\mathbf{y}}+\|\Delta_\tau\eta_\varepsilon\|_{W^{3/2,2}_\mathbf{y}})\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_I\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] using also the trace theorem (together with \(\Delta_\tau\overline{\mathbf{v}}_\varepsilon\circ\bfvarphi=\Delta_\tau\partial_t\eta_\varepsilon\mathbf{n}\) in \(\omega\)). Note that we can express \(\mathbf{A}_{\zeta}=a(x,\zeta,\nabla \zeta)\) hence we find that \[\begin{align} \tau\abs{\Delta_\tau \mathbf{A}_{\zeta}}&=\abs{a(x,\zeta(t+\tau,x),\nabla\zeta(t+\tau,x))-a(x,\zeta(t,x),\nabla\zeta(t,x))} \\ &\leq \abs{a(x,\zeta(t+\tau,x),\nabla\zeta(t,x))-a(x,\zeta(t,x),\nabla\zeta(t,x))} \\ &\quad + \abs{a(x,\zeta(t,x),\nabla\zeta(t+\tau,x))-a(x,\zeta(t,x),\nabla\zeta(t,x))}, \end{align}\] and so checking the definition of \(\mathbf{A}_{\zeta_\epsilon}\) we find \[\abs{\Delta_\tau \mathbf{A}_{\zeta_\epsilon}} \lesssim (1+\abs{\nabla \zeta_\epsilon})(\abs{ \Delta_\tau \nabla \zeta_\epsilon}+\abs{\Delta_\tau \zeta_\epsilon}),\] where the constant depends on \(\Omega\) and \(\norm{\zeta_\epsilon}_\infty\). Similarly we can estimate the other terms. Using again Lemma 1, 8 –10 and \({\zeta_\varepsilon}\in X_{\tt shell}(0,T)\) we estimate further that \[\begin{align} (II)^\varepsilon&\lesssim \int_I\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^4_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\nabla\overline{\bfvarphi}_\varepsilon\|_{L^4_\mathbf{x}}\, \mathrm{d} \mathbf{x}+\int_I\|\Delta_\tau{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\nabla\overline{\bfvarphi}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{W^{1,2}_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{W^{1,4}_\mathbf{y}}\,\mathrm{d} t+\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\lesssim \int_I\Big(\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2+\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\Big)\,\mathrm{d} t,\\ (III)_\varepsilon&\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}\big)\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\Big(\|\Delta_\tau \nabla \overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}^2+\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\Big)\,\mathrm{d} t, \end{align}\] where we used the fact that \(\partial_t\nabla_\mathbf{y}\zeta_\varepsilon\in L^\infty_t(W^{1,2}_\mathbf{y})\). We also have \[\begin{align} (IV)^\varepsilon& \lesssim \sup_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\boldsymbol{\varphi}}_{\varepsilon}\|_{L^2_\mathbf{x}}+\sup_I\|\partial_t\Delta_\tau\eta_{\varepsilon}\|_{L^2_\mathbf{y}}\|\phi_\varepsilon\|_{L^2_\mathbf{y}}\\ &\lesssim \sup_I\Big(\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}+\|\partial_t\Delta_\tau\eta_\varepsilon\|_{L^2_\mathbf{y}}\Big)\|\Delta_\tau\eta_\varepsilon\|_{W^{2s,2}_\mathbf{y}}\\ &\lesssim\sup_I\Big(\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2+\|\partial_t\Delta_\tau\eta_\varepsilon\|_{L^2_\mathbf{y}}^2+\| \Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\Big), \end{align}\] and \[\begin{align} (V)^\varepsilon &\lesssim \int_I\|\partial_t\Delta_\tau \eta_\varepsilon\|_{L^2_\mathbf{y}}\left(\|\partial_t\Delta_\tau \eta_\varepsilon\|_{W^{2s,2}_\mathbf{y}} + \Vert \partial_t\zeta_\varepsilon\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{L^1_\mathbf{y}} \right)\,\mathrm{d} t+\int_I\|\Delta_\tau g\|_{L^2_\mathbf{y}}\| \Delta_\tau\eta_\varepsilon\|_{W^{2s,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\Big(\|\partial_t\Delta_\tau\eta_\varepsilon\|_{W^{1/2,2}_\mathbf{y}}^2+\|\Delta_\tau g\|_{L^2_\mathbf{y}}^2+\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}^2\Big)\,\mathrm{d} t\\ &\lesssim \int_I\Big(\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\Delta_\tau g\|_{L^2_\mathbf{y}}^2+\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\Big)\,\mathrm{d} t, \end{align}\] again by the Trace theorem. For the pressure term we have \[\begin{align} (VI)^\varepsilon&\lesssim \int_I\|\pi_\varepsilon\|_{L^6_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\|\overline{\bfvarphi}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\quad + \int_I\|\pi\|_{L^4_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^4_\mathbf{y}}(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}})\|\nabla\overline{\bfvarphi}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\big(\|\pi_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\overline{\bfvarphi}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\big)\,\mathrm{d} t\lesssim \int_I\big(\|\pi_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\Delta_\tau\eta_\varepsilon\|_{W^{1+2s}_\mathbf{y}}^2\big)\,\mathrm{d} t\\ &\lesssim \int_I\big(\|\pi_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\big)\,\mathrm{d} t. \end{align}\] We can now decompose \((VII)^\varepsilon\) into the sum of \[\begin{align} (VII)^\varepsilon_1&:= \int_I\int_\Omega \Delta_\tau\Big(J_{\zeta_\varepsilon} \nabla \overline{\mathbf{v}}_\varepsilon\cdot \partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon}\Big)\cdot\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t, \\(VII)^\varepsilon_2&:= \int_I\int_\Omega\Delta_\tau\big(\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}_\varepsilon\big)\cdot\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t,\\ (VII)^\varepsilon_3&:=- \int_I\int_\Omega\Delta_\tau\big( J_{\zeta_\varepsilon} \bff\circ \bfPsi_{\zeta_\varepsilon}\big)\cdot\overline{\boldsymbol{\varphi}}_\varepsilon\, \mathrm{d} \mathbf{x}\,\mathrm{d} t, \end{align}\] while further decomposing \((VII)^\varepsilon_2\) into the sum of \((VII)^\varepsilon_{2,1}\), \((VII)^{\varepsilon}_{2,2}\) and \((VII)^{\varepsilon}_{2,3}\) (defining them in an obvious manner differentiating in time by product rule). We estimate by Lemma 1, 8 , 10 and \({\zeta_\varepsilon}\in X_{\tt shell}(0,T)\), \[\begin{align} (VII)^\varepsilon_1&\lesssim\int_I \|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}\|\partial_t\nabla\zeta_\epsilon\|_{L^4_\mathbf{y}}\|\overline{\boldsymbol{\varphi}}_\epsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t+\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}\|\partial_t\Delta_\tau\zeta_\epsilon\|_{L^3_\mathbf{y}}\|\overline{\bfvarphi}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\quad +\int_I(\|\Delta_\tau\nabla\overline{\mathbf{v}}_\epsilon\|_{L^2_\mathbf{x}}+\|\nabla\overline{\mathbf{v}}_\epsilon\|_{L^2_\mathbf{x}})\|\overline{\boldsymbol{\varphi}}_{\epsilon}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}(1+\|\partial_t^2\zeta_\varepsilon\|_{L^3_\mathbf{y}})\|\overline{\boldsymbol{\varphi}}_\epsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t+\int_I\|\Delta_\tau\nabla\overline{\mathbf{v}}_\epsilon\|_{L^2_\mathbf{x}}\|\overline{\boldsymbol{\varphi}}_\epsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ & \lesssim \int_I\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^{2}_\mathbf{x}}(1+\|\partial_t^2\zeta_\varepsilon\|_{L^3_\mathbf{y}})(\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{W^{1,2}_\mathbf{y}}+\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\, \nabla{\zeta_\varepsilon}\|_{L^2_\mathbf{y}})\,\mathrm{d} t\\ &\quad +\int_{I^\tau}\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_{-h}^s\Delta_h^s\partial_t\eta_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_{I}\|\nabla \overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}^4\,\mathrm{d} t+\int_I\|\partial_t^2\zeta_\varepsilon\|_{L^3_\mathbf{y}}^4\,\mathrm{d} t+\int_I\|\Delta_\tau\eta_\epsilon\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\Delta_\tau\overline{\mathbf{v}}_\epsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] where we use the embedding \[\label{1embed} \zeta_\varepsilon\in W^{2,\infty}(I; L^2(\omega))\cap L^\infty(I;W^{4,2}(\omega))\hookrightarrow W^{\frac{7}{3},2}(I; L^3(\omega))\hookrightarrow W^{2,4}(I; L^3(\omega)),\tag{36}\] Furthermore, \[\begin{align} (VII)^\varepsilon_{2,1}&\lesssim \int_I\|\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\|_{L^{6}_\mathbf{x}}\|\nabla \overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\|\overline{\boldsymbol{\varphi}}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{W^{1,2}_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2(\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{W^{1,2}_\mathbf{y}}+\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\, \nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^2_\mathbf{y}})\,\mathrm{d} t\\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\leq\,\int_I\big(\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^{2}+\|\nabla \overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}^{4}\big)\,\mathrm{d} t, \end{align}\] \[\begin{align} (VII)^\varepsilon_{2,2}&\lesssim \int_I\|\mathbf{B}_{\zeta_\varepsilon}^\tau\|_{L^{\infty}_\mathbf{x}}\|\Delta_\tau\nabla \overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^4_\mathbf{x}}\|\overline{\boldsymbol{\varphi}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}^\tau\|_{L^{\infty}_\mathbf{y}})\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\,\int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_{I }\big(\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}^{4}+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{4}\big)\,\mathrm{d} t, \end{align}\] \[\begin{align} (VII)^\varepsilon_{2,3}&\lesssim \int_I\|\mathbf{B}_{\zeta_\varepsilon}^\tau\|_{L^{\infty}_\mathbf{x}}\|\nabla \overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\|\overline{\bfvarphi}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}^\tau\|_{L^{\infty}_\mathbf{y}})\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\,\int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_{I }\big(\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}^{4}+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{4}\big)\,\mathrm{d} t. \end{align}\] Finally we see that \[\begin{align} (VIII)^\varepsilon&\lesssim \int_I\|\overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}\big(\| \partial_t\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta_\varepsilon\|_{L^{2}_\mathbf{y}}+\|\Delta_\tau\Delta_{-h}^s\Delta_h^s\eta\, \partial_t{\zeta_\varepsilon}\|_{L^2_\mathbf{y}})\,\mathrm{d} t\\&\quad +\sup_I\|\overline{\mathbf{v}}_\varepsilon\|_{L^{2}_\mathbf{x}}\|\Delta_{-h}^s\Delta_h^s\Delta_\tau\eta\|_{L^2_\mathbf{x}}\\ &\quad +\int_I\Big(\|\partial_t\Delta_\tau\zeta_\varepsilon\|_{L^3_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}+\|\partial_t\zeta_\varepsilon^\tau\|_{L^\infty_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\Big)\|\Delta_\tau\eta_\varepsilon\|_{W^{2s,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim 1+ \int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_I\|\Delta_\tau\eta_\varepsilon\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t+\sup_I\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2, \end{align}\] As all estimates are uniformly with respect to \(h\) we have finished the proof. ◻

Proof of Proposition 10. We start working with system 26 . Observe that \(( \eta_\varepsilon, \overline{\mathbf{v}}_\varepsilon)\), as solutions of 26 , has sufficient regularity such that one can use \((\partial_t \eta_\varepsilon, \overline{\mathbf{v}}_\varepsilon)\) as a test-function. Hence in particular the following energy inequality is satisfied: \[\begin{align} \nonumber &\sup_I \int_\omega\big(\vert \partial_t\eta_\varepsilon\vert^2 + \vert \Delta_{\mathbf{y}}\eta_\varepsilon\vert^2 \big)\, \mathrm{d}\mathbf{y} + \sup_I \int_\Omega\vert \overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x} +\varepsilon\int_I\int_\omega|\partial_t\nabla_{\mathbf{y}}\eta_\varepsilon|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+ \int_I\int_\Omega\vert \nabla \overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_\omega\big(\vert \eta_*\vert^2 + \vert \Delta_{\mathbf{y}}\eta_0\vert^2 \big)\, \mathrm{d}\mathbf{y} + \int_I\int_\omega|g|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t + \int_\Omega\vert \overline{\mathbf{v}}_0\vert^2\, \mathrm{d} \mathbf{x} + \int_I\int_\Omega\vert \overline{\bff}\vert^2 \, \mathrm{d} \mathbf{x}\,\mathrm{d} t, \label{34681} \end{align}\tag{37}\] where we used 27 and the hidden constant is independent of \(T\) and \(\varepsilon\). In particular we find that \(\norm{\eta_\epsilon}_\infty\leq C_0\), where \(C_0\) is independent of \(\varepsilon\) but may depend on the initial data via 37 .

Now we want to differentiate the equation in time and test with \(\partial_t\overline{\mathbf{v}}_\varepsilon+\bfG\) in the momentum equation (the corresponding test-function for the structure equation is then \(\partial_t^2\eta_\varepsilon+\mathbf{n}^\intercal\bfG|_{\partial\Omega}\circ\bfvarphi\)), where \(\bfG=\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon\). However, since this procedure is not well-defined at the current stage, we instead replace time derivatives by temporal difference quotients with parameter \(0<\tau\ll1\). Due to the discrete product rule, time shifted functions of the form \(f^\tau:=f(\cdot+\tau)\) appear frequently. Our pair of test-functions is \[\begin{align} (\partial_t\Delta_\tau\eta_\varepsilon+\mathbf{n}^\intercal\bfG|_{\partial\Omega}\circ\bfvarphi\,,\,\Delta_\tau\overline{\mathbf{v}}_\varepsilon+\bfG),\qquad \bfG=\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}^\tau_\varepsilon. \end{align}\]

