1 Introduction↩︎

This paper is concerned with sign-changing solutions of the equation\[\label{eq:eq0} u \partial_x u - \partial_{yy} u = f\tag{1}\] in the rectangular domain \(\Omega := (x_0, x_1) \times (-1,1)\), where \(f\) is an external source term. A natural solution to 1 with a null source term \(\mathbf{f}= 0\) is the linear shear flow \(\mathbf{u}(x,y) := y\), which changes sign across the horizontal line \(\{ y = 0 \}\). We are interested in strong solutions to 1 which are close (with respect to an appropriate norm) to this linear shear flow \(\mathbf{u}\). Our purpose is to construct such solutions by perturbing the lateral boundary data at \(x=x_0\) or \(x=x_1\) or the source term \(\mathbf{f} = 0\).

Since such solutions will change sign across a line \(\{ u = 0 \}\) lying within \(\Omega\), a key feature of this work is that 1 must be seen as a quasilinear forward-backward parabolic problem in the horizontal direction. Thus, to ensure the existence of a solution, one must be particularly careful as to how one enforces the lateral perturbations. More precisely, the problem is forward parabolic in the domain above the line \(\{u=0\}\), in which \(u>0\), and therefore we shall prescribe a boundary condition on \(\Sigma_0 := \{x=x_0\}\cap\{u>0\}\); and backward parabolic in the domain below the line \(\{u=0\}\), and we shall prescribe a boundary condition on \(\Sigma_1 := \{x=x_1\}\cap \{u<0\}\).

Our purpose is to construct strong solutions of this system, in a high regularity functional space. However, one key difficulty of our work lies in the fact that, even when the source term \(f\) is smooth, say in \(C^\infty_0(\Omega)\), solutions to 1 have singularities in general. Actually, this feature is already present at the linear level, i.e.for the equation \(y\partial_x u -\partial_{xx}u=f\). For this linear equation, we prove that if \(f\) is smooth, the associated weak solution to the linear system inherits the regularity of \(f\) if and only if \(f\) satisfies orthogonality conditions (i.e.the scalar product of \(f\) with some identified profiles must vanish). We also describe the singularities that appear when these orthogonality conditions are not satisfied. In the nonlinear setting, the equivalent of these orthogonality conditions is the following (informal) statement: when \(f\) is smooth and satisfies a smallness assumption in a suitable functional space, strong solutions exist if and only if \(f\) belongs to a finite codimensional space.

Due to the forward-backward nature of the problem, we must choose the lateral perturbations and the source term in a particular product space. We therefore introduce the vector space \[\label{eq:E} \begin{align} \Big\{ (f,\delta_0,\delta_1) & \in C^\infty_c(\Omega)\times C^\infty([0,1]) \times C^\infty([-1,0]) ; \quad \delta_i(0) = \partial_y \delta_i(0) = \partial_y^2 \delta_i(0) = 0 \\ & \text{and } \delta_i((-1)^i) = \partial_y^2 \delta_i((-1)^i) = 0 \text{ for } i = 0, 1 \Big\} \end{align}\tag{2}\] and \(\mathcal{H}\), the Hilbert space defined as its completion with respect to the following norm, associated with the corresponding canonical scalar product \[\label{eq:H-norm} \| (f,\delta_0,\delta_1) \|_{\mathcal{H}} := \| f \|_{H^1_xH^1_y} + \| \partial_y^3 f \|_{L^2} + \sum_{i \in \{0,1\}} \|\delta_i\|_{H^5} + \|(\partial_y^2 \delta_i) / y \|_{H^1(|y| \:\mathrm{d}y)}.\tag{3}\] For all \(\eta>0\), we denote by \(B_\eta\) the open ball of radius \(\eta\) in \(\mathcal{H}\).

We establish the existence and uniqueness of solutions in the following anisotropic Sobolev space \[\label{def:Q1} {Q^1}:= L^2((x_0,x_1); H^5(-1,1)) \cap H^{5/3}((x_0,x_1);L^2(-1,1)).\tag{4}\] We first state a result concerning the well-posedness in \({Q^1}\) of the linear version of 1 around the linear shear flow, up to two orthogonality conditions:

We then turn towards the nonlinear problem 1 . The main result of this paper is the following nonlinear generalization of Theorem 1 for small enough perturbations.