Thereby, we derive from 26 that \[\begin{align} &\sup_I \int_\omega\big(\vert \partial_t\Delta_\tau\eta_\varepsilon\vert^2 + \vert \Delta_{\mathbf{y}}\Delta_\tau\eta_\varepsilon\vert^2 \big)\, \mathrm{d}\mathbf{y} + \sup_I \int_\Omega\vert \Delta_\tau\overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x}\nonumber \\ &\quad + \varepsilon\int_I\int_\omega\vert \partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\vert^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t + \int_I\int_\Omega\vert \Delta_\tau\nabla \overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &\lesssim \underbrace{\int_\omega\big(\vert \partial_t\Delta_\tau\eta_\varepsilon(0)\vert^2 + \vert \Delta_{\mathbf{y}}\Delta_\tau\eta_{\varepsilon}(0)\vert^2\big)\, \mathrm{d}\mathbf{y}}_{1st} +\underbrace{ \int_\Omega\vert \Delta_\tau\overline{\mathbf{v}}_\varepsilon(0)\vert^2\, \mathrm{d} \mathbf{x}}_{2nd}\nonumber \\ &\quad + \underbrace{\int_I\int_\omega\Delta_\tau g \,\partial_t\Delta_\tau\eta_\varepsilon\, \mathrm{d}\mathbf{y}\,\mathrm{d} t}_{3rd} -\underbrace{\varepsilon\int_I\int_\omega\partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon \cdot\nabla_{\mathbf{y}}(\mathbf{n}^\intercal\bfG\circ\bfvarphi)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t}_{4th}\nonumber \\ & \quad -\underbrace{\int_I\int_\omega\partial_t^2\Delta_\tau\eta_\varepsilon\,\mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t}_{5th} -\underbrace{\int_I\int_\omega\Delta_\tau\Delta_{\mathbf{y}}^2\eta_\varepsilon\, \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t}_{6th}\nonumber \\ &\quad + \underbrace{\int_I\int_\omega\Delta_\tau g\, \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t}_{7th} + \underbrace{\int_I\int_\Omega\Big(\partial_t J^\tau_{\zeta_\varepsilon}\frac{\Delta_\tau\overline{\bfv}_\varepsilon}{2}-\Delta_\tau J_{\zeta_\varepsilon}\partial_t\overline{\bfv}_\varepsilon \Big)\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon \, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{8th}\nonumber \\ & \quad + \underbrace{ \int_I\int_\Omega\Delta_\tau\mathbf{h}_{\zeta_\varepsilon}\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon \, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{9th} - \underbrace{\int_I\int_\Omega \Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon \, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{10th}\nonumber \\ &\quad +\underbrace{\int_I\int_\Omega\pi_\varepsilon\mathop{\mathrm{div}}(\Delta_\tau \mathbf{B}_{\zeta_\varepsilon}^\intercal(\Delta_\tau\overline{\mathbf{v}}_\varepsilon+\bfG))\, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{11th} +\underbrace{\int_I\int_\Omega(\Delta_\tau\bfh_{\zeta_\varepsilon}-\Delta_\tau J_{\zeta_\varepsilon}\partial_t\overline{\bfv}_\varepsilon)\cdot \bfG \, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{12th}\nonumber \\ &\quad -\underbrace{\int_I\int_\Omega J_{_{\zeta_\varepsilon}}^\tau\partial_t\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot \bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{13th} -\underbrace{\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\nabla\bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{14th}\nonumber \\ &\quad -\underbrace{\int_I\int_\Omega\bfA_{\zeta_\varepsilon}\nabla\Delta_\tau\overline{\mathbf{v}}_\varepsilon:\nabla\bfG \, \mathrm{d} \mathbf{x}\,\mathrm{d} t}_{15th}. \label{est} \end{align}\tag{38}\] We proceed to estimate the terms on the right in 38 . Notice that, different from the plate case in [19], here \(\bfG\) does not have zero boundary values leading to the extra terms of the form \(\bfG\circ\bfvarphi\) in 38 which we are now going to estimate. It holds \[\begin{align} \bfG\circ \bfvarphi&=(\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal)\circ\bfvarphi\,\partial_t\eta^\tau_\varepsilon\bfn = (\Delta_\tau J_{\zeta_\varepsilon} J_{\zeta_\varepsilon}^{-1}+1)\, \partial_t\eta_\varepsilon^\tau\, \mathbf{n} \approx \Delta_\tau J_{\zeta_\varepsilon} (1+{\zeta_\varepsilon}) \partial_t\eta^\tau_\varepsilon\bfn \end{align}\] on \(\omega\) using the fact that \(\nabla\bfPsi_{\zeta_\varepsilon}\circ\bfvarphi\bfn=\bfn\) and 11 . Note that the 1st 2nd and 3rd integral on the right hand side of 38 can be treated directly by the data. Hence we obtain starting from the estimate of the 4th integral: \[\begin{align} \nonumber \text{4th}&\lesssim\varepsilon^2\int_I\|\partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+c(\varepsilon)\int_I\|\nabla_{\mathbf{y}}\bfG\|^2_{L^{2}_\mathbf{y}}\,\mathrm{d} t\nonumber\\ &\lesssim\varepsilon^2\int_{I^\tau}\|\partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\,\mathrm{d} t+c(\varepsilon)\int_I\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|^2_{L^{2}_\mathbf{x}}\|{\zeta_\varepsilon}\|^2_{L^{\infty}_\mathbf{y}}\|\partial_t\eta^\tau_\varepsilon\|^2_{L^{\infty}_\mathbf{y}}\,\mathrm{d} t\nonumber\\ &\quad +c(\varepsilon)\int_I\|\Delta_\tau{\zeta_\varepsilon}\|^2_{L^{2}_\mathbf{y}}\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|^2_{L^{\infty}_\mathbf{x}}\| \partial_t\eta^\tau_\varepsilon\|_{L^\infty_\mathbf{y}}\,\mathrm{d} t+c(\varepsilon)\int_I\|\Delta_\tau{\zeta_\varepsilon}\|^2_{L^{\infty}_\mathbf{y}}\|{\zeta_\varepsilon}\|^2_{L^{\infty}_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta^\tau_\varepsilon\|^2_{L^{2}_\mathbf{y}}\,\mathrm{d} t\nonumber\\ &\lesssim \varepsilon^2\int_{I^\tau}\|\partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\|_{L^{2}_\mathbf{y}}^2\,\mathrm{d} t+c(\varepsilon)\int_{I^\tau}\|\partial_t\Delta_{\mathbf{y}}\eta_\varepsilon\|^2_{L^{2}_\mathbf{y}}\,\mathrm{d} t, \end{align}\] uniformly in \(\bfn\) and \(\varepsilon\) and where \(I^\tau=(0,T+\tau)\). We proceed with the 5th term \[\begin{align} -\text{5th}&=\int_I\int_\omega \partial_t^2\Delta_\tau\eta_\varepsilon\mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\approx -\int_I\int_\omega \partial_t^2\eta_\varepsilon^\tau\,\Delta_\tau\partial_t\eta_\varepsilon\Delta_\tau{\zeta_\varepsilon} (1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t-\int_I\int_\omega \partial_t\Delta_\tau\eta_\varepsilon\partial_t^2{\zeta_\varepsilon}\partial_t\eta^\tau_\varepsilon (1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad -\int_I\int_\omega\partial_t\Delta_\tau\eta_\varepsilon \partial_t\eta^\tau_\varepsilon |\partial_t{\zeta_\varepsilon}|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+\int_\omega \partial_t\Delta_\tau\eta_\varepsilon\partial_t{\zeta_\varepsilon}\partial_t\eta^\tau_\varepsilon(1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\bigg|_{0}^{T} \\ &\lesssim \int_{I^\tau}\big(\|\partial_t^2\eta_\varepsilon\|_{L^2_\mathbf{y}}^2 +\|\partial_t^2\eta_\varepsilon\|_{L^2_\mathbf{y}}\|\partial_t\eta_\varepsilon\|_{L^\infty_\mathbf{y}}\big)\,\mathrm{d} t +\sup_{I}\|\partial_t\Delta_\tau\eta_\varepsilon\|_{L^2_\mathbf{y}}\|\partial_t\eta_\varepsilon^\tau\|_{L^2_\mathbf{y}}\\ &\lesssim \int_{I^\tau}\big(\|\partial_t^2\eta_\varepsilon\|_{L^2_\mathbf{y}}^2 +\|\partial_t\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\big)\,\mathrm{d} t+\delta\sup_{I}\|\partial_t\Delta_\tau\eta_\varepsilon\|_{L^2_\mathbf{y}}^2+c(\delta), \end{align}\] using \(\zeta_\epsilon\in X_{\tt shell}(0,T)\) and 37 . Similarly, it holds for the 6th term that \[\begin{align} &\int_I\int_\omega \Delta_\tau\Delta_{\mathbf{y}}^2\eta_\varepsilon\mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t \\&\approx\int_I\int_\omega \Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\partial_t\Delta_{\mathbf{y}}\eta_\varepsilon^\tau\,\partial_t{\zeta_\varepsilon}(1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t +\int_I\int_\omega \Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\partial_t\Delta_{\mathbf{y}}{\zeta_\varepsilon} \partial_t\eta^\tau_\varepsilon(1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad +2\int_I\int_\omega \Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\partial_t\nabla_{\mathbf{y}} {\zeta_\varepsilon}\cdot\partial_t\nabla_{\mathbf{y}}\eta^\tau_\varepsilon(1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+\int_I\int_\omega\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\partial_t\eta_\varepsilon\partial_t{\zeta_\varepsilon}\Delta_{\mathbf{y}}{\zeta_\varepsilon}\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad +2\int_I\int_\omega\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon \partial_t\nabla_{\mathbf{y}}\eta^\tau\partial_t{\zeta_\varepsilon}\cdot\nabla_{\mathbf{y}}{\zeta_\varepsilon}\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+2\int_I\int_\omega\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon \partial_t\eta^\tau_\varepsilon\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\cdot\nabla_{\mathbf{y}}{\zeta_\varepsilon}\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\lesssim \int_I(\|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2+\|\partial_t\eta^\tau_\varepsilon\|_{L^2_\mathbf{y}}^2)\,\mathrm{d} t+\int_I \|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}\|\partial_t\eta^\tau_\varepsilon\|_{L^\infty_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\quad +\int_I \|\Delta_\tau\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta^\tau_\varepsilon\|_{L^4_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^4_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_{I^\tau} \|\partial_t\Delta_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] and, for the 7th term \[\begin{align} \text{7th}=\int_I\int_\omega\Delta_\tau g\, \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t &\approx \int_I\int_\omega\Delta_\tau g \partial_t\eta^\tau_\varepsilon\partial_t{\zeta_\varepsilon}(1+{\zeta_\varepsilon})\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\lesssim\int_{I^\tau}\|\partial_tg\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\int_{I^\tau}\|\partial_t\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Now we look into the remaining terms in 38 . The 8th integral on the right-hand side can be estimated based on 11 and the assumption on \({\zeta_\varepsilon}\), \[\begin{align} \text{8th}=\int_I\int_\Omega\Big(\partial_t J^\tau_{\zeta_\varepsilon}\frac{\Delta_\tau\overline{\bfv}_\varepsilon}{2}-\Delta_\tau J_{\zeta_\varepsilon}\partial_t\overline{\bfv}_\varepsilon \Big)\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon \, \mathrm{d} \mathbf{x}\,\mathrm{d} t\lesssim \,\int_{I^\tau}\|\partial_t\overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] The 9th integral on the right hand side of 38 decomposes into \[\begin{align} ({\tt II})^{\bfh}&:=- \int_I\int_\Omega\Delta_\tau\big( J_{\zeta_\varepsilon} \nabla \overline{\mathbf{v}}_\varepsilon\cdot \partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon}\big)\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t, \\({\tt III })^{\bfh}&:=- \int_I\int_\Omega\Delta_\tau\big(\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}_\varepsilon\big)\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ ({\tt IV})^{\bfh}&:= \int_I\int_\Omega\Delta_\tau\big( J_{\zeta_\varepsilon} \bff\circ \bfPsi_{\zeta_\varepsilon}\big)\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] Clearly, the last term is of lower order and it holds \[\begin{align} ({\tt IV})^{\bfh} &\lesssim \int_{I^\tau}\big(\|\partial_t\bff\circ \bfPsi_{\zeta_\varepsilon}\|^2_{L^2_\mathbf{x}}+\|\bff\circ \bfPsi_{\zeta_\varepsilon}\|_{W^{1,2}_\mathbf{x}}^2\big)\,\mathrm{d} t+\int_{I^\tau}\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] We further have \[\begin{align} ({\tt III })^{\bfh}&=- \int_I\int_\Omega\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon^\tau~\overline{\mathbf{v}}^\tau_\varepsilon\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t- \int_I\int_\Omega\mathbf{B}_{\zeta_\varepsilon}\Delta_\tau\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}^\tau_\varepsilon\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad - \int_I\int_\Omega\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon~\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=:({\tt III })^{\bfh}_1+({\tt III })^{\bfh}_2+({\tt III })^{\bfh}_3. \end{align}\] First of all we have \[\begin{align} ({\tt III })^{\bfh}_1&\lesssim\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^6_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\overline{\mathbf{v}}^\tau_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{W^{1,2}_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\| \Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}_\varepsilon^\tau\|^4_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t, \end{align}\] where we used that \(\partial_t{\zeta_\varepsilon} \in L^\infty(I; W^{2,2}(\omega))\).

We have by 8 , 10 and \({\zeta_\varepsilon}\in X_{\tt shell}(0,T)\) that for arbitrary \(\delta>0\), \[\begin{align} ({\tt III })^{\bfh}_2&\lesssim \int_I\|\mathbf{B}_{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{x}}\|\Delta_\tau\nabla \overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}^\tau_\varepsilon\|_{L^6_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^3_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}})\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\overline{\mathbf{v}}^\tau_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{1}{2}}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{3}{2}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\|\overline{\mathbf{v}}^\tau_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\leq\,\delta \int_{I}\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{4}\,\mathrm{d} t+c(\delta)\int_{I^\tau}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{8}\,\mathrm{d} t, \end{align}\] as well as \[\begin{align} ({\tt III })^{\bfh}_3&\lesssim \int_I\|\mathbf{B}_{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{x}}\|\nabla \overline{\bfv}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}})\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{3}{2}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\big(\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{4}+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{8}\big)\,\mathrm{d} t. \end{align}\] We proceed similarly for \(({\tt II})^{\bfh}\) obtaining \[\begin{align} ({\tt II})^{\bfh}=&- \int_I\int_\Omega\Delta_\tau J_{\zeta_\varepsilon} \nabla \overline{\mathbf{v}}_\varepsilon\cdot \partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon}\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_{\zeta_\varepsilon}^\tau \Delta_\tau\nabla \overline{\mathbf{v}}_\varepsilon\cdot (\partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon})^\tau\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_{\zeta_\varepsilon}^\tau \nabla \overline{\mathbf{v}}^\tau_\varepsilon\cdot \Delta_\tau(\partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon})\cdot\Delta_\tau\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ =:&({\tt II})^{\bfh}_1+({\tt II})^{\bfh}_2+({\tt II})^{\bfh}_3. \end{align}\] Now \(({\tt II})^{\bfh}_1\) and \(({\tt II})^{\bfh}_2\) can be estimated directly using 11 and 10 . We have \[\begin{align} ({\tt II})^{\bfh}_1+({\tt II})^{\bfh}_2&\lesssim \int_I\|\partial_t{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t+\int_I\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\big(\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{2}+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{2}\big)\,\mathrm{d} t. \end{align}\] Finally, using again 36 we see that \[\begin{align} ({\tt II})^{\bfh}_3 &\lesssim \int_I\|\nabla \overline{\mathbf{v}}^\tau_\varepsilon\|_{L^2_\mathbf{x}}\|\partial_t\Delta_\tau{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t+\int_I\|\nabla \overline{\mathbf{v}}^\tau\|_{L^2_\mathbf{x}}\|(\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon})^\tau\|_{L^4_\mathbf{x}}\|\partial_t{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\mathbf{v}}^\tau_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\partial_t\Delta_\tau{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t+\int_I\|\nabla\overline{\mathbf{v}}^\tau_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\partial_t\nabla_\mathbf{y}\zeta_\varepsilon\|_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\Delta_\tau\overline{\bfv}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}(1+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{4})\,\mathrm{d} t+c(\delta)\int_I\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}^4\,\mathrm{d} t. \end{align}\] Lastly, for \(0<s\leq \frac{1}{4}\) using the embedding \[\label{ineq2} \zeta_\varepsilon\in W^{2,\infty}(I; L^2(\omega))\cap L^\infty(I;W^{4,2}(\omega))\hookrightarrow W^{\frac{3-s}{2},2}(I; W^{s+2,2}(\omega))\hookrightarrow W^{1,4}(I; W^{s+2,2}(\omega)),\tag{39}\] we have \[\begin{align} \int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\,\mathrm{d}\mathbf{x}\,\mathrm{d}t&\leq \int_I\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\|\partial_t{\zeta_\varepsilon}\|_{W^{2+s,2}_\mathbf{y}}^4\,\mathrm{d} t+\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^4\,\mathrm{d} t . \end{align}\] Next we consider now the 11th term that is the pressure term: \[\begin{align} \text{11th}=\int_I\int_\Omega&\overline{\pi}_\varepsilon^\tau\mathop{\mathrm{div}}(\Delta_\tau \mathbf{B}_{\zeta_\varepsilon}^\intercal(\Delta_\tau\overline{\mathbf{v}}_\varepsilon+\bfG))\, \mathrm{d} \mathbf{x}\,\mathrm{d} t. \end{align}\] We split the 11th term into two parts. First \[\begin{align} &\int_I\int_\Omega\overline{\pi}_\varepsilon^\tau\mathop{\mathrm{div}}(\Delta_\tau \mathbf{B}_{\zeta_\varepsilon}^\intercal\Delta_\tau\overline{\mathbf{v}}_\varepsilon)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\pi}_\varepsilon\|_{L^4_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon \|_{L^4_\mathbf{x}}\,\mathrm{d} t+\int_I\|\overline{\pi}_\varepsilon \|_{L^4_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^4_\mathbf{x}}\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\pi}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\Delta_\tau\zeta_\varepsilon\|_{W^{2,2}_\mathbf{y}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\pi}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] The first term can eventually be absorbed while we estimate the second one using Lemma 3. Next we estimate \[\begin{align} &\int_I\int_\Omega\overline{\pi}_\varepsilon^\tau\mathop{\mathrm{div}}(\Delta_\tau \mathbf{B}_{\zeta_\varepsilon}^\intercal\bfG)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\pi}_\varepsilon\|_{L^6_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}})\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\quad + \int_I\|\overline{\pi}_\varepsilon\|_{L^6_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}\|\partial_t\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^2_\mathbf{y}}\|\nabla^2_\mathbf{y}\zeta_\varepsilon\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\quad + \int_I\|\overline{\pi}_\varepsilon\|_{L^6_x}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|^2_{L^6_\mathbf{y}}(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}})\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\big(\|\overline{\pi}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{W^{1,2}_\mathbf{x}}^2\big)\,\mathrm{d} t. \end{align}\] Combining the two terms and using Lemma 3 we finally get \[\begin{align} \text{11th} \lesssim \delta\int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_{I^\tau}\big(\mathcal{E}_\eta^{5}+1\big)\,\mathrm{d} t+\int_I\|\bff\circ\bfPsi_{\zeta_\varepsilon}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\]

Estimates for the divergence-corrector \(G\), terms 12 until 15.

It remains to estimate the terms related to \(\bfG=\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\). For the 12th and 13th integral in 38 , we decompose it into (using the definition of \(\bfh_\zeta\)) \[\begin{align} ({\tt I})^{\bfh,\bfG}&:= -\int_I\int_\Omega\Delta_\tau\big(J_{\zeta_\varepsilon}\partial_t \overline{\mathbf{v}}_\varepsilon\big)\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ ({\tt II})^{\bfh,\bfG}&:=- \int_I\int_\Omega\Delta_\tau\big( J_{\zeta_\varepsilon} \nabla \overline{\mathbf{v}}_\varepsilon\cdot \partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon}\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t \\({\tt III })^{\bfh,\bfG}&:=- \int_I\int_\Omega\Delta_\tau\big(\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}_\varepsilon\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ ({\tt IV})^{\bfh,\bfG}&:= \int_I\int_\Omega\Delta_\tau\big( J_{\zeta_\varepsilon} \bff\circ \bfPsi_{\zeta_\varepsilon}\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] Note that the last term is of lower order and we obtain that \[\begin{align} ({\tt IV})^{\bfh,\bfG} &\lesssim \int_{I^\tau}\big(\|\bff\circ \bfPsi_{\zeta_\varepsilon}\|_{L^2_\mathbf{x}}^2+\|\partial_t\bff\circ \bfPsi_{\zeta_\varepsilon}\|_{L^2_\mathbf{x}}^2+\|\nabla\bff\circ \bfPsi_{\zeta_\varepsilon}\|^2_{L^2_\mathbf{x}}\big)\,\mathrm{d} t+\int_{I^\tau}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] With the embedding 39 , we now perform the following estimate: \[\begin{align} ({\tt III })^{\bfh,\bfG}&=- \int_I\int_\Omega\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}_\varepsilon\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t- \int_I\int_\Omega\mathbf{B}_{\zeta_\varepsilon}^\tau\Delta_\tau\nabla \overline{\bfv}_\varepsilon~\overline{\mathbf{v}}_\varepsilon\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad - \int_I\int_\Omega\mathbf{B}^\tau_{\zeta_\varepsilon}\nabla \overline{\bfv}_\varepsilon^\tau~\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}^2\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}\,\mathrm{d} t+\int_I\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}^2\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}\,\mathrm{d} t\\ &\quad +\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^2_\mathbf{x}}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^6_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_{I}\|\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{W^{1+s,2}_\mathbf{y}}^4\,\mathrm{d} t+c(\delta)\int_{I^\tau}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^4\,\mathrm{d} t+\delta\int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] and decompose \[\begin{align} ({\tt II})^{\bfh,\bfG}=&- \int_I\int_\Omega\Delta_\tau J_{\zeta_\varepsilon} \nabla \overline{\mathbf{v}}_\varepsilon\cdot \partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon}\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_{\zeta_\varepsilon}^\tau \Delta_\tau\nabla \overline{\mathbf{v}}_\varepsilon\cdot (\partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon})^\tau\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_{\zeta_\varepsilon}^\tau \nabla \overline{\mathbf{v}}_\varepsilon^\tau\cdot \Delta_\tau(\partial_t \bfPsi_{\zeta_\varepsilon}^{-1}\circ \bfPsi_{\zeta_\varepsilon})\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ =:&({\tt II})^{\bfh,\bfG}_1+({\tt II})^{\bfh,\bfG}_2+({\tt II})^{\bfh,\bfG}_3. \end{align}\] Then we have by 8 , 10 and 11 \[\begin{align} ({\tt II})^{\bfh,\bfG}_1+({\tt II})^{\bfh,\bfG}_2 &\lesssim \int_I(\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}+\|\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}})\|\partial_t\zeta_\varepsilon\|_{L^\infty_\mathbf{y}}^2\|\Delta_\tau\nabla\zeta_\varepsilon\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Finally, as for \(({\tt II})^{\bfh}_3\) above,\[\begin{align} ({\tt II})^{\bfh,\bfG}_3 &\lesssim\,\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\partial_t\zeta_\varepsilon\|_{L^4_\mathbf{y}}\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^{20}_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^5_\mathbf{x}}\,\mathrm{d} t+\int_I\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\partial_t\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^6_\mathbf{y}}^2\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{19}{10}}\|\Delta_\tau\partial_t\zeta_\varepsilon\|_{W^{\frac{1}{2},2}_\mathbf{y}}\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{W^{1,2}_{\mathbf{y}}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{10}}\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\nabla_\mathbf{y}\zeta_\varepsilon\|_{W^{1,2}_\mathbf{y}}^2\,\mathrm{d} t\\ &\lesssim \int_{I}(1+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^4)\,\mathrm{d} t+\int_I|\|\partial_t^2\zeta_\varepsilon\|_{W^{\frac{1}{2},2}_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] where we used the interpolation inequality: \[\label{ineq1} \|\overline{\mathbf{v}}_\varepsilon\|_{L^5_\mathbf{x}}\lesssim \|\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{10}}\|\overline{\mathbf{v}}\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{9}{10}}.\tag{40}\] Now we estimate the 12th term that was defined as \(({\tt I})^{\bfh,\bfG}\). Taking an integration by parts we have \[\begin{align} \text{12th} &= ({\tt I})^{\bfh,\bfG}=-\int_I\int_\Omega\Delta_\tau J_{\zeta_\varepsilon}\partial_t\overline{\mathbf{v}}_\varepsilon\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t-\int_I\int_\Omega J^\tau_{\zeta_\varepsilon}\partial_t\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=-\int_I\int_\Omega\Delta_\tau J_{\zeta_\varepsilon}\partial_t\overline{\mathbf{v}}_\varepsilon\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega \partial_t J^\tau_{\zeta_\varepsilon}\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega J^\tau_{\zeta_\varepsilon}\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot \partial_t\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad -\int_\Omega J^\tau_{\zeta_\varepsilon}\Delta_\tau\overline{\mathbf{v}}_\varepsilon\cdot \bfG\, \mathrm{d} \mathbf{x}\bigg|_0^T = ({\tt I})^{\bfh,\bfG}_1+({\tt I})^{\bfh,\bfG}_2+({\tt I})^{\bfh,\bfG}_3+({\tt I})^{\bfh,\bfG}_4. \end{align}\] The first two terms on the right hand side above are bounded by \[\begin{align} ({\tt I})^{\bfh,\bfG}_1+({\tt I})^{\bfh,\bfG}_2 &\lesssim \int_I\|\partial_t\zeta_\varepsilon\|_{L^\infty_\mathbf{y}}(\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}+\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}})\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{W^{1,2}_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] The last integral in \(({\tt I})^{\bfh,\bfG}\) can be estimated as \[\begin{align} ({\tt I})^{\bfh,\bfG}_4&\lesssim \sup_{I^\tau}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}\|\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{4}_\mathbf{y}}\\ &\lesssim \sup_{I^\tau}\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{1/4}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{3/4}\\ &\lesssim \delta\sup_{I^\tau}\Big(\|\Delta_\tau\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{2} \Big)+c(\delta)\sup_{I^\tau}\|\overline{\mathbf{v}}_\varepsilon\|^{2}_{L^2_\mathbf{x}}. \end{align}\]

Control of the term \(({\tt I})^{\bfh,\bfG}_3\).

This is the most complicated term. For this we compute \(\partial_t\bfG\). Since \(\mathbf{B}_{\zeta_\varepsilon}=\mathrm{cof}(\nabla\bfPsi_{\zeta_\varepsilon})\), there is a bilinear mapping \(\mathcal{A}:\mathbb{R}^{3\times 3}\times\mathbb{R}^{3\times 3}\rightarrow\mathbb{R}^{3\times 3}\) such that \[\begin{align} \partial_t\bfG&=\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\partial_t\overline{\mathbf{v}}_\varepsilon^\tau+\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau+\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t\Delta_\tau\big(\mathcal{A}(\nabla \bfPsi_{\zeta_\varepsilon}^\intercal,\nabla\bfPsi_{\zeta_\varepsilon}^\intercal)\big)\overline{\mathbf{v}}_\varepsilon^\tau\label{dtg}\\ &= \mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\partial_t\overline{\mathbf{v}}_\varepsilon^\tau+\partial_t\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\overline{\mathbf{v}}_\varepsilon^\tau+2\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}( \Delta_\tau\nabla \bfPsi_{\zeta_\varepsilon},\partial_t\nabla\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\nonumber\\ &\quad +\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}( \partial_t\Delta_\tau\nabla \bfPsi_{\zeta_\varepsilon},\nabla\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau=: A_1+A_2+A_3.\nonumber \end{align}\tag{41}\] The most critical term in 41 is the last one. Writing \(\mathcal{A}=\sum_{i,j=1}^3\mathcal{A}_{ij}\), with \(\mathcal{A}_{ij}:\mathbb{R}^3\times\mathbb{R}^3\rightarrow\mathbb{R}^{3\times 3}\) acting on the column vectors of the matrices we obtain that \[\begin{align} A_3&=-\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}( \partial_t\Delta_\tau\nabla \bfPsi_{\zeta_\varepsilon},\nabla\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=-\sum_{i,j=1}^3\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau \Delta_{\tau}\mathbf{v}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau\partial_i \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=-\sum_{i,j=1}^3\int_I\int_{\partial\Omega}J_{{\zeta_\varepsilon}}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon\,n^i\,\mathrm d\mathcal{H}^2\,\mathrm{d} t\\ &\quad +\sum_{i,j=1}^3\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_{ij}\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad +\sum_{i,j=1}^3\int_I\int_\Omega\partial_iJ_{\zeta_\varepsilon}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad +\sum_{i,j=1}^3\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau}\partial_i\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad +\sum_{i,j=1}^3\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\partial_i\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad +\sum_{i,j=1}^3\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\cdot\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\big(\mathcal{A}_{ij}( \partial_t\Delta_\tau \bfPsi_{\zeta_\varepsilon},\partial_j\bfPsi_{\zeta_\varepsilon})\big)\partial_i\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=:J_0+J_1+\dots J_5. \end{align}\] Using the estimates for \(\bfPsi_{\zeta_\varepsilon}\) (cf. 8 , 10 and 11 ) as well as \(\zeta_\varepsilon \in X_{\tt shell}(0,T)\), we have by the trace embedding \(W^{1,2}(\Omega)\hookrightarrow W^{1/2,2}(\partial\Omega)\hookrightarrow L^4(\partial\Omega)\) \[\begin{align} J_0&\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}^3\big)\|\partial_t^2{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{L^4(\partial\Omega)}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4(\partial\Omega)}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta) \int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Similarly, it holds \[\begin{align} J_1&\lesssim \int_I\lVert\partial_t^2{\zeta_\varepsilon}\rVert_{L^2_\mathbf{y}}\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{L^4_\mathbf{x}}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{L^4_\mathbf{x}}\lVert\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] where we used the fact that \(\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\in L^\infty(I\times \omega)\). Moreover, \[\begin{align} J_2&\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}^3\big)\|\partial_t^2{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta) \int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} J_3&\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}^3\big)\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}\|\Delta_{\tau}\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_x}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta) \int_I\big(\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}^4+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^4\big)\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta) \bigg(\int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^4\,\mathrm{d} t+1\bigg), \end{align}\] where we used the embedding 36 . Finally, we have \[\begin{align} J_4&\lesssim\int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}^2)\|\partial_t^2{\zeta_\varepsilon}\|_{L^2_\mathbf{y}}\|\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{L^{4}_\mathbf{x}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\|_{L^{\infty}_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\, \mathrm{d} \mathbf{x}+c(\delta)\int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t, \end{align}\] as well as \[\begin{align} J_5&\lesssim\int_I(1+\|\nabla_{\mathbf{y}}{\zeta_\varepsilon}\|_{L^\infty_\mathbf{y}}^3)\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon \|_{L^{6}_\mathbf{x}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\|\partial_t^2{\zeta_\varepsilon}\|_{L^{3}_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\Big(\|\partial_t^2{\zeta_\varepsilon}\|_{L^3_\mathbf{y}}^{4}+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{4}\Big)\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\, \mathrm{d} \mathbf{x}+c(\delta)\bigg(\int_I\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{4}\,\mathrm{d} t+1\bigg), \end{align}\] using again \({\zeta_\varepsilon}\in X_{\tt shell}(0,T)\subset W^{2,4}(I; L^3(\omega))\).

We continue the estimates of the terms involving \(\partial_t\bfG\) from 41 . For the first term in 41 , we have \[\begin{align} A_1&=\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau} \overline{\mathbf{v}}_\varepsilon\cdot \mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}\partial_t\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{L^4_\mathbf{x}}\lVert\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^4_\mathbf{y}}\lVert\partial_t\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{1,2}_\mathbf{y}}\lVert\partial_t\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\lVert\partial_t\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] where we used \(\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\in L^\infty(I; W^{1,2}(\omega))\). For the second term in 41 , we have \[\begin{align} A_2&=\int_I\int_\Omega J_{\zeta_\varepsilon}^\tau\Delta_{\tau} \overline{\mathbf{v}}_\varepsilon\cdot \Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\partial_t\mathbf{B}_{\zeta_\varepsilon}\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^6_\mathbf{y}}^2\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{W^{1,2}_\mathbf{x}}\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{1,2}_\mathbf{y}}^2\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}^2\,\mathrm{d} t+\int_{I^\tau}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] The integral related to the third term on the right hand side of 41 can be estimated in a similar way, thereby we finished estimating \(({\tt I})^{\bfh,\bfG}\).

The last two integrals related to \(\nabla \bfG\) in 38 , namely the 14th and 15th term .

For these terms we will be using repeatedly 8 and 10 . As \(\bfG=\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\), we have \[\begin{align} \text{14th}&=\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\nabla\bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &=\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:(\nabla\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau+\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\nabla\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau+\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\nabla\overline{\mathbf{v}}_\varepsilon^\tau)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t. \end{align}\] Using again the fact that \(\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\in L^\infty(I\times \omega)\), we derive that \[\begin{align} &\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\nabla\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^6_\mathbf{y}}^2\lVert\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{1,2}_\mathbf{y}}^2\big(\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2+\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{W^{1,2}_\mathbf{x}}^2\big)\,\mathrm{d} t\lesssim \int_{I^\tau}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] By using the embedding, \[\zeta_\varepsilon\in W^{2,\infty}(I; L^2(\omega))\cap L^\infty(I;W^{4,2}(\omega))\hookrightarrow W^{\frac{5}{4},2}(I;W^{2,4}(\omega))\hookrightarrow W^{1,2}(I; W^{2,4}(\omega)),\] and 40 we continue by estimating \[\begin{align} &\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\nabla\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^{20}_\mathbf{y}}\|\nabla\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}\|\nabla_\mathbf{y}\zeta_\varepsilon\|_{L^\infty_\mathbf{y}}\|\Delta_\tau\nabla_\mathbf{y}^2\zeta_\varepsilon\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^5_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\Delta_\tau\nabla_\mathbf{y}\zeta_\varepsilon\|_{W^{1,2}_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^{\frac{19}{10}}\|\Delta_\tau\nabla_\mathbf{y}^2\zeta_\varepsilon\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^{\frac{1}{10}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\nabla_\mathbf{y}^2\zeta_\varepsilon\|_{L^4_\mathbf{y}}^2\,\mathrm{d} t+\int_I(1+\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^4)\,\mathrm{d} t. \end{align}\] We finally have \[\begin{align} &\int_I\int_\Omega\Delta_\tau\bfA_{\zeta_\varepsilon}\nabla\overline{\mathbf{v}}_\varepsilon:\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\nabla\overline{\mathbf{v}}_\varepsilon^\tau\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}^2\big(\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2+\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{W^{1,2}_\mathbf{x}}^2\big)\lVert\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{s+1, 2}_\mathbf{y}}^4\,\mathrm{d} t+\int_{I^\tau}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^4\,\mathrm{d} t, \end{align}\] where for \(0<s\leq \frac{1}{4}\) we use the embedding 39 . This concludes the estimate of the 14th term.

For the 15th and last integral in 38 , we have \[\begin{align} \text{15th}&=\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\nabla\Delta_\tau\overline{\mathbf{v}}_\varepsilon:\nabla\bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &=\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon:(\nabla\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau+\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\nabla\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau+\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\nabla\overline{\mathbf{v}}_\varepsilon^\tau)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t. \end{align}\] Again we estimate the above terms one by one. First, we have \[\begin{align} &\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon:\nabla\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\,\mathrm{d} t\\ &\lesssim \int_I\lVert\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^3_\mathbf{y}}\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{1,2}_\mathbf{y}}\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Next based on the embedding \[\zeta_\varepsilon\in W^{2,\infty}(I; L^2(\omega))\cap L^\infty(I;W^{4,2}(\omega))\hookrightarrow W^{\frac{4}{3},2}(I; W^{2,3}(\omega))\hookrightarrow W^{1,4}(I; W^{2,3}(\omega)),\] we estimate \[\begin{align} &\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon:\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\nabla\mathbf{B}_{\zeta_\varepsilon}^\intercal\overline{\mathbf{v}}_\varepsilon^\tau\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\Delta_\tau\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\rVert_{L^3_\mathbf{y}}\lVert\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_{I^\tau}\lVert\partial_t\nabla_{\mathbf{y}}^2{\zeta_\varepsilon}\rVert_{L^3_\mathbf{y}}^4\,\mathrm{d} t+c(\delta)\int_{I^\tau}\lVert\overline{\mathbf{v}}_\varepsilon\rVert_{W^{1,2}_\mathbf{x}}^4\,\mathrm{d} t. \end{align}\] Finally, using 39 again we also have \[\begin{align} &\int_I\int_\Omega\bfA_{\zeta_\varepsilon}^\tau\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon:\mathbf{B}_{\zeta_\varepsilon}^{-\intercal}\Delta_\tau\mathbf{B}_{\zeta_\varepsilon}^\intercal\nabla\overline{\mathbf{v}}_\varepsilon^\tau\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\nabla\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^2_\mathbf{x}}\lVert\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}^2\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{L^\infty_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\lVert\partial_t\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}\lVert\Delta_\tau\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{s+1,2}_\mathbf{y}}\lVert\nabla\overline{\mathbf{v}}_\varepsilon^\tau\rVert_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\lVert\Delta_\tau\nabla\overline{\mathbf{v}}_\varepsilon\rVert_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}_\varepsilon^\tau\|_{W^{1,2}_\mathbf{x}}^4\,\mathrm{d} t+c(\delta)\int_I\lVert\partial_t\nabla_{\mathbf{y}}{\zeta_\varepsilon}\rVert_{W^{s+1,2}_\mathbf{y}}^4\,\mathrm{d} t. \end{align}\] We have finished estimating the terms depending on \(\nabla\bfG\). Now all terms in 38 are estimated.

Putting 38 and 34 together and using 33 we have shown that for \(0<\varepsilon\ll 1\), \[\begin{align} &\int_{I}\|\Delta_\tau\eta_\varepsilon\|_{W^{s+2,2}_\mathbf{y}}^2\,\mathrm{d} t+\varepsilon\sup_I\|\Delta_\tau\eta_\varepsilon\|_{W^{s+1,2}_\mathbf{y}}^2+\sup_I \int_\omega\big(\vert \partial_t\Delta_\tau\eta_\varepsilon\vert^2 + \vert \Delta_{\mathbf{y}}\Delta_\tau\eta_\varepsilon\vert^2\big)\, \mathrm{d}\mathbf{y} \\ &\quad + \sup_I \int_\Omega\vert \Delta_\tau\overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x} + \varepsilon\int_I\int_\omega\vert \partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\vert^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t + \int_I\int_\Omega\vert \Delta_\tau\nabla \overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \\ &\lesssim \delta\int_{I^\tau}\|\Delta_{\tau}\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2\, \mathrm{d} \mathbf{x}+\delta\Big(\sup_{I^\tau}\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\sup_I\|\Delta_\tau\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2+\sup_I\|\Delta_\tau\partial_t\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\Big)\\ &\quad +\int_I(\mathcal{E}^5_{\eta_\varepsilon}(s)+1)\,\mathrm{d}s+\int_I H(s)\,\mathrm{d}s+\varepsilon^2\int_I\|\partial_t\Delta_\tau\nabla_{\mathbf{y}}\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t\\ &\quad+\|\partial_t\Delta_\tau\eta_\varepsilon(0)\|_{L^2_\mathbf{y}}^2+ \|\Delta_\mathbf{y}\Delta_\tau\eta(0)\|_{L^2_\mathbf{y}}^2 +\|\Delta_\tau \overline{\mathbf{v}}_\varepsilon(0)\|_{L^2_\mathbf{x}}^2+\varepsilon\|\Delta_\tau\eta_\varepsilon(0)\|_{W^{1+s,2}_\mathbf{y}}^2. \end{align}\] Finally, recalling 32 we find that \[\begin{align} &\limsup_{\tau\to 0}\|\partial_t\Delta_\tau\eta_\varepsilon(0)\|_{L^2_\mathbf{y}}^2+ \|\Delta_\mathbf{y}\Delta_\tau\eta(0)\|_{L^2_\mathbf{y}}^2 +\|\Delta_\tau \overline{\mathbf{v}}_\varepsilon(0)\|_{L^2_\mathbf{x}}^2)\leq \limsup_{\tau\to 0}\dashint_0^\tau \mathcal{E}_{\eta_\varepsilon}\, \,\mathrm{d}s \\ &\quad \leq c\left(1+\|\overline{\mathbf{v}}_0\|_{W^{2,2}_\mathbf{x}}+\|\eta_0\|_{W^{4,2}_\mathbf{y}}+\|\eta_*\|_{W^{2,2}_\mathbf{y}}+\|\overline{\bff}_0\|_{L^2_\mathbf{x}}+\|g_0\|_{L^2_\mathbf{y}}\right). \end{align}\] Hence we can pass to the limit \(\tau\rightarrow0\) obtaining for \(0<\varepsilon\ll 1\) \[\begin{align} &\int_{I}\|\partial_t\eta_\varepsilon\|_{W^{s+2,2}_\mathbf{y}}^2\,\mathrm{d} t +\sup_I \int_\omega\big(\vert \partial_t^2\eta_\varepsilon\vert^2 + \vert \Delta_{\mathbf{y}}\partial_t\eta_\varepsilon\vert^2\big)\, \mathrm{d}\mathbf{y}\\ & \quad + \sup_I \int_\Omega\vert \partial_t\overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x} + \sup_I \int_\Omega\vert \nabla\overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x} + \int_I\int_\Omega\vert \partial_t\nabla \overline{\mathbf{v}}_\varepsilon\vert^2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \\ &\lesssim \delta\sup_{I}\Big(\|\overline{\mathbf{v}}_\varepsilon\|_{W^{1,2}_\mathbf{x}}^2+\|\partial_t\overline{\mathbf{v}}_\varepsilon\|_{L^2_\mathbf{x}}^2+\|\partial_t^2\eta_\varepsilon\|_{L^2_\mathbf{y}}^2\Big)+\int_I(\mathcal{E}^5_{\eta_\varepsilon}(s)+1)\,\mathrm{d}s+\int_I H(s)\,\mathrm{d}s\\ &\quad +1+\|\overline{\mathbf{v}}_0\|_{W^{2,2}_\mathbf{x}}+\|\eta_0\|_{W^{4,2}_\mathbf{y}}+\|\eta_*\|_{W^{2,2}_\mathbf{y}}+\|\overline{\bff}_0\|_{L^2_\mathbf{x}}+\|g_0\|_{L^2_\mathbf{y}}+\|\eta_*\|_{W^{1+s,2}_\mathbf{y}}^2, \end{align}\]where we used 27 and the embedding \[W^{1,2}(I;L^2(\Omega))\hookrightarrow L^\infty(I;L^2(\Omega)).\]