Our original motivation stems from fluid mechanics. Indeed, the stationary Prandtl equation, which describes the behavior of a fluid with small viscosity in the vicinity of a wall, reads \[\label{Prandtl} \begin{cases} u \partial_x u + v \partial_y u - \partial_{yy} u = - \partial_x p_E, \\ u_{|y=0} = v_{|y=0}=0,\\ \lim_{y \to \infty } u(x,y)=u_E(x), \end{cases}\tag{5}\] where \(u_E(x)\) (resp.\(p_E(x)\)) is the trace of an outer Euler flow (resp.pressure) on the wall, and satisfies \(u_E \partial_x u_E = - \partial_x p_E\).

As long as \(u\) remains positive, 5 can be seen as a nonlocal, nonlinear diffusion type equation, the variable \(x\) being the evolution variable. Using this point of view, Oleinik (see e.g.[5]) proved the local well-posedness of a solution to 5 when the equation 5 is supplemented with a boundary data \(u_{|x=0}=u_0\), where \(u_0(y)>0\) for \(y>0\) and such that \(u_0'(0)>0\). Let us mention that such positive solutions exist globally when \(\partial_x p_E\leq 0\), but are only local when \(\partial_x p_E>0\). More precisely, when \(\partial_x p_E=1\) for instance, for a large class of boundary data \(u_0\), there exists \(x^*>0\) such that \(\lim_{x\to x^*} \partial_y u(x,0)=0\). Furthermore, the solution may develop a singularity at \(x=x^*\), known as Goldstein singularity [6]–[8]. Downstream of the singularity, the solution \(u\) is expected to change sign, taking negative values in the vicinity of the boundary and positive values in the vicinity of \(+\infty\), the usual convention being that \(u_E(x)>0\). Such solutions are called “recirculating solutions”, and the zone where \(u<0\) is called a recirculation bubble.

In the recent preprint [9], Sameer Iyer and Nader Masmoudi prove a priori estimates in high regularity norms for smooth solutions to the Prandtl equation 5 , in the vicinity of explicit self-similar recirculating flows, called Falkner-Skan profiles. The latter are given by \[\begin{align} u(x,y) & = x^m f'(\zeta), \\ v(x,y) & = - y^{-1} \zeta f(\zeta) - \frac{m-1}{m+1} y^{-1} \zeta^2 f'(\zeta), \end{align}\] where \(\zeta := (\frac{m+1}{2})^{\frac{1}{2}} y x^{\frac{m-1}{2}}\) is the self-similarity variable, \(m\) is a real parameter and \(f\) is the solution to the Falkner-Skan equation \[f''' + f f'' + \beta (1- (f')^2) = 0,\] where \(\beta = \frac{2m}{m+1}\), subject to the boundary conditions \(f(0) = f'(0) = 0\) and \(f'(+\infty) = 1\). Such flows correspond to an outer Euler velocity field \(u_E(x) = x^m\). For some particular values of \(m\) (or, equivalently, \(\beta\)), these formulas provide physical solutions to 5 which exhibit recirculation (see [10]). Obtaining a priori estimates for recirculating solutions to the Prandtl system 5 is very difficult, due to the combination of several difficulties (forward-backward system, nonlocality of the transport term \(v\partial_y u\), loss of derivative).