We can now absorb the \(\delta\)-terms obtaining the estimate\[\begin{align} \label{eq:124611} \mathcal{E}_{\eta_\varepsilon}(t)+\int_0^t\|\partial_t\nabla\overline{\mathbf{v}}_\varepsilon\|^2_{L^2_\mathbf{x}}\,\mathrm{d}s+\int_0^t\|\partial_t{\eta_\varepsilon}\|_{W^{2+s,2}_\mathbf{y}}^2\,\mathrm{d}s\leq\,c_0+c_1\int_0^t H(s)\,\mathrm{d}s+c_2\int_0^t \mathcal{E}_{\eta_\varepsilon}^{5}(s)\,\mathrm{d}s. \end{align}\tag{42}\] Note that this estimate is independent from \(\varepsilon\) which allows us to extend the existence time in Proposition 9 to an \(\varepsilon\)-independent time \(T_{\tt max}\). Now we pass to the limit \(\varepsilon\to 0\) in 42 which finishes the proof. ◻

Proof. We can now apply the local Grönwall’s lemma from [27] (assuming that \(T_0(\mathcal{E}(0)+1)<1\)) means of \(\mathcal{E}(0)\) and \(\int_0^{T_0}H(s)\,\mathrm{d}s\). In particular, the left-hand side can be made (much) smaller than \(R^2\), if we choose \(T_{\tt max}\) and \(R\) accordingly, i.e. \[\begin{align} \label{eq:094608} \mathcal{E}_\eta(t)+\int_0^t\|\partial_t\nabla\overline{\mathbf{v}}\|^2_{L^2_\mathbf{x}}\,\mathrm{d}s+\int_0^t\|\partial_t\eta\|_{W^{2+s,2}_\mathbf{y}}^2\,\mathrm{d}s\leq\,\delta (T_{\tt max},R) R^2. \end{align}\tag{43}\]

Arguing exactly as in the proof of Lemma 3 (but taking the supremum in time instead of the integral) we obtain \[\begin{align} \sup_I\Big(\|\overline{\bfv}\|_{W^{2,2}_\mathbf{x}(\Omega)}^2+\|\overline{\pi}\|_{W_\mathbf{x}^{1,2}(\Omega)}^2\Big)&\lesssim \sup_I\Big(1+\mathcal{E}_\eta^{5}+\|\bff\circ\bfPsi_{\zeta}\|_{L^2_\mathbf{x}}^2\Big), \end{align}\] which is uniformly bounded by 43 .

Based on equation 23 , we get now further\[\begin{align} \sup_I\int_\omega|\Delta_{\mathbf{y}}^2 \eta|^2\, \mathrm{d}\mathbf{y} &\lesssim \sup_I\int_\omega\left(|\partial_t^2 \eta|^2+|g|^2\right)\, \mathrm{d}\mathbf{y}+\sup_I\|\nabla_{\mathbf{y}}\zeta\|_{L^8(\omega)}^4\|\nabla\overline{\mathbf{v}}\|_{L^4(\partial\Omega)}^2+\sup_I\|\nabla_{\mathbf{y}}\zeta\|_{L^4(\omega)}^2\|\overline{\pi}\|_{L^4(\partial\Omega)}^2\\ &\lesssim \sup_I\int_\omega\left(|\partial_t^2 \eta|^2+|g|^2\right)\, \mathrm{d}\mathbf{y}+\sup_I\|\nabla_{\mathbf{y}}\zeta\|_{W^{1,2}(\omega)}^4\|\nabla\overline{\mathbf{v}}\|_{W^{\frac{1}{2},2}(\partial\Omega)}^2\\ &\quad +\sup\|\nabla_{\mathbf{y}}\zeta\|_{W^{1, 2}(\omega)}^2\|\overline{\pi}\|_{W^{\frac{1}{2},2}(\partial\Omega)}^2\\ &\lesssim \sup_I\int_\omega\left(|\partial_t^2 \eta|^2+|g|^2\right)\, \mathrm{d}\mathbf{y}+\sup_I\|\nabla\overline{\mathbf{v}}\|_{W^{1,2}(\Omega)}^2+\sup_I\|\overline{\pi}\|_{W^{1,2}(\Omega)}^2, \end{align}\] as well as \[\begin{align} \int_I\|\Delta_{\mathbf{y}}^2 \eta\|^2_{W^{1/2,2}(\omega)}&\lesssim \int_I\|\partial_t^2 \eta\|^2_{W^{1/2,2}(\omega)}+\int_I\int_\Omega|\nabla^2\overline{\bfv}|^2\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega|\nabla\overline{\pi}|^2\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\|g\|^2_{W^{1/2,2}(\omega)}\,\mathrm{d} t, \end{align}\] where \[\begin{align} \|\partial_t^2\eta\|_{W^{1/2,2}_\mathbf{y}}\lesssim \|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}, \end{align}\] where the last term is controlled by Proposition 10 yielding a uniform control of \(\mathcal{E}(t)\). We conclude that \((\eta,\mathbf{v},\pi)\in X_{\tt shell}\times X_{\tt fluid}\) with \(\|\eta\|_{X_{\tt shell}(0,T_{\tt max})}\leq R\). ◻

Proof. We transform \(\mathbf{v}_2\) and \(\pi_2\) to the domain of the pair \((\mathbf{v}_1,\eta_1)\) (that is \(\Omega_{\zeta_1}\)) by setting \[\begin{align} \underline{\mathbf{v}}_2:=\mathbf{v}_2\circ\bfPsi_{\zeta_2-\zeta_1},\quad \underline\pi_2:=\pi_2\circ\bfPsi_{\zeta_2-\zeta_1}, \quad \underline{\mathbf{f}}_2:=\mathbf{f}_2\circ\bfPsi_{\zeta_2-\zeta_1}, \end{align}\] where the Hanzawa transform \(\boldsymbol{\Psi}_{\zeta_2-\zeta_1} : \Omega_{\zeta_1} \rightarrow\Omega_{\zeta_2}\) is defined similarly as in 7 . Next we introduce a suitable Bogovskij operator for our setting: \[\mathop{\mathrm{Bog}}_{\zeta_1}(f):=\mathop{\mathrm{Bog}}(f\chi_{\Omega_{\zeta_1}}),\] where \(\mathop{\mathrm{Bog}}\) is defined in Theorem 2, depending on \(\norm{\zeta_1}_{L^\infty_\mathbf{y}}=:L\) and \(\norm{\nabla_{\mathbf{y}}\zeta_1}_{L^\infty_\mathbf{y}}=:C_L\) only.

To obtain the difference estimate, we test the equation for \((\mathbf{v}_1-\underline\mathbf{v}_2,\eta_1-\eta_2)\) by the pair \((\mathbf{v}_1-\underline\mathbf{v}_2+\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2),\partial_t(\eta_1-\eta_2))\). We find that \[\begin{align} &\tfrac{1}{2}\int_{\Omega_{\zeta_1(t)}}|\mathbf{v}_1(t)-\underline{\mathbf{v}}_2(t)|^2\, \mathrm{d} \mathbf{x}+\int_0^t \int_{\Omega_{\zeta_1( s )}}|\nabla(\mathbf{v}_1-\underline{\mathbf{v}}_2)|^2\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\tfrac{1}{2}\int_\omega|\partial_t(\eta_1-\eta_2)(t)|^2\, \mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_\omega|\Delta_{\mathbf{y}}(\eta_1-\eta_2)(t)|^2\, \mathrm{d}\mathbf{y}\nonumber\\ &\leq \tfrac{1}{2}\int_{\Omega_{\zeta_1(0)}}|\mathbf{v}_1(0)-\underline{\mathbf{v}}_2(0)|^2\, \mathrm{d} \mathbf{x}+\tfrac{1}{2}\int_\omega|\partial_t(\eta_1-\eta_2)(0)|^2\, \mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_\omega|\Delta_{\mathbf{y}}(\eta_1-\eta_2)(0)|^2\, \mathrm{d}\mathbf{y}\nonumber\\ &\quad-\int_0^t\int_{\Omega_{\zeta_1( s )}}\big((\mathbb{I}_{3\times 3}-J_{\zeta_2-\zeta_1})\partial_t \underline{\mathbf{v}}_2\big)\cdot\big(\mathbf{v}_1- \underline\mathbf{v}_2+ \mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}\big(J_{\zeta_2-\zeta_1}\nabla \underline{\mathbf{v}}_2 \partial_t \bfPsi_{\zeta_2-\zeta_1}^{-1}\circ \bfPsi_{\zeta_2-\zeta_1}\big)\cdot\big(\mathbf{v}_1- \underline\mathbf{v}_2+ \mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}\nabla \underline{\mathbf{v}}_2 \underline{\mathbf{v}}_2\cdot\big(\mathbf{v}_1- \underline\mathbf{v}_2+ \mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber \\ &\quad+\tfrac{1}{2} \int_0^t\int_{\partial\Omega_{\zeta_1( s )}}\bfn\circ\bfvarphi^{-1}_ {\zeta_1}\cdot\partial_t\eta_1 \mathbf{n}_{\zeta_1 }\circ\bfvarphi^{-1}_ {\zeta_1}|\underline\mathbf{v}_2 |^2 \mathrm d\mathcal{H}^2\,\mathrm{d}s\nonumber\\ &\quad- \int_0^t\int_{\partial\Omega_{\zeta_1( s )}}\bfn\circ\bfvarphi^{-1}_ {\zeta_1}\cdot\partial_t\eta_2 \mathbf{n}_{\zeta_1 }\circ\bfvarphi^{-1}_ {\zeta_1}| \mathbf{v}_1 |^2 \mathrm d\mathcal{H}^2\,\mathrm{d}s\nonumber \\ &\quad +\int_0^t\int_{\Omega_{\zeta_1( s )}}\big( \mathbf{B}_{\zeta_2-\zeta_1}- \mathbb{I} _{3\times 3}\big)\nabla \underline{\mathbf{v}}_2 \underline{\mathbf{v}}_2\cdot\big(\mathbf{v}_1- \underline\mathbf{v}_2+ \mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+ \int_0^t\int_{\Omega_{\zeta_1( s )}}(\mathbf{v}_1-\underline\mathbf{v}_2) \cdot \partial_t\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s- \int_{\Omega_{\zeta_1(t)}}(\mathbf{v}_1-\underline\mathbf{v}_2) \cdot \mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\, \mathrm{d} \mathbf{x}\nonumber\\ &\quad+ \int_{\Omega_{\zeta_1(0)}}(\mathbf{v}_1-\underline\mathbf{v}_2)(0) \cdot \mathrm{Bog}_{\zeta_1(0)}(\mathop{\mathrm{div}}\underline\mathbf{v}_2(0))\, \mathrm{d} \mathbf{x} -\int_0^t\int_{\Omega_{\zeta_1( s )}}\nabla(\mathbf{v}_1- \underline\mathbf{v}_2):\nabla\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber \\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}\big(\mathbf{A}_{\zeta_2-\zeta_1}-\mathbb{I}_{3\times 3}\big)\nabla\underline\mathbf{v}_2:\nabla(\mathbf{v}_1-\underline \mathbf{v}_2+\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2))\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}\big(\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}\big)\underline \pi_2:\nabla(\mathbf{v}_1-\underline \mathbf{v}_2+\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline\mathbf{v}_2))\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t \int_\omega ( g_1-g_2)\partial_t(\eta_1-\eta_2)\, \mathrm{d}\mathbf{y}\,\mathrm{d}s +\int_0^t \int_{\Omega_{\zeta_1( s )}} (\mathbf{v}_1-\underline \mathbf{v}_2)\cdot (\mathbf{f}_1-\underline{\mathbf{f}}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t \int_{\Omega_{\zeta_1( s )}} (\mathbb{I}_{3\times 3}-J_{ \zeta_2-\zeta_1})\underline{\mathbf{f}}_2\cdot (\mathbf{v}_1-\underline \mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s+\int_0^t\int_{\Omega_{\zeta_1}}(\mathbb{I}_{3\times 3}-J_{\zeta_2-\zeta_1})\underline{\mathbf{f}}_2\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline \mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}\mathbf{v}_1\otimes \mathbf{v}_1:\nabla\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline \mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s+\int_0^t\int_{\Omega_{\zeta_1( s )}}\mathbf{v}_1\cdot\nabla\mathbf{v}_1\cdot\underline \mathbf{v}_2\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\int_0^t\int_{\Omega_{\zeta_1( s )}}(\mathbf{f}_1-\underline{\mathbf{f}}_2)\mathrm{Bog}_{\zeta_1}(\mathop{\mathrm{div}}\underline \mathbf{v}_2)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &=:\sum_{i=1}^{22} R_{i}.\nonumber \end{align}\] Except for \(R_{5}\), \(R_{10}\) and \(R_{11}\), all terms can be estimated as in [3] and are bounded by\[\begin{align} & \delta\int_0^t\left(\lVert\nabla(\mathbf{v}_1-\underline \mathbf{v}_2)\rVert_{L^{2}(\Omega_{\zeta_1(s)})}^2\right)\,\mathrm{d}s+C(\delta)\int_0^t\|\partial_t\eta_1-\partial_t\eta_2\|_{L^2(\omega)}^2\,\mathrm{d}s\\ &+C(\delta)\int_0^t\left(\lVert g_1-g_2\rVert_{L^2(\omega)}^2+\lVert \mathbf{f}_1-\underline {\mathbf{f}}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2\right)\,\mathrm{d}s \\ &+C(\delta)\lVert \mathbf{v}_1(0)-\underline \mathbf{v}_2(0)\rVert_{L^2(\Omega_{\zeta_1(0)})}^2 \\ &+C(\delta) \int_0^t\Big(\lVert\underline {\mathbf{f}}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2+\lVert\partial_t\underline \mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2+\lVert\underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1(s)})}^2+\lVert\underline \pi_2\rVert_{W^{1,2}(\Omega_{\zeta_1(s)})}^2\Big)\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2\,\mathrm{d}s \\ &+C(\delta) \int_0^t\Big(1+\lVert\underline {\mathbf{f}}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2+\lVert\partial_t\underline \mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2+\lVert\underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1(s)})}^2\Big)\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^{2}(\omega)}^2\,\mathrm{d}s \\ &+C(\delta)\int_0^t\left(1+\lVert\underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1(s)})}^2+\lVert\partial_t\eta_2\rVert_{W^{2,2}(\omega)}^2\right)\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2\,\mathrm{d}s, \end{align}\] which is bounded by (taking \(g_1=g_2\) and \(\mathbf{f}_1=\underline{\mathbf{f}}_2\) and \(\underline\mathbf{v}_2(0)=\mathbf{v}_1(0)\circ\bfPsi_{\zeta_2(0)-\zeta_1(0)}\)) \[\begin{align} & \delta\int_0^t\left(\lVert\nabla(\mathbf{v}_1-\underline \mathbf{v}_2)\rVert_{L^{2}(\Omega_{\zeta_1(s)})}^2\right)\,\mathrm{d}s+C(\delta)\int_0^t\|\partial_t\eta_1-\partial_t\eta_2\|_{L^2(\omega)}^2\,\mathrm{d}s \\ &\quad +C(\delta,R) \int_0^t\Big(\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2+\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^{2}(\omega)}^2+\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2\Big)\,\mathrm{d}s \\ &\leq \delta\int_0^t\lVert\nabla(\mathbf{v}_1-\underline \mathbf{v}_2)\rVert_{L^{2}(\Omega_{\zeta_1(s)})}^2\,\mathrm{d}s +C(\delta) T_{\tt max}\sup_t\|\partial_t\eta_1-\partial_t\eta_2\|_{L^2(\omega)}^2\\ &\quad +C(\delta,R)T_{\tt max} \sup_t\Big(\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2+\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^{2}(\omega)}^2+\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}^2\Big). \end{align}\] On the other hand we have by continuity of \(\mathrm{Bog}_{\zeta_1}\) that \[\begin{align} R_5 & \lesssim \int_0^t\lVert\nabla\underline \mathbf{v}_2\rVert_{L^4(\Omega_{\zeta_1(s)})}\lVert\partial_t\bfPsi_{\zeta_2-\zeta_1}^{-1}\circ\bfPsi_{\zeta_2-\zeta_1}\rVert_{L^2(\Omega_{\zeta_1(s)})}\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{L^4(\Omega_{\zeta_1(s)})}\,\mathrm{d}s \\ & \lesssim \int_0^t\lVert \underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1(s)})}\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^2(\omega)}\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{L^4(\Omega_{\zeta_1(s)})}\,\mathrm{d}s \\ & \leq\delta\int_0^t\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1(s)})}^2\,\mathrm{d}s+C(\delta)\int_0^t\lVert\underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1(s)})}^2\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^2(\omega)}^2\,\mathrm{d}s. \end{align}\] We rewrite \(\mathop{\mathrm{div}}\underline\mathbf{v}_2\) as \[\mathop{\mathrm{div}}\underline\mathbf{v}_2=\mathbb{I}_{3\times 3}:\nabla\underline\mathbf{v}_2=(\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}):\nabla\underline\mathbf{v}_2,\] and thereby have \[\begin{align} R_{10}&=\int_0^t\int_{\Omega_{\zeta_1( s )}}(\mathbf{v}_1-\underline \mathbf{v}_2)\cdot\mathrm{Bog}_{\zeta_1}\big((\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}):\partial_t\nabla\underline\bfv_2\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\\ &\quad -\int_0^t\int_{\Omega_{\zeta_1( s )}}(\mathbf{v}_1-\underline \mathbf{v}_2)\cdot\mathrm{Bog}_{\zeta_1}\big(\partial_t\mathbf{B}_{\zeta_2-\zeta_1}:\nabla\underline{\bfv}_2\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\\ &=:R_{10}^1+R_{10}^2. \end{align}\] By 8 and continuity of \(\mathrm{Bog}\), we see that \[\begin{align} R_{10}^1&\leq \int_0^t\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{L^6(\Omega_{\zeta_1( s )})}\lVert(\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1})\partial_t\nabla \underline \mathbf{v}_2\rVert_{W^{-1,\frac{6}{5}}(\Omega_{\zeta_1( s )})}\,\mathrm{d}s\\ &\lesssim \int_0^t\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1( s )})}\lVert\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}\rVert_{W^{1,2}(\Omega_{\zeta_1( s ))}}\lVert\partial_t\underline \mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1( s )})}\,\mathrm{d}s\\ &\leq \delta \int_0^t\lVert \mathbf{v}_1-\underline\mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1( s )})}^2\,\mathrm{d}s+C(\delta)\int_0^t\lVert\partial_t\underline \mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1( s )})}^2\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2\,\mathrm{d}s, \end{align}\] as well as (using the Piola identity, the definition of \(\mathbf{B}_{\zeta_2-\zeta_1}\) and 11 ) \[\begin{align} R_{10}^2&\leq \int_0^t\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{L^6(\Omega_{\zeta_1( s )})}\lVert \partial_t\mathbf{B}_{\zeta_2-\zeta_1}:\nabla\underline{\bfv}_2\rVert_{W^{-1,\frac{6}{5}}(\Omega_{\zeta_1( s )})}\,\mathrm{d}s\\ &\leq \int_0^t\lVert\mathbf{v}_1-\underline \mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1( s )})}\|\partial_t(\zeta_2-\zeta_1)\|_{L^2(\omega)}\|\nabla\underline \mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1( s )})}\,\mathrm{d}s\\ &\leq \delta \int_0^t\lVert \mathbf{v}_1-\underline\mathbf{v}_2\rVert_{W^{1,2}(\Omega_{\zeta_1( s )})}^2\,\mathrm{d}s+C(\delta)\int_0^t\lVert\underline \mathbf{v}_2\rVert_{W^{2,2}(\Omega_{\zeta_1( s )})}^2\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^2(\omega)}^2\,\mathrm{d}s. \end{align}\] Note that we also took into account that \(\zeta_1\) and \(\zeta_2\) belong to the class \(L^\infty(I;C^{0,1}(\omega))\).

Finally, we estimate \[\begin{align} R_{11} &\leq \lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}\lVert\mathrm{Bog}_{\zeta_1}\left((\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}):\nabla\underline\mathbf{v}_2\right)\rVert_{L^2(\Omega_{\zeta_1(s)})} \\ &\lesssim \lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{L^2(\Omega_{\zeta_1(s)})}\lVert\mathrm{Bog}_{\zeta_1}\left((\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}):\nabla\underline\mathbf{v}_2\right)\rVert_{W^{1,6/5}(\Omega_{\zeta_1(s)})} \\ &\leq \delta\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2+C(\delta)\lVert(\mathbb{I}_{3\times 3}-\mathbf{B}_{\zeta_2-\zeta_1}):\nabla\underline\mathbf{v}_2\rVert_{ L^{\frac{6}{5}}(\Omega_{\zeta_1(s)})}^2 \\ &\leq \delta\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2+C(\delta)\lVert\nabla\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2\lVert\nabla_{\mathbf{y}}(\zeta_1-\zeta_2)\rVert_{L^3(\omega)}^2 \\ &\leq \delta\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2+C(\delta)\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^{\frac{4}{3}}\lVert\zeta_1-\zeta_2\rVert_{L^{2}(\omega)}^{\frac{2}{3}} \\ &\leq \delta\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2+\nu\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2+C(\delta,\nu)\lVert\zeta_1-\zeta_2\rVert_{L^{2}(\omega)}^2 \\ &\leq \delta\lVert\mathbf{v}_1-\underline\mathbf{v}_2\rVert_{ L^2(\Omega_{\zeta_1(s)})}^2+\nu\lVert\zeta_1-\zeta_2\rVert_{W^{2,2}(\omega)}^2+C(\delta,\nu)\int_0^t\lVert\partial_t(\zeta_1-\zeta_2)\rVert_{L^{2}(\omega)}^2\,\mathrm{d}s. \end{align}\]

Combining everything yields (choosing \(T_{\tt max},\delta,\nu\) small enough) \[\begin{align} &\tfrac{1}{2}\sup_{0\leq t\leq T_{\tt max}}\int_{\Omega_{\zeta_1(t)}}|\mathbf{v}_1(t)-\underline{\mathbf{v}}_2(t)|^2\, \mathrm{d} \mathbf{x}+\int_0^{T_{\tt max}} \int_{\Omega_{\zeta_1(s)}}|\nabla(\mathbf{v}_1-\underline{\mathbf{v}}_2)|^2\, \mathrm{d} \mathbf{x}\,\mathrm{d}s\nonumber\\ &\quad+\sup_{0\leq t\leq T_{\tt max}}\bigg(\tfrac{1}{2}\int_\omega|\partial_t(\eta_1-\eta_2)(t)|^2\, \mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_\omega|\Delta_{\mathbf{y}}(\eta_1-\eta_2)(t)|^2\, \mathrm{d}\mathbf{y}\bigg)\nonumber\\ &\leq \tfrac{1}{2}\sup_{0\leq t\leq T_{\tt max}}\bigg(\tfrac{1}{2}\int_\omega|\partial_t(\zeta_1-\zeta_2)(t)|^2\, \mathrm{d}\mathbf{y}+\tfrac{1}{2}\int_\omega|\Delta_{\mathbf{y}}(\zeta_1-\zeta_2)(t)|^2\, \mathrm{d}\mathbf{y}\bigg),\nonumber \end{align}\] which gives the claim. ◻

Proof of Theorem 2. As a consequence of Proposition 11 the weak solution obtained in Proposition 10 is unique. Hence the mapping \(\zeta\mapsto \eta\) is well-defined. By Corollary 3, it maps the convex set \[B^{ X_{\tt shell}(0,T_{\tt max})}_R(0)\cap \{\zeta\in X_{\tt shell}(0,T_{\tt max}):\,\zeta(0)=\eta_0\}\] into itself. Again, by Proposition 11 the mapping \(\zeta\mapsto \eta\) is a contraction with respect to the topology from \[W^{1,\infty}(0,T_{\tt max};L^2(\omega))\cap L^{\infty}(0,T_{\tt max};W^{2,2}(\omega)).\] By Banach’s fixed-point theorem, we obtain the existence of the solution for the system 19 –21 . ◻

Proof. For that purpose consider the test function pair for the fluid-structure system 19 –21 as follows: \[\begin{align} (\overline{\boldsymbol{\varphi}}^\eta,\phi^\eta)=\left(\Delta_{-h}\Delta_h \partial_t\eta-\mathscr K_{{\eta}}(\Delta_{-h}\Delta_h \partial_t\eta),\, \mathscr F_{\eta}^{\mathrm{div}}(\Delta_{-h}\Delta_h \partial_t\eta-\mathscr K_{{\eta}}(\Delta_{-h}\Delta_h \partial_t\eta))\circ\bfPsi_{\eta}\right). \end{align}\] where \(\mathscr F_{\eta}^{\mathrm{div}}\) is the corresponding solenoidal extension introduced in Lemma 1 and for \(h>0\), \[\Delta_hv(\mathbf{y}):=h^{-1}(v(\mathbf{y}+h\boldsymbol{e}_\alpha)-v(\mathbf{y}))\] is the difference quotient for space variable in direction \(\boldsymbol{e}_\alpha\) with \(\alpha\in\{1,2\}\). Similarly to 35 , we have\[\begin{align} &\int_I \int_{\omega} |\Delta_h\Delta_{\mathbf{y}}\partial_t\eta|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\nonumber\\ &=\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\partial_t\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega\partial_t\bfA_{\eta}\nabla\overline{\mathbf{v}}:\nabla\overline{\boldsymbol{\varphi}} ^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega\bfA_{\eta}\nabla\partial_t\overline{\mathbf{v}}:\nabla\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\quad -\int_I\frac{\mathrm d}{\,\mathrm{d} t}\left(\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}+\int_\omega\partial_t^2\eta\cdot\phi^\eta\, \mathrm{d}\mathbf{y}\right)\,\mathrm{d} t+\int_I\int_\omega(\partial_t^2\eta\, \partial_t\phi^\eta+\partial_t g \,\phi^\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\nonumber\\ &\quad +\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\partial_t\mathbf{B}_{\eta}^\intercal \overline{\boldsymbol{\varphi}}^\eta)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_I\int_\Omega\partial_t\bfh_{\eta}\cdot \overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \nonumber\\ &\quad+\int_I\int_\Omega\left(-\partial_t J_{\eta} \overline{\mathbf{v}}\cdot \partial_t\overline{\boldsymbol{\varphi}}+\frac{\mathrm d}{\,\mathrm{d} t}\left(\partial_tJ_{\eta}\overline{\mathbf{v}}\cdot \overline{\boldsymbol{\varphi}}^\eta\right)-\partial_t(\partial_t J_{\eta}\overline{\mathbf{v}})\cdot \overline{\boldsymbol{\varphi}}^\eta\right)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &=:(I)^\eta+...+(VII)^\eta+(VIII)^\eta.\label{mom:reg2D} \end{align}\tag{45}\] We first estimate \((I)^\eta\) as follows: \[\begin{align} (I)^\eta&\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\bfvarphi}^\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\left(\|\partial_t\Delta_{-h}\Delta_h\partial_t\eta\|_{L^2_\mathbf{y}}+\|\Delta_{-h}\Delta_h\partial_t\eta\,\partial_t\eta\|_{L^2_\mathbf{y}}\right)\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\left(\|\partial_t^2\eta\|_{W^{2s,2}_\mathbf{y}}+\|\Delta_{-h}\Delta_h\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\right)\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}(\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}})\,\mathrm{d} t\nonumber\\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t.\label{later2} \end{align}\tag{46}\] Note that \(\nabla\eta\in L^\infty_{t,\mathbf{y}}\), then we have \(|\partial_t\bfA_\eta|\lesssim |\partial_t\nabla\eta|+|\partial_t\eta|\). Using Lemma 1 and 10 we estimate further \[\begin{align} (II)^\eta&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\nabla\overline{\bfvarphi}^{\eta}\|_{L^4_\mathbf{x}}\, \mathrm{d} \mathbf{x}+\int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^4_\mathbf{y}}\|\nabla\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{W^{1,2}_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{W^{1+2s,4}_\mathbf{y}}\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{8}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{5}{8}+s}\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\Big(1+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\Big)(1+\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2)\,\mathrm{d} t,\label{later1} \end{align}\tag{47}\] where we used \(\partial_t\eta\in L^\infty_t(L^2_\mathbf{y})\) and \[\|\partial_t\eta\|_{L^4_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{7}{8}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{8}},\] \[\|\partial_t\eta\|_{W^{1+2s,4}_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{8}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{5}{8}+s}.\] As before (but now using that \(\nabla_{\mathbf{y}}\eta\in L^\infty_{t,\mathbf{y}}\) by the 1D Sobolev embedding),\[\begin{align} (III)^\eta &\lesssim \int_I\Big(\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^{2}_\mathbf{y}}^2\Big)\,\mathrm{d} t,\\ (IV)^\eta &\lesssim\sup_I\Big(\|\partial_t\mathbf{v}\|_{L^2_\mathbf{x}}^2+\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^{2}_\mathbf{y}}^2\Big),\\ (V)^\eta &\lesssim \int_I\Big(\|\partial_t\mathbf{v}\|_{W^{1,2}_\mathbf{x}}^2+\|\partial_tg\|_{L^2_\mathbf{y}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^{2}_\mathbf{y}}^2\Big)\,\mathrm{d} t. \end{align}\] We now treat \((VII)^\eta\) and \((VIII)^\eta\) before dealing with the more subtle \((VI)^\eta\). Similar to how \((VII)^\eta\) was treated in 35 , we can again decompose \((VII)^\eta\) into the sum of \((VII)^{\eta}_1,\dots, (VII)^{\eta}_3\), where \[\begin{align} (VIII)^\eta&:=- \int_I\int_\Omega\partial_t J_\eta \overline{\mathbf{v}}\cdot \partial_t\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_\Omega\partial_tJ_{\eta} \overline{\mathbf{v}}\cdot \overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\bigg|_0^T+\int_I\int_\Omega\partial_t(\partial_t J_{\eta}\overline{\mathbf{v}})\cdot \overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t,\\ (VII)^\eta_1&:= \int_I\int_\Omega J_\eta\nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_{\eta}^{-1}\circ \bfPsi_{\eta}\cdot \partial_t\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega \partial_t\Big(J_{\eta} \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_{\eta}^{-1}\circ \bfPsi_{\eta}\cdot\overline{\boldsymbol{\varphi}}^\eta\Big)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t, \\(VII)^\eta_2&:=- \int_I\int_\Omega\partial_t\big(\mathbf{B}_{\eta}\nabla \overline{\bfv}~\overline{\mathbf{v}}\big)\cdot\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t,\\ (VII)^\eta_3&:= \int_I\int_\Omega\partial_t\big( J_\eta\bff\circ \bfPsi_{\eta}\big)\cdot\overline{\boldsymbol{\varphi}}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t. \end{align}\] Arguing as before (and using again \(\nabla_{\mathbf{y}}\eta\in L^{\infty}_{t,\mathbf{y}}\)) we have\[\begin{align} (VII)^{\eta}_3&\lesssim \int_I\Big(\|\partial_t\overline{\bff}\|_{L^2_\mathbf{x}}^2+\|\overline{\bff}\|_{L^2_\mathbf{x}}^2\Big)\,\mathrm{d} t+\int_I(1+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2)\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{2}\,\mathrm{d} t. \end{align}\] Arguing as in 46 we obtain \[\begin{align} (VIII)^\eta&\lesssim \int_I\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\bfvarphi}^\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t+\sup_I\|\partial_t\eta\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\\ &\quad +\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}+\sup_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{2s,4}_\mathbf{y}}\\ &\quad+\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\eta\|_{W^{2s,4}_\mathbf{y}}\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{W^{2s,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_I\left(\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2+\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2\right)\,\mathrm{d} t+\sup_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\\ &\quad +\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2(\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2+\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2)\,\mathrm{d} t. \end{align}\] Now we estimate the remaining terms in 45 : \[\begin{align} (VII)^\eta_1&\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\overline{\bfvarphi}^\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t+\sup_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\overline{\bfvarphi}^\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}(\|\partial_t^2\Delta_{-h}^s\Delta_h^s\eta\|_{L^2_\mathbf{y}}+\|\partial_t\Delta_{-h}^s\Delta_h^s\eta\,\partial_t\eta\|_{L^2_\mathbf{y}})\,\mathrm{d} t \\ &\quad +\sup_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{1-s}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^s\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\left(\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\right)\,\mathrm{d} t\\&\quad +\sup_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{7}{4}-s}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}+s}\,\mathrm{d} t \\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I(\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2+1)\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t\\&\quad +\delta\sup_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2+\varepsilon\sup_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2+c(\delta, \varepsilon). \end{align}\] Note that \(|\partial_t\mathbf{B}_\eta|\lesssim |\partial_t\nabla\eta|+|\partial_t\eta|\), thereby we have \[\begin{align} (VII)_2^\eta&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^3_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^6_\mathbf{x}}\,\mathrm{d} t\\ &\quad +\int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}} +\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{3}{4}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{3}{2}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{7}{8}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{8}+s}\,\mathrm{d} t\\ &\quad+\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{4}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{2}{3}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{\frac{4}{3}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{5}{6}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{6}+s}\,\mathrm{d} t\\ &\quad +\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+1)(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2)\,\mathrm{d} t, \end{align}\] where the \(1D\) interpolation inequalities: \[\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\lesssim \|\partial_t\nabla_{\mathbf{y}}\eta\|_{W^{-1,2}_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{W^{1,2}_\mathbf{y}}^{\frac{3}{4}},\qquad \|\partial_t\eta\|_{L^\infty_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{4}},\] \[\|\partial_t\eta\|_{W^{2s,4}_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{7}{8}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{8}+s},\qquad \|\partial_t\eta\|_{W^{2s,6}_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{5}{6}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{6}+s},\] and \(2D\) interpolation inequalities: \[\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\lesssim \|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{\frac{1}{2}}, \qquad \|\overline{\mathbf{v}}\|_{L^3_\mathbf{x}}\lesssim \|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{2}{3}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{\frac{1}{3}}.\]

We finally return to treat \((VI)^\eta\). For this, we use the weak formulation of the coupled system to represent the pressure integrals involved in 45 as \[\begin{align} &\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\partial_t\mathbf{B}_\eta^\intercal \overline{\bfvarphi}^\eta)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t =\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\mathbf{B}_\eta^\intercal\boldsymbol{\Upsilon}^\eta)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &=\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_I\int_\Omega\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega\bfh_\eta(\overline{\mathbf{v}})\cdot\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &\quad +\int_I\int_\omega(\partial_t^2\eta+\Delta_{\mathbf{y}}^2\eta)\bfn^\intercal \boldsymbol{\Upsilon}^\eta\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t -\int_I\int_\omega g\, \bfn^\intercal \boldsymbol{\Upsilon}^\eta\circ\bfvarphi \, \mathrm{d}\mathbf{y}\,\mathrm{d} t,\label{for2dlater} \end{align}\tag{48}\] where \(\boldsymbol{\Upsilon}^\eta=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal\overline{\bfvarphi}^\eta\) and \[\begin{align} \boldsymbol{\Upsilon}^\eta\circ\bfvarphi&=\mathbf{B}_\eta^{-\intercal}\circ\bfvarphi \partial_t\mathbf{B}_\eta^\intercal\circ\bfvarphi \Delta_{\mathbf{y}}^{1/4}\partial_t\eta \bfn= J_\eta ^{-1}\partial_t J_\eta \Delta_{\mathbf{y}}^{1/4}\partial_t\eta \bfn \\ &\approx (1+\eta)\partial_t\eta \Delta_{\mathbf{y}}^{1/4}\partial_t\eta\bfn, \end{align}\] by 11 and 12 . We derive from Lemma 1 that \[\begin{align} &\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \lesssim \int_I(\partial_t\overline{\mathbf{v}}\,\partial_t\eta+\partial_t\overline{\mathbf{v}}\,\partial_t\nabla\eta)\overline{\bfvarphi}^\eta\,\mathrm{d} t \\ &\lesssim \int_I(\|\partial_t\eta\|_{L^\infty_\mathbf{y}}+\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}})\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{-h}^s\Delta_h^s\eta\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\Big(\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\Big)\,\mathrm{d} t, \end{align}\] Using again 18 and 8 –10 , we see that the critical term in \(|\partial_t\nabla\mathbf{B}_\eta|\) is \(\partial_t\nabla^2\eta\), then \[\begin{align} &\int_I\int_\Omega\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\sim \int_I\int_\Omega\nabla\overline{\mathbf{v}}\left(\partial_t\nabla^2\eta+\nabla^2\eta\partial_t\nabla\eta\right)\overline{\bfvarphi}^\eta\,\mathrm{d} t+\int_I\int_\Omega\nabla\overline{\mathbf{v}}\partial_t\nabla\eta\nabla\overline{\bfvarphi}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &:=I_{\bfA}+II_{\bfA}+III_{\bfA}. \end{align}\] We see that \[\begin{align} I_{\bfA}&\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla^2\eta\|_{L^4_\mathbf{y}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\nabla^2\eta\|_{W^{\frac{1}{4},2}_\mathbf{y}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\leq \delta\int_I\|\partial_t\nabla^2\eta\|_{W^{\frac{1}{4},2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Moreover, we have \[\begin{align} II_\bfA&\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\nabla^2\eta\|_{L^4_\mathbf{y}}\|\partial_t\nabla\eta\|_{L^\infty_\mathbf{y}}\|\overline{\bfvarphi}^\eta\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\nabla^2\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\eta\|_{W^{3,2}_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{7}{8}-s}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{\frac{1}{8}+s}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I(1+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2)(1+\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2)\,\mathrm{d} t. \end{align}\] Finally we notice that \(III_\bfA\) has been estimated in 47 .

Recalling the definition of \(\bfh_{\eta}\) in 18 , we have \[\begin{align} &\int_I\int_\Omega\bfh_\eta(\overline{\mathbf{v}})\cdot\boldsymbol{\Upsilon}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &\sim \int_I\int_\Omega \nabla\overline{\mathbf{v}}\,\overline{\mathbf{v}}\partial_t\nabla\eta\overline{\bfvarphi}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_I\int_\Omega\nabla\overline{\mathbf{v}}\partial_t\eta\,\partial_t\nabla\eta\,\overline{\bfvarphi}^\eta\,\mathrm{d} t+\int_I\int_\Omega\overline{\bff}\partial_t\nabla\eta\overline{\bfvarphi}^\eta\,\mathrm{d} t. \end{align}\] The first integral has been estimated in \((VII)_2^\eta\). We treat the remaining integrals below: \[\begin{align} \int_I\int_\Omega\nabla\overline{\mathbf{v}}\partial_t\eta\,\partial_t\nabla\eta\,\overline{\bfvarphi}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t&\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\eta\|_{W^{2s,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\eta\|_{W^{2s,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^4)\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Fnally using \(\overline{\bff}\in L^\infty_t(L^2_\mathbf{x})\), we have \[\begin{align} \int_I\int_\Omega\overline{\bff}\partial_t\nabla\eta\overline{\bfvarphi}^\eta\,\mathrm{d} t&\lesssim \int_I\|\overline{\bff}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\Delta_{-h}^s\Delta_h^s\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\left(\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\right)\,\mathrm{d} t. \end{align}\]

We continue the estimate of the solid part in 48 . We have \[\begin{align} \int_I\int_\omega\partial_t^2\eta\mathbf{n}^\intercal\boldsymbol{\Upsilon}^\eta\circ\bfvarphi \, \mathrm{d}\mathbf{y}\,\mathrm{d} t&\leq \int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\Delta_\mathbf{y}^{1/4}\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\nonumber\\ &\leq \int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] For the integral depends on \(\Delta_{\mathbf{y}}^2\eta\) we take an integration by parts and obtain \[\begin{align} &\int_I\int_\omega\Delta_{\mathbf{y}}^2\eta \mathbf{n}^\intercal\boldsymbol{\Upsilon}^\eta\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &=-\int_I\int_\omega\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta\nabla_{\mathbf{y}}\eta \partial_t\eta(\Delta_{-h}^s\Delta_h^s\partial_t\eta-\mathscr K_\eta(\Delta_{-h}^s\Delta_h^s\partial_t\eta))\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad-\int_I\int_\omega\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta (1+\eta)\partial_t\nabla_{\mathbf{y}}\eta(\Delta_{-h}^s\Delta_h^s\partial_t\eta-\mathscr K_\eta(\Delta_{-h}^s\Delta_h^s\partial_t\eta))\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\quad -\int_I\int_\omega\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta (1+\eta)\partial_t\eta \Delta_{-h}^s\Delta_h^s\partial_t\nabla_{\mathbf{y}}\eta\, \mathrm{d}\mathbf{y}\,\mathrm{d} t \\ &\lesssim \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\|\partial_t\eta\|_{L^{2}_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,\infty}_\mathbf{y}}\,\mathrm{d} t+ \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\,\mathrm{d} t \\ &\quad + \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\partial_t\eta\|_{L^{4}_\mathbf{y}}\|\partial_t\eta\|_{W^{1+2s,4}_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t+ \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t+\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}+s}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I\left(1+\|\partial_t\eta\|^2_{W^{2,2}_\mathbf{y}}\right)\Big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+1\Big)\,\mathrm{d} t. \end{align}\] Finally, we also have \[\begin{align} \int_I\int_\omega g\mathbf{n}^\intercal\boldsymbol{\Upsilon}^\eta\circ\bfvarphi \, \mathrm{d}\mathbf{y}\,\mathrm{d} t\lesssim \int_I\|g\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{W^{2s,\infty}_\mathbf{y}}\,\mathrm{d} t\lesssim \int_I\|g\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Thus we finish the estimate of 45 . ◻

Proof of Theorem 12. Based on Theorem 2, which also hold for \(n=2\) by constant elongation. Hence there exists a local strong solution for short times. In particular \(\partial_t^2\eta,\partial_t \overline{\mathbf{v}}\) have well defined initial conditions and \[\begin{align} \label{eq:initial} \int_\omega\abs{\Delta_\tau\partial_t\eta(0)}^2\, \mathrm{d}\mathbf{y}+\int_{\Omega} \abs{\Delta_\tau\overline{\mathbf{v}}(0)}^2\, \mathrm{d} \mathbf{x}<\infty \end{align}\tag{49}\] uniformly \(h\). The goal is to obtain estimates which do not blow up in time and to extend the local solution to a global one. The regularity of the strong solution does not suffice to differentiate 19 –21 in time and use the test function pair: \[(\partial_t^2\eta+\mathbf{n}^\intercal \bfG\circ\bfvarphi, \quad \partial_t\overline{\mathbf{v}}+\bfG),\] where \(\bfG=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal\overline{\mathbf{v}}\). Hence the following computations are formal. They do hold nevertheless as it can be easily checked that the very same computations can be performed for the respective version using finite in time-differences, in particular because of 49 . While in the 3D case we took the effort to perform all estimates on the finite differences in the 2D case we chose to use derivatives as this allows to follow the argument better. Hence (formally) using \((\partial_t^2\eta+\mathbf{n}^\intercal \bfG\circ\bfvarphi, \,\, \partial_t\overline{\mathbf{v}}+\bfG),\) as test function on the (formally) time-differentiated system implies \[\begin{align} &\sup_I \int_\omega\big(\vert \partial_t^2\eta\vert^2 + \vert \partial_t\Delta_{\mathbf{y}}\eta\vert^2 \big)\, \mathrm{d}\mathbf{y} + \sup_I \int_\Omega\vert \partial_t\overline{\mathbf{v}}\vert^2\, \mathrm{d} \mathbf{x} + \sup_I \int_\Omega\vert \nabla\overline{\mathbf{v}}\vert^2\, \mathrm{d} \mathbf{x} + \int_I\int_\Omega\vert \partial_t\nabla \overline{\mathbf{v}}\vert^2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &\lesssim \int_\omega\big(\vert \partial_t^2\eta(0)\vert^2 + \vert \Delta_{\mathbf{y}}\eta_*\vert^2 \big)\, \mathrm{d}\mathbf{y}+ \int_\Omega\vert \partial_t\overline{\mathbf{v}}(0)\vert^2\, \mathrm{d} \mathbf{x} + \int_\Omega\vert \nabla \overline{\mathbf{v}}_0\vert^2\, \mathrm{d} \mathbf{x}\nonumber\\ & \quad + \int_I\int_\omega\partial_tg \,\partial_t^2\eta\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+ \int_I\int_\Omega \partial_t\bfA_\eta\nabla\overline{\mathbf{v}}\cdot\partial_t\nabla\overline{\mathbf{v}} \, \mathrm{d} \mathbf{x}\,\mathrm{d} t+ \int_I\int_\Omega\big(\partial_t\mathbf{h}_\eta-\partial_t J_\eta\partial_t\overline{\mathbf{v}}\big)\cdot\partial_t\overline{\mathbf{v}} \, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\quad +\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\partial_t \mathbf{B}_\eta^\intercal(\partial_t\overline{\mathbf{v}}+\bfG))\, \mathrm{d} \mathbf{x}\,\mathrm{d} t +\int_I\int_\Omega\big(\partial_t\bfh_\eta-\partial_t J_\eta\partial_t\overline{\mathbf{v}}\big)\cdot \bfG \, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\quad -\int_I\int_\Omega J_\eta\partial_t^2\overline{\mathbf{v}}\cdot \bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t -\int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega\bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:\nabla\bfG \, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\quad +\int_I\int_\omega\partial_t g\, \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t -\int_I\int_\omega\partial_t^3\eta\,\mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t -\int_I\int_\omega\partial_t\Delta^2_\mathbf{y}\eta\, \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t, \label{dt3468a2d} \end{align}\tag{50}\] analogously to 38 . Here \(\partial_t\overline{\mathbf{v}}(0)\) and \(\partial_t^2\eta (0)\) can be computed from the data using the equation at \(t=0\). Now we make use of another test in order to improve the spatial regularity of \(\eta\).

Estimate of the solid part in 50 . Note that \(\partial_t\mathbf{B}_\eta^\intercal\circ\bfvarphi \mathbf{n}=0\) does not hold in the case of shells in general. Thus the integrals in the last line of 50 appear additionally to those arsing in the case of plates from [19]. For this recalling that \(\bfG\circ\bfvarphi\sim |\partial_t\eta|^2(1+\eta)\bfn\) we have \[\begin{align} &\int_I\int_\omega\partial_tg \mathbf{n}^\intercal\bfG\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\lesssim \int_I\|\partial_tg\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^4_\mathbf{y}}^2\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_tg\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\,\mathrm{d} t\lesssim \int_I\|\partial_tg\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Taking an integration by parts in time, we see that \[\begin{align} &\int_I\int_\omega \partial_t^3\eta\partial_t\eta\partial_t\eta(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\nonumber\\ &=-2\int_I\int_\omega |\partial_t^2\eta|^2\partial_t\eta(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t-\int_I\int_\omega\partial_t^2\eta|\partial_t\eta|^3\, \mathrm{d}\mathbf{y}\,\mathrm{d} t +\int_\omega \partial_t^2\eta|\partial_t\eta|^2(1+\eta)\, \mathrm{d}\mathbf{y}\bigg|_{s=0}^{s=T}\nonumber \\ &\lesssim \int_I\|\partial_t^2\eta\|^2_{L^4_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t+\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^6_\mathbf{y}}^3\,\mathrm{d} t +\sup_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\nonumber\\ &\lesssim \int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t^2\eta\|_{W^{1/2,2}_\mathbf{y}} \,\mathrm{d} t+\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^2\|\partial_t\eta\|_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t +\sup_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}} \|\partial_t\nabla\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\nonumber\\ &\lesssim c(\delta)\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t\nonumber\\ &\quad +\delta\sup_I\big(\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2 +\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\big)+c(\delta),\label{etaestimate} \end{align}\tag{51}\] where we used the trace theorem and the embedding \(W^{1/4,2}_\mathbf{y}\hookrightarrow L^4_\mathbf{y}\) and the following \(1D\) interpolation inequality \[\|\partial_t^2\eta\|_{W^{\frac{1}{4},2}_\mathbf{y}}\lesssim \|\partial_t^2\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t^2\eta\|_{W^{\frac{1}{2},2}_\mathbf{y}}^{\frac{1}{2}}.\] Similarly, for the last integral in 50 after taking integration by parts in space we just write the high-order terms here: \[\begin{align} &\int_I\int_\omega \partial_t\Delta_{\mathbf{y}}^2\eta|\partial_t\eta|^2(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &\lesssim \int_I\int_\omega |\partial_t\Delta_{\mathbf{y}}\eta|^2\partial_t\eta\, \mathrm{d}\mathbf{y}\,\mathrm{d} t +\int_I\int_\omega \partial_t\Delta_{\mathbf{y}}\eta|\partial_t\nabla_{\mathbf{y}}\eta|^2\, \mathrm{d}\mathbf{y}\,\mathrm{d} t+\int_I\int_\omega\partial_t\Delta_{\mathbf{y}}\eta|\partial_t\eta|^2\nabla_{\mathbf{y}}^2\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t \\ &\lesssim \int_I \big(\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}} \|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^4_\mathbf{y}} +\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}}^2\big)\,\mathrm{d} t+\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^4_\mathbf{y}}^2\|\Delta_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{L^2_\mathbf{y}} \|\partial_t\eta\|_{W^{1/4,2}_\mathbf{y}}^2\,\mathrm{d} t+\delta\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{W^{1/4,2}_\mathbf{y}} +\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\,\mathrm{d} t\\ &\quad +\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\|\eta\|_{W^{3,2}_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim c(\delta)\int_I\|\partial_t\eta\|^2_{W^{2,2}_\mathbf{y}} \big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+1\big)\,\mathrm{d} t+\delta\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{W^{1/4,2}_\mathbf{y}}+\delta\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Note that the last term can be handled by means of 44 . Now we estimate the remaining integrals on the right hand side of 50 in order.

We note that \[\int_I\int_\omega\partial_t g\partial_t^2\eta\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\lesssim \int_I\|\partial_tg\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t.\] Then we estimate further that \[\begin{align} &\int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}\cdot\partial_t\nabla\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim\int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{L^{2}_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Recalling \(\bfh_\eta\) in 18 , we decompose the integral \(\int_I\int_\Omega (\partial_t\bfh_\eta-\partial_tJ_\eta\partial_t\overline{\mathbf{v}})\cdot\partial_t\overline{\mathbf{v}}\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\) into \[\begin{align} ({\tt I})^{\bfh}&:=- \int_I\int_\Omega\partial_t J_\eta|\partial_t \overline{\mathbf{v}}|^2\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ ({\tt II})^{\bfh}&:=- \int_I\int_\Omega\partial_t\big( J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta\big)\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t, \\({\tt III })^{\bfh}&:=- \int_I\int_\Omega\partial_t\big(\mathbf{B}_\eta\nabla \overline{\bfv}~\overline{\mathbf{v}}\big)\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ ({\tt IV})^{\bfh}&:= \int_I\int_\Omega\partial_t\big( J_\eta \bff\circ \bfPsi_\eta\big)\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t, \end{align}\] where the last term clearly is of lower order. As for \((\mathrm{I})^{\bfh}\) it holds that \[\begin{align} (\mathrm{I})^{\bfh}& \lesssim\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|^2_{L^4_\mathbf{x}}\,\mathrm{d} t\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] For \(({\tt III })^{\bfh}\), we have \[\begin{align} ({\tt III })^{\bfh}&=- \int_I\int_\Omega\partial_t\mathbf{B}_\eta\nabla \overline{\bfv}~\overline{\mathbf{v}}\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t- \int_I\int_\Omega\mathbf{B}_\eta\partial_t\nabla \overline{\bfv}~\overline{\mathbf{v}}\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t- \int_I\int_\Omega\mathbf{B}_\eta\nabla \overline{\bfv}~\partial_t\overline{\mathbf{v}}\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=:({\tt III })^{\bfh}_1+({\tt III })^{\bfh}_2+({\tt III })^{\bfh}_3, \end{align}\] where, by 8 and 10 , \[\begin{align} ({\tt III })^{\bfh}_1&\lesssim \int_I\|\partial_t\mathbf{B}_\eta\|_{L^{\infty}_\mathbf{x}}\|\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^{3/2}\|\overline{\bfv}\|_{L^{2}_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\partial_t\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\big(1+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{2}+\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\big)\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt III })^{\bfh}_2&\lesssim \int_I\|\mathbf{B}_\eta\|_{L^{\infty}_\mathbf{x}}\|\partial_t\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}} \big)\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{3/2}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{1/2}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\partial_t\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t, \end{align}\] as well as \[\begin{align} ({\tt III })^{\bfh}_3&\lesssim \int_I\|\mathbf{B}_\eta\|_{L^{\infty}_\mathbf{x}}\|\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim \int_I\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\big)\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\leq\,\delta \int_I\|\partial_t\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t. \end{align}\]

We proceed similarly for \(({\tt II})^{\bfh}\) obtaining \[\begin{align} ({\tt II})^{\bfh}=&- \int_I\int_\Omega\partial_t J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_\eta \partial_t\nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &- \int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t(\partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta)\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ =:&({\tt II})^{\bfh}_1+({\tt II})^{\bfh}_2+({\tt II})^{\bfh}_3. \end{align}\] We first see that \[\begin{align} ({\tt II})^{\bfh}_1&\lesssim \int_I\|\partial_t\eta\|_{L_\mathbf{y}^\infty}^2\|\nabla\overline{\bfv}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\bfv}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\|\nabla\overline{\bfv}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\bfv}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^{2}\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\big). \end{align}\] Now we use a trick frequently used in [19] to write the integral over \(\Omega\) as an iterated integral and use 1D embeddings. This is more complicated for shells than for plates. Fortunately, the function \(\partial_t\bfPsi_\eta\) is supported in \(S_L\), where coordinates \((s,y)\in(-L,L)\times\omega\) exists together with a smooth diffeomorphism \(\Lambda :(-L,L)\times \partial\Omega\rightarrow S_L\), cf. Section 2.3. Moreover, we can parametrise \(\partial\Omega\) via the function \(\bfvarphi:\omega\rightarrow\partial\Omega\). The function \[\label{mapneed} \Lambda^\bfvarphi:(-L,L)\times \omega\rightarrow S_L,\quad (s,y)\mapsto \Lambda(s,\bfvarphi(y))\tag{52}\] is also a smooth diffeomorphism. We estimate \((\tt II)_2^{\bfh}\) as follows using also 10 and ?? that \[\begin{align} \label{eq:2405} \begin{aligned} (\tt II)_2^{\bfh}&\lesssim \int_I\int_{-L}^0\|\partial_t\nabla\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t \\ &\lesssim \int_I\int_{-L}^0\|\partial_t\nabla\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{W^{1,2}_\mathbf{y}}^{\frac{1}{2}}\mathrm dz\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{\frac{3}{2}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\,\mathrm{d} t \\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{aligned} \end{align}\tag{53}\] transforming back to \(S_L\) in the penultimate step.

Finally, using the interpolation of \(L^\infty_\mathbf{y}\) between \(L^2_\mathbf{y}\) and \(W^{1,2}_\mathbf{y}\) we obtain arguing as in 53 that \[\begin{align} ({\tt II})^{\bfh}_3&=\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t^2 \bfPsi_\eta^{-1}\circ \bfPsi_\eta\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t\nabla \bfPsi_\eta^{-1}\circ \bfPsi_\eta\partial_t\bfPsi_\eta\cdot\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\int_{-L}^0\|\nabla \overline{\mathbf{v}}\circ \Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ \Lambda^\bfvarphi\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t \\ &\quad +\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\int_{-L}^0\|\nabla \overline{\mathbf{v}}\circ \Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ \Lambda^\bfvarphi\|_{L^2_\mathbf{y}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\,\mathrm dz\,\mathrm{d} t \\&\quad +\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{1/2}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\,\mathrm{d} t+\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+1\big)\big(\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{2}+\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^{2}+\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\big)\,\mathrm{d} t. \end{align}\]

It remains to estimate the terms in 50 related to \(\bfG=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}\). We now consider \(\int_I\int_\Omega(\partial_t\bfh_\eta-\partial_t J_\eta\partial_t\overline{\mathbf{v}})\cdot \bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\), which is again decomposed as \[\begin{align} ({\tt I})^{\bfh,\bfG}&:=- \int_I\int_\Omega\partial_tJ_\eta\partial_t \overline{\mathbf{v}}\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ ({\tt II})^{\bfh,\bfG}&:=- \int_I\int_\Omega\partial_t\big( J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t, \\({\tt III })^{\bfh,\bfG}&:=- \int_I\int_\Omega\partial_t\big(\mathbf{B}_\eta\nabla \overline{\bfv}~\overline{\mathbf{v}}\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ ({\tt IV})^{\bfh,\bfG}&:= \int_I\int_\Omega\partial_t\big( J_\eta \bff\circ \bfPsi_\eta\big)\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] By 8 , 10 and ?? , we have \[\begin{align} ({\tt I})^{\bfh,\bfG}&\lesssim \int_I\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\|\partial_t \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{3/4}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{1/4}\|\partial_t \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{1/4}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{3/4}\,\mathrm{d} t\\ &\lesssim \int_I\big(\|\partial_t \Delta_{\mathbf{y}}\eta\|^2_{L^2_\mathbf{y}}+ \|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\big)\,\mathrm{d} t. \end{align}\] Then with the obvious sub-decompositions into \(({\tt III })^{\bfh,\bfG}_1,({\tt III })^{\bfh,\bfG}_2\) and \(({\tt III })^{\bfh,\bfG}_3\) as well as \(({\tt II})^{\bfh,\bfG}_1,({\tt II})^{\bfh,\bfG}_2\) and \(({\tt II})^{\bfh,\bfG}_3\). It holds by 10 and ?? that \[\begin{align} ({\tt III })^{\bfh,\bfG}_1&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}^2\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\big)\|\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{L^{2}_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^{2}_\mathbf{y}}^{\frac{3}{2}}\|\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\|\overline{\bfv}\|_{L^{2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{L^{2}_\mathbf{y}}\|\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt III })^{\bfh,\bfG}_2&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}^2\big)\|\partial_t\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|^{\frac{3}{4}}_{L^{2}_\mathbf{y}}\|\partial_t\eta\|_{L^{2}_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\nabla \overline{\bfv}\|_{L^{2}_\mathbf{x}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\nabla \overline{\bfv}\|_{L^{2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta) \int_I(1+\|\partial_t\Delta_{\mathbf{y}}\eta\|^2_{L^{2}_\mathbf{y}})\|\overline{\bfv}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt III })^{\bfh,\bfG}_3&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}^2\big)\|\nabla \overline{\bfv}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^{2}_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\eta\|_{L^{2}_\mathbf{y}}^{\frac{1}{4}}\|\mathbf{v}\|_{W^{1,2}_\mathbf{x}}^{\frac{3}{2}}\|\mathbf{v}\|_{L^{2}_\mathbf{x}}^{\frac{1}{2}}\|\partial_t\mathbf{v}\|_{W^{1,2}_\mathbf{x}}^{\frac{1}{2}}\|\partial_t\mathbf{v}\|_{L^{2}_\mathbf{x}}^{\frac{1}{2}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\nabla \overline{\bfv}\|_{L^{2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{2}{3}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\nabla \overline{\bfv}\|_{L^{2}_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\left(\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2+\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2+1\right)\,\mathrm{d} t, \end{align}\] as well as \[\begin{align} ({\tt II})^{\bfh,\bfG}_1&\lesssim \int_I\|\partial_t \eta\|^2_{L^\infty_\mathbf{y}} \|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t \eta\|_{L^2_\mathbf{y}}^{7/4} \|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|^{5/4}_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\,\mathrm{d} t+\int_I(1+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2)\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt II})^{\bfh,\bfG}_2&\lesssim \int_I \|\partial_t\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] and, finally, using the trace embedding \(W^{3/4,2}(\Omega)\hookrightarrow W^{1/4,2}(\partial\Omega)\hookrightarrow L^4(\partial\Omega)\) and arguing as in 53 , we have \[\begin{align} ({\tt II})^{\bfh}_3&=\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t^2 \bfPsi_\eta^{-1}\circ \bfPsi_\eta\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t\nabla \bfPsi_\eta^{-1}\circ \bfPsi_\eta\partial_t\bfPsi_\eta\cdot\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\int_{-L}^0\|\nabla \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{\infty}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{\infty}}\,\mathrm dz\,\mathrm{d} t \\ &\quad +\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{x}}\|\partial_t\eta\|_{L^\infty_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{\infty}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{\infty}}\,\mathrm{d} t \\ &\lesssim \int_I\int_{-L}^0\|\nabla \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{2}}\,\mathrm dz\,\mathrm{d} t \\ &\quad +\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{x}}^{5/4}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{x}}^{7/4}\,\mathrm{d} t \\ &\lesssim \int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{2}}\,\mathrm{d} t+\int_I\|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{x}}^{7/4}\,\mathrm{d} t \\ &\lesssim \int_I\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+1\big)\big(\|\partial_t^2\eta\|^2_{L^2_\mathbf{y}}+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L_\mathbf{y}^{2}}^{2}\big)\,\mathrm{d} t+\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t. \end{align}\]

Now we estimate \(-\int_I\int_\Omega J_\eta\partial_t^2 \overline{\mathbf{v}}\cdot \bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\). For this, we integrate by parts in time obtaining that \[\begin{align} -\int_I\int_\Omega J_\eta\partial_t^2 \overline{\mathbf{v}}\cdot \bfG\, \mathrm{d} \mathbf{x}\,\mathrm{d} t&=-\int_I\int_\Omega\partial_t\big(J_\eta\partial_t \overline{\mathbf{v}}\cdot \bfG\big)\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\int_I\int_\Omega\partial_tJ_\eta\partial_t \overline{\mathbf{v}}\cdot \bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\&\quad +\int_I\int_\Omega J_\eta\partial_t \overline{\mathbf{v}}\cdot \partial_t\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] Note that the second term coincides with \(({\tt I})^{\bfh,\bfG}\) estimate above, while we have (using boundedness of \(\eta\) due to ?? ) \[\begin{align} &-\int_I\int_\Omega\partial_t\big(J_\eta\partial_t \overline{\mathbf{v}}\cdot \bfG\big)\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\&\lesssim \|\eta\|_{L^\infty((0,T_0)\times\omega)}\sup_I\Big(\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\Big)\\ &\lesssim \sup_I\Big(\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\eta\|^{1/4}_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|^{3/4}_{L^{2}_\mathbf{y}}\Big)\\ &\lesssim \delta\sup_I\big(\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|^{2}_{L^{2}_\mathbf{y}}\big)+c(\delta). \end{align}\] Control of the integral \(\int_I\int_\Omega J_\eta\partial_t \overline{\mathbf{v}}\cdot \partial_t\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\). Since \(\mathbf{B}_\eta=\mathrm{cof}(\nabla\bfPsi_\eta)\) there is a linear mapping \(\mathcal{A}:\mathbb{R}^{2\times 2}\rightarrow\mathbb{R}^{2\times 2}\) such that \[\begin{align} \partial_t\bfG&=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\partial_t\overline{\mathbf{v}}+\partial_t\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\mathcal{A}(\nabla\partial_t^2 \bfPsi_\eta\big)\overline{\mathbf{v}}. \end{align}\] Writing \(\mathcal{A}=\sum_{i=1}^2\mathcal{A}_{i}\) with \(\mathcal{A}_{i}:\mathbb{R}^2\rightarrow\mathbb{R}^{2\times 2}\) acting on the column vectors of the matrices we obtain \[\begin{align} \nonumber &\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}( \partial_t^2\nabla \bfPsi_\eta\big)\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t=\sum_{i=1}^2\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2\partial_i \bfPsi_\eta\big)\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\nonumber\\ &=\sum_{i=1}^2\int_I\int_\Omega\partial_i J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2 \bfPsi_\eta\big)\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\sum_{i=1}^2\int_I\int_\Omega J_\eta\partial_t\partial_i\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2 \bfPsi_\eta\big)\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\nonumber\\ &\quad +\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\partial_i\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2 \bfPsi_\eta\big)\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+\sum_{i=1}^2\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2 \bfPsi_\eta\big)\partial_i\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\nonumber\\ &\quad +\sum_{i=1}^2\int_I\int_{\partial\Omega}J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t^2\bfPsi_\eta\big)\overline{\mathbf{v}}\, n^i\,\mathrm d\mathscr H^1\,\mathrm{d} t\nonumber\\ &=J_1+\dots J_5.\label{eq:2308} \end{align}\tag{54}\] We have by 8 , 10 , 11 and ?? that \[\begin{align} J_1&\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^{4}_\mathbf{x}}\big(1+\|\nabla_{\mathbf{y}}\eta\|^2_{L^\infty_\mathbf{y}}\big)\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/2}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2/3}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^{4/3}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2/3}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+ c(\delta)\int_I\big(\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+\|\partial_t^2\eta\|_{L^{2}_\mathbf{y}}^{2}\big)\,\mathrm{d} t, \end{align}\] and, arguing as in 53 , \[\begin{align} J_2&\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}^{1/2}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}^{1/2}\,\mathrm dz\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+ c(\delta)\int_I\|\partial_t^2\eta\|_{L^{2}_\mathbf{y}}^{2}\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+1\big)\,\mathrm{d} t, \end{align}\] as well as \[\begin{align} J_3&\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{\infty}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}^2\eta\|_{L^2_\mathbf{y}}\big)\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{2}_\mathbf{y}}^{1/2}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}^{1/2}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}^{1/2}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}^{1/2}\,\mathrm dz\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/2}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+ c(\delta)\int_I\big(\|\partial_t^2\eta\|_{L^{2}_\mathbf{y}}^{2}+\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{2}\big)\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+1\big)\,\mathrm{d} t, \end{align}\] and \[\begin{align} J_4&\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{\infty}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{2}_\mathbf{y}}^{1/2}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}^{1/2}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{W^{1,2}_\mathbf{y}}\,\mathrm dz\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{1/2}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+ c(\delta)\int_I\big(\|\partial_t^2\eta\|_{L^{2}_\mathbf{y}}^{2}+\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{2}\big)\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+1\big)\,\mathrm{d} t. \end{align}\] Finally, it holds \[\begin{align} J_5&\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^{4}(\partial\Omega)}\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^4(\partial\Omega)}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{3/4,2}_\mathbf{x}}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{3/4,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{3/4}\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/4}\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{3/4}\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{1/4}\,\mathrm{d} t\\ &\lesssim\delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+ c(\delta)\int_I\big(\|\partial_t^2\eta\|_{L^{2}_\mathbf{y}}^{2}+\|\partial_t\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}^{2}\big)\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}+1\big)\,\mathrm{d} t. \end{align}\] As for \(\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\) we argue similarly 54 to terms \[\begin{align} J_1'&:=\sum_{i=1}^2\int_I\int_\Omega\partial_i J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t \bfPsi_\eta\big)\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ J_2'&:=\sum_{i=1}^2\int_I\int_\Omega J_\eta\partial_t\partial_i\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t\bfPsi_\eta\big)\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ J_3'&=\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\partial_i\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t \bfPsi_\eta\big)\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ J_4'&:=\sum_{i=1}^2\int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t \bfPsi_\eta\big)\partial_i\partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t,\\ J_5'&:=\sum_{i=1}^2\int_I\int_{\partial\Omega}J_\eta\partial_t\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\mathcal{A}_i( \partial_t\bfPsi_\eta\big)\partial_t\overline{\mathbf{v}}\, n^i\,\mathrm d\mathscr H^1\,\mathrm{d} t. \end{align}\] They can be estimated analogously to \(J_1,\dots,J_5\) above obtaining by using the function \(\Lambda^\bfvarphi\): \[\begin{align} J_1'&\lesssim\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t, \end{align}\] \[\begin{align} J_2'+J_4'&\lesssim\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{1/2}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{3/2}\,\mathrm{d} t\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t, \end{align}\] \[\begin{align} J_3'&\lesssim\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}\big(1+\|\eta\|_{W^{2,2}_\mathbf{y}}\big)\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t, \end{align}\] \[\begin{align} J_5'&\lesssim\int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{L^4(\partial\Omega)}^{2}\,\mathrm{d} t\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1/4,2}(\partial\Omega)}^{2}\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{3/4,2}_\mathbf{x}}^{2}\,\mathrm{d} t\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{2}\,\mathrm{d} t. \end{align}\] Finally, we have \[\begin{align} \int_I\int_\Omega J_\eta\partial_t\overline{\mathbf{v}}\partial_t\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t &\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\nonumber\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\nonumber\\ &\lesssim \delta \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t,\label{similar} \end{align}\tag{55}\] using again the argument from 53 .

Estimate of two integrals depending on \(\nabla\boldsymbol{G}\) in 50 . By the definition of \(\bfG=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal\overline{\mathbf{v}}\), we see that \[\begin{align} \int_I\int_\Omega \partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t=\int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:(\nabla\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\partial_t\nabla\mathbf{B}_\eta\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\nabla\overline{\mathbf{v}})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] We have the estimate \[\begin{align} \int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t&\lesssim\int_I\|\partial_t\nabla_\mathbf{y}\eta\|_{L^\infty_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\nabla^2_\mathbf{y}\eta\|_{L^4_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^4_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\Delta_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{3}{2}}\|\nabla^2_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\eta\|_{W^{3,2}_\mathbf{y}}^{\frac{1}{4}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\,\mathrm{d} t\\ &\leq \delta\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I(1+\|\partial_t\Delta_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^2)(1+\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2)\,\mathrm{d} t. \end{align}\] Using the map \(\Lambda^\bfvarphi\) in 52 , we have\[\begin{align} &\int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:\mathbf{B}_\eta^{-\intercal}\partial_t\nabla\mathbf{B}_\eta\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\int_{-L}^0\|\partial_t\nabla_\mathbf{y}\eta\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}^2\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t\\ &\lesssim \int_I\int_{-L}^0\|\partial_t\nabla_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\nabla^2_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\|\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\nabla\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\mathrm dz\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\nabla_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\nabla^2_\mathbf{y}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{2}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{3}{2}}\,\mathrm{d} t\\ &\leq \int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\,\mathrm{d} t+\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\] Also we estimate: \[\begin{align} \int_I\int_\Omega\partial_t\bfA_\eta\nabla\overline{\mathbf{v}}:\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\nabla\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t&\lesssim \int_I\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t\\ &\lesssim \int_I(1+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2)\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\]

Now for the second integral on \(\nabla\bfG\), we have \[\begin{align} \int_I\int_\Omega\bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:\nabla\bfG\,\mathrm{d}\mathbf{x}\,\mathrm{d}t=\int_I\int_\Omega \bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:(\nabla\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\partial_t\nabla\mathbf{B}_\eta\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\nabla\overline{\mathbf{v}})\,\mathrm{d}\mathbf{x}\,\mathrm{d}t. \end{align}\] We estimate the above integrals as below: \[\begin{align} &\int_I\int_\Omega\bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:\nabla\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t \\& \lesssim \int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\nabla_{\mathbf{y}}^2\eta\|_{L^6_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^6_\mathbf{y}}\|\overline{\mathbf{v}}\|_{L^6_\mathbf{x}}\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\nabla_{\mathbf{y}}^2\eta\|_{L^2_\mathbf{y}}^{\frac{2}{3}}\|\nabla_{\mathbf{y}}^2\eta\|_{W^{1,2}_\mathbf{y}}^{\frac{1}{3}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{1}{3}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{2}{3}}\|\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{1}{3}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{2}{3}}\,\mathrm{d} t \\ &\leq\delta\int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\nabla_{\mathbf{y}}^2\eta\|_{W^{1,2}_\mathbf{y}}^{\frac{2}{3}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{4}{3}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^{\frac{4}{3}}\,\mathrm{d} t \\ &\leq \delta\int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+\mu\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta, \mu)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} &\int_I\int_\Omega\bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:\mathbf{B}_\eta^{-\intercal}\partial_t\nabla\mathbf{B}_\eta\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\lesssim \int_I\int_{-L}^0\|\nabla\partial_t\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^2_\mathbf{y}}\|\partial_t\nabla^2_\mathbf{y}\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^\bfvarphi\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t\\ &\lesssim \int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla^2_\mathbf{y}\eta\|_{L^2_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t\\ &\leq \delta\int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\eta\|_{W^{2,2}_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} \int_I\int_\Omega\bfA_\eta\nabla\partial_t\overline{\mathbf{v}}:\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta\nabla\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t&\lesssim \int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\,\mathrm{d} t \\ &\leq\delta\int_I\|\nabla\partial_t\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\,\mathrm{d} t. \end{align}\]

Estimates for the pressure term. Finally, we are concerned with the pressure term in 50 . We use the weak formulation of the system 19 –21 to represent the pressure integrals involved in 50 as follows: \[\begin{align} &\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\partial_t \mathbf{B}_\eta^\intercal(\partial_t\overline{\mathbf{v}}+\bfG))\, \mathrm{d} \mathbf{x}\,\mathrm{d} t=\int_I\int_\Omega\pi\mathop{\mathrm{div}}(\mathbf{B}_\eta^\intercal\boldsymbol{\Theta}^\eta)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &=\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\boldsymbol{\Theta}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t+\int_I\int_\Omega\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\boldsymbol{\Theta}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t-\int_I\int_\Omega \big(\bfh_\eta(\overline{\mathbf{v}})-J_\eta\partial_t\overline{\mathbf{v}}\big)\cdot\boldsymbol{\Theta}^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber \\ &\quad +\int_I\int_\omega(\partial_t^2\eta+\Delta_{\mathbf{y}}^2\eta) \mathbf{n}^\intercal\boldsymbol{\Theta}^\eta\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t-\int_I\int_\omega g\, \mathbf{n}^\intercal\boldsymbol{\Theta}^\eta\circ\bfvarphi\, \mathrm{d}\mathbf{y}\,\mathrm{d} t,\label{later} \end{align}\tag{56}\] where \[\begin{align} &\boldsymbol{\Theta}^\eta:=\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal\partial_t\overline{\mathbf{v}}+\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal\bfG=:\boldsymbol{\Theta}^\eta_1+\boldsymbol{\Theta}^\eta_2. \end{align}\] We note that \(\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\boldsymbol{\Theta}^\eta_2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\) can be done similarly as 55 , while (after an integration by parts) we can procced as for the terms \(J_1',\dots, J_5'\) considered above to bound \(\int_I\int_\Omega J_{\eta}\partial_t\overline{\mathbf{v}}\cdot\boldsymbol{\Theta}^\eta_1\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\). In order to proceed we argue yet another time as in 53 obtaining \[\begin{align} \int_I\int_\Omega\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\boldsymbol{\Theta}_1^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t&\lesssim \int_I\int_{-L}^0\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}^2\eta\|_{L^2_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t \\ &\quad + \int_I\int_{-L}^0\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}^2\eta\|_{L^{2}_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{\infty}_\mathbf{y}}\mathrm dz\,\mathrm{d} t \\ &\quad +\int_I\int_{-L}^0\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim \delta\int_I\|\partial_t\overline{\mathbf{v}}\|^2_{W^{1,2}_\mathbf{y}}\,\mathrm{d} t+c(\delta)\int_I\|\nabla\overline{\mathbf{v}}\|^2_{L^2_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2, \end{align}\] on account of 8 , 10 and ?? . Similarly, we have \[\begin{align} \int_I\int_\Omega\bfA_\eta\nabla\overline{\mathbf{v}}:\nabla\boldsymbol{\Theta}_2^\eta\, \mathrm{d} \mathbf{x}\,\mathrm{d} t&\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\big(1+\|\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}^2\,\mathrm{d} t \\ &\quad +\int_I\int_{-L}^0\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{2}_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{\infty}_\mathbf{y}}\big(1+\|\nabla_{\mathbf{y}}^2\eta\|_{L^2_\mathbf{y}}\big)\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}^2\,\mathrm dz\,\mathrm{d} t \\ &\quad +\int_I\int_{-L}^0\|\nabla\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{2}_\mathbf{y}}\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^{\infty}_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^{\infty}_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}^2\eta\|_{L^{2}_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t \\ &\lesssim \int_I\|\nabla\overline{\mathbf{v}}\|_{L^2_\mathbf{x}}^2\big(\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2+1\big)\,\mathrm{d} t. \end{align}\] Finally, \[\begin{align} &\int_I\int_\Omega\big(\bfh_\eta(\overline{\mathbf{v}})-J_\eta\partial_t\overline{\mathbf{v}}\big)\cdot\boldsymbol{\Theta}^\eta_1\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\&= -\int_I\int_\Omega J_\eta\partial_t \overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal \partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t-\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta \cdot\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal \partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad -\int_I\int_\Omega\mathbf{B}_\eta\nabla \overline{\bfv}~\overline{\mathbf{v}}\cdot\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal \partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+ \int_I\int_\Omega J_\eta \bff \circ \bfPsi_\eta\cdot\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal \partial_t\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=:({\tt J})^1_1+({\tt J})^2_1+({\tt J})^3_1+({\tt J})^4_1, \end{align}\] where the first term is the most critical one. However, it can be estimates as \(J_1',\dots, J_5'\) considered above. Note that \({\tt(J)}_1^4\) is trivial such that we only estimate the remainig two integrals. It holds (using once more@eq:210and212 , 10 and ?? and employ the argument from 53 ) \[\begin{align} {\tt (J)}_1^2&\lesssim\int_I\int_{-L}^0 \|\nabla \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L_\mathbf{y}^2} \|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\| \partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I \|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}} \|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\| \partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\leq\delta \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} {\tt (J)}_1^3&\lesssim\int_I\int_{-L}^0 \|\nabla \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}} \|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\| \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\| \partial_t\overline{\mathbf{v}}\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I \|\nabla \overline{\mathbf{v}}\|_{L^2_\mathbf{x}} \|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\| \overline{\mathbf{v}}\|_{L^2_\mathbf{x}}\| \partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\leq\delta \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\] Similarly, \[\begin{align} &\int_I\int_\Omega\big(\bfh_\eta(\overline{\mathbf{v}})-J_\eta\partial_t\overline{\mathbf{v}}\big)\cdot\boldsymbol{\Theta}^\eta_2\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\\ &= -\int_I\int_\Omega J_\eta\partial_t \overline{\mathbf{v}}\cdot(\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal)^2 \overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t-\int_I\int_\Omega J_\eta \nabla \overline{\mathbf{v}}\cdot \partial_t \bfPsi_\eta^{-1}\circ \bfPsi_\eta \cdot(\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal)^2 \overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &\quad -\int_I\int_\Omega\mathbf{B}_\eta\nabla \overline{\bfv}~\overline{\mathbf{v}}\cdot(\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal)^2 \overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t+ \int_I\int_\Omega J_\eta \bff \circ \bfPsi_\eta\cdot(\mathbf{B}_\eta^{-\intercal}\partial_t\mathbf{B}_\eta^\intercal)^2\overline{\mathbf{v}}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t\\ &=:({\tt J})^1_2+({\tt J})^2_2+({\tt J})^3_2+({\tt J})^4_2, \end{align}\] where \[\begin{align} ({\tt J})^1_2&\lesssim\int_I\int_{-L}^0\|\partial_t\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\,\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \delta \int_I\|\partial_t\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^{2}\,\mathrm{d} t+c(\delta)\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt J})^2_2&\lesssim\int_I\int_{-L}^0\|\nabla \overline{\bfv}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}^2\| \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_x}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{2}\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt J})^3_2&\lesssim\int_I\int_{-L}^0\|\nabla \overline{\bfv}\circ\Lambda^{\bfvarphi}\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}}^2\| \overline{\mathbf{v}}\circ\Lambda^{\bfvarphi}\|_{L^\infty_\mathbf{y}}^2\mathrm dz\,\mathrm{d} t\\ &\lesssim\int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}\,\mathrm{d} t\\ &\lesssim \int_I\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t, \end{align}\] \[\begin{align} ({\tt J})^4_2 &\lesssim\int_I\|\overline{\bff}\|_{L^{2}_\mathbf{x}}\|\partial_t\nabla_\mathbf{y}\eta\|_{L^\infty_\mathbf{y}}^2\|\overline{\mathbf{v}}\|_{L^{2}_\mathbf{x}}\,\mathrm{d} t\lesssim \int_I\big(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+\|\overline{\bff}\|_{L^{2}_\mathbf{x}}^2\big)\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t. \end{align}\]

Finally we deal with the integrals in 56 coming from the boundary of \(\bfG\). on account of \(\nabla\bfPsi_\zeta\bfn=\bfn\) and 11 we have \[\begin{align} \boldsymbol{\Theta}^\eta\circ\bfvarphi&=\mathbf{B}_\eta^{-\intercal}\circ\bfvarphi\,\partial_t\mathbf{B}_\eta^\intercal\circ\bfvarphi \,\partial_t^2\eta\mathbf{n}+\mathbf{B}_\eta^{-\intercal}\bfvarphi\,\partial_t\mathbf{B}_\eta^\intercal\circ\bfvarphi \bfG\circ\bfvarphi\mathbf{n}\nonumber\\ &=J_\eta^{-1}\partial_t J_\eta \partial_t^2\eta \mathbf{n}+ J_\eta^{-1} J_\eta^{-1} \partial_t J_\eta \partial_t J_\eta \partial_t\eta\mathbf{n}\\ &\approx \partial_t\eta\partial_t^2\eta(1+\eta)\mathbf{n}+(1+\eta)^2|\partial_t\eta|^2\partial_t\eta\mathbf{n}. \end{align}\] Then we realize that \(\int_I\int_\omega|\partial_t^2\eta|^2\partial_t\eta(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\) can be estimated similarly as in 51 . We find that \[\begin{align} \int_I\int_\omega\partial_t^2\eta(1+\eta)^2|\partial_t\eta|^3\, \mathrm{d}\mathbf{y}\,\mathrm{d} t&\lesssim \int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^6_\mathbf{y}}^3\,\mathrm{d} t \\ &\lesssim \int_I\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^2\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t \\ &\lesssim \int_I\left(\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\right)\,\mathrm{d} t, \end{align}\] where we used the \(1D\) interpolation inequality: \[\|\partial_t\eta\|_{L^6_\mathbf{y}}\lesssim \|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{2}{3}}\|\partial_t\eta\|_{W^{1,2}_\mathbf{y}}^{\frac{1}{3}}.\] We obtain further by using ?? that \[\begin{align} \int_I\int_\omega\Delta_{\mathbf{y}}^2\eta (1+\eta)^2|\partial_t\eta|^3\, \mathrm{d}\mathbf{y}\,\mathrm{d} t& =-\int_I\int_{\omega}\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta \cdot\nabla_{\mathbf{y}}\big((1+\eta)^2|\partial_t\eta|^3\big)\, \mathrm{d} \mathbf{x}\,\mathrm{d} t\nonumber\\ &\lesssim \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\big(\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}^2 +\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^\infty_\mathbf{y}}\|\partial_t\eta\|_{L^2_y}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}\big)\,\mathrm{d} t\nonumber \\ &\lesssim \int_I\|\eta\|_{W^{3,2}_\mathbf{y}}\big(\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}} +\|\partial_t\eta\|_{L^2_\mathbf{y}}^{1/4}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{3/4}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{3/4}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{1/4}\big)\,\mathrm{d} t\nonumber \\ &\lesssim\,\delta\int_I\|\eta\|_{W^{3,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2 \,\mathrm{d} t.\label{needagain} \end{align}\tag{57}\] As for the remaining integral containing \(\Delta_{\mathbf{y}}^2\eta\) we procced by a formal computation which can be made rigorous by projecting \(\eta\) on a finite sum of Fourier modes. We obtain \[\begin{align} &\int_I\int_\omega\Delta_{\mathbf{y}}^2\eta\,\partial_t^2\eta\partial_t\eta(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t =\tfrac{1}{2}\int_I\int_\omega\Delta_{\mathbf{y}}^2\eta\,\partial_t|\partial_t\eta|^2(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\ &=-\tfrac{1}{2}\int_I\int_\omega\Delta_{\mathbf{y}}^2\partial_t\eta\,|\partial_t\eta|^2(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t-\tfrac{1}{2}\int_I\int_\omega\Delta_{\mathbf{y}}^2\eta\,|\partial_t\eta|^2\partial_t(1+\eta)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t\\&\quad +\frac{1}{2}\int_\omega\Delta_{\mathbf{y}}^2\eta\,|\partial_t\eta|^2(1+\eta)\, \mathrm{d}\mathbf{y}\bigg|_0^T\\ &=:(\tt{I})_1+(\tt{I})_2+(\tt{I})_3, \end{align}\] where (using ?? ) \[\begin{align} ({\tt I})_1&=-\frac{1}{2} \int_I\int_\omega\Delta_{\mathbf{y}}\partial_t\eta\,\Delta_{\mathbf{y}}\big(|\partial_t\eta|^2(1+\eta)\big)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t \\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}}^2 \,\mathrm{d} t+\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^3_\mathbf{y}}\|\partial_t\eta\|_{L^6_\mathbf{y}}\,\mathrm{d} t\\&\quad +\int_I\|\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}^2\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\,\mathrm{d} t\\ &\lesssim \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{7/4,4}_\mathbf{y}}\,\mathrm{d} t+\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{4}{3}}\|\partial_t\Delta_{\mathbf{y}}\eta\|^{\frac{2}{3}}_{W^{1/4,2}_\mathbf{y}}\|\partial_t\eta\|_{W^{1/2,2}_\mathbf{y}}\,\mathrm{d} t\\&\quad +\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\,\mathrm{d} t \\ &\lesssim \delta \int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{W^{1/4,2}_\mathbf{y}}^2\,\mathrm{d} t+c(\delta)\int_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2(\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2+1)\,\mathrm{d} t, \end{align}\] Notice that we have \[\begin{align} ({\tt I})_2&=\int_I\int_\omega\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta\cdot\nabla_{\mathbf{y}}\big(|\partial_t\eta|^2\partial_t(1+\eta)\big)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t=3\int_I\int_\omega\nabla_{\mathbf{y}}\Delta_{\mathbf{y}}\eta|\partial_t\eta|^2\partial_t\nabla_{\mathbf{y}}\eta\, \mathrm{d}\mathbf{y}\,\mathrm{d} t, \end{align}\] which has been estimated in the second part of 57 . For \(({\tt I})_3\), we have \[\begin{align} ({\tt I})_3&=\int_\omega\Delta_{\mathbf{y}}\eta\,\Delta_{\mathbf{y}}\big(|\partial_t\eta|^2(1+\eta)\big)\, \mathrm{d}\mathbf{y}\bigg|_0^T \\ &\lesssim \sup_I\|\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}+ \sup_I\|\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2\|\partial_t\eta\|^2_{L^\infty_\mathbf{y}}+ \sup_I\|\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^4_\mathbf{y}}^2\\ &\lesssim \sup_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{4}}+\sup_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{2}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{1}{2}}+\sup_I\|\partial_t\eta\|_{L^2_\mathbf{y}}^{\frac{3}{4}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{\frac{5}{4}}\\ &\lesssim\delta \sup_I\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^2+c(\delta). \end{align}\] Moreover, we have \[\begin{align} &\int_I\int_\omega g \left(\partial_t\eta\partial_t^2\eta(1+\eta)+(1+\eta)^2|\partial_t\eta|^3\right)\, \mathrm{d}\mathbf{y}\,\mathrm{d} t \\&\leq\int_I\|g\|_{L^2_\mathbf{y}}\big(\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}+\|\partial_t\eta\|_{L^2_\mathbf{y}}\|\partial_t\eta\|_{L^\infty_\mathbf{y}}^2\big)\,\mathrm{d} t\\ &\lesssim \int_I\|g\|_{L^2_\mathbf{y}}\left(\|\partial_t^2\eta\|_{L^2_\mathbf{y}}\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}+\|\partial_t\eta\|_{L^2_\mathbf{y}}^2\|\partial_t\nabla_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}\right)\,\mathrm{d} t\\ &\lesssim \int_I\left(\|g\|_{L^2_\mathbf{y}}^2+1\right)\left(1+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{L^2_\mathbf{y}}^{2}+\|\partial_t^2\eta\|_{L^2_\mathbf{y}}^2\right)\,\mathrm{d} t, \end{align}\] using that \(g\in W^{1,2}_tL^2_\mathbf{y}\).

Now setting \[\begin{align} \mathcal{E}(t)&=\|\partial_t^2\eta(t)\|_{L^2_\mathbf{y}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta(t)\|_{L^2_\mathbf{y}}^2+\|\partial_t\mathbf{v}(t)\|_{L^2_\mathbf{x}}^2+\|\nabla\mathbf{v}(t)\|_{L^2_\mathbf{x}}^2 ,\\ H(t)&=\|\bff(t)\|^2_{L^2_\mathbf{x}}+\|\partial_t\bff(t)\|^2_{L^2_\mathbf{x}}+\|g(t)\|^2_{L^2_\mathbf{y}}+\|\partial_tg(t)\|^2_{L^2_\mathbf{y}}, \end{align}\] collecting the previous estimates, choosing \(\delta\) small enough and using 44 , we have shown that \[\begin{align} &\mathcal{E}(t)+\int_0^t\big(\|\eta\|_{W^{3,2}_\mathbf{y}}^2+\|\partial_t\Delta_{\mathbf{y}}\eta\|_{W^{1/4,2}_\mathbf{y}}^2+\|\partial_t\nabla\overline{\mathbf{v}}\|^2_{L^2_\mathbf{x}}\big)\,\mathrm{d} t\\&\lesssim \mathcal{E}(0)+1+\int_0^tH(s)\,\mathrm{d}s+\int_0^t(1+\|\overline{\mathbf{v}}\|_{W^{1,2}_\mathbf{x}}^2)\mathcal{E}(s)\,\mathrm{d}s. \end{align}\] We now apply Grönwall’s lemma to complete the proof. ◻