In the present paper, we have chosen to focus on a different type of difficulty, and to consider the toy-model 1 , which differs from 5 through the lack of the nonlinear transport term \(v \partial_y u\) and its associated difficulties and the exclusion of the zones close to the wall and far from the wall. For the model 1 , a priori estimates are easy to derive, see [4]. The difficulty lies elsewhere, as explained previously. Indeed, in order to construct a sequence of approximate solutions satisfying the a priori estimates, we need to ensure that the orthogonality conditions are satisfied all along the sequence. The core of the proof is to keep track of these orthogonality conditions, and to analyze their dependency on the sequence itself. This strategy is presented in greater detail in an early version of our work [1], to which we will refer throughout the text. For the Prandtl system 5 , this difficulty has recently been tackled by Sameer Iyer and Nader Masmoudi in [11], building upon their a priori estimates of [9] and the ideas developed in [1]. The methodology developed in the present paper can be adapted to many other settings, as explained in the latest version of our work, see [12]. A natural extension of 1 is the 1d nonlinear Fokker-Planck equation in a bounded domain: consider equations of the type \[\label{eq:FP} \begin{cases} v\partial_x g + b[g]\partial_v g - a[g]\partial_{vv} g=0, & (x,v)\in \Omega,\\ g_{|\Sigma_i}=\delta_i,\\ g_{|v=\pm 1}=0. \end{cases}\tag{6}\] Note that in the above system, the vertical variable \(z\) has been relabeled as \(v\), as is customary in kinetic models. The coefficients \(a\) and \(b\) are nonlocal functionals of the unknown \(g\). A typical example (see [13]) is \[v\partial_x g = \rho^\alpha \partial_v (T\partial_v g + g(v-u))\] where \(\alpha\in [0,1]\) and \(\rho, u, T\) are respectively the local density, velocity and temperature, namely \[\begin{align} \rho (x)&=\int_\Omega g(x,v)\:\mathrm{d}v,\\ \rho(x) u(x)&=\int_\Omega v g(x,v)\:\mathrm{d}v,\\ \rho(x)|u(x)|^2 + \rho(x) T(x)&=\int_\Omega|v|^2 g(x,v)\:\mathrm{d}v. \end{align}\] We believe that under suitable assumptions on the operators \(a\) and \(b\), the methods presented here could allow us to prove the existence of strong solutions of 6 in the vicinity of smooth changing-sign solution.

2 The case of the linear shear flow↩︎

We begin with the study of the linear system ?? , for which we give a notion of weak solution:

Weak solutions in the above sense are known to exist since the work of Fichera [14] (which concerns generalized versions of ?? , albeit with vanishing boundary data). Uniqueness dates back to [15] by Baouendi and Grisvard. A natural question is then to consider strong solutions, i.e.solutions for which ?? holds almost everywhere. The main result on this topic is due to Pagani [16] (see also [17]), who introduced the space \[Z^0:=\{ u\in L^2(\Omega), y\partial_x u\in L^2(\Omega),\;\partial_y u \in L^2 (\Omega)\text{ and } \partial_{yy} u \in L^2(\Omega)\}.\] It can easily be proved (see [1]) that \(Z^0\) is continuously embedded in \[\label{def:qnot} {Q^{0}}:=H^{2/3}_x L^2_y\cap L^2_x H^2_y.\tag{8}\]

Let us now observe that equation ?? is stable by differentiation with respect to \(x\), at least formally. More precisely, if \(u\) is a smooth (say, \(H^1_x H^1_y\)) solution of ?? , with smooth data \((f,\delta_0,\delta_1)\), then \(\partial_x u\) is also a solution of ?? with the data \((\partial_x f, \Delta_0, \Delta_1)\), with \[\label{eq:def-Di} \Delta_i(y):= \frac{f(x_i,y) + \partial_y^2 \delta_i(y)}{y}.\tag{9}\] As a consequence, if \(u\in H^1_x H^1_y\), then \(u\) satisfies the a priori estimate \[\label{eq:apriori-ux} \| \partial_x u \|_{L^2_x H^1_y} \lesssim \| \partial_x f \|_{L^2_x(H^{-1}_y)} + \| \Delta_0 \|_{L^2(\Sigma_0, |y| \:\mathrm{d}y)} + \| \Delta_1 \|_{L^2(\Sigma_1, |y| \:\mathrm{d}y)}.\tag{10}\] However, generally, the unique weak solution \(u\in Z^0\) corresponding to smooth data \((f,\delta_0,\delta_1)\) does not belong to \(H^1_x H^1_y\). Indeed, assume that \((f,\delta_0,\delta_1)\in C^\infty_c(\Omega)\times C^\infty_c((0,1))\times C^\infty_c((-1,0))\)⁴ , and consider the unique solution \(w\in Z^0\) of \[\label{eq:ux} \begin{cases} y \partial_x w - \partial_{yy} w = \partial_x f,\\ w_{\rvert \Sigma_i} = \Delta_i, \\ w_{\rvert y = \pm 1} = 0. \end{cases}\tag{11}\] Attempting to reconstruct \(u\) from \(w\), we define \[\tilde{u}:= \begin{cases} \delta_0 + \int_{x_0}^x w\quad \text{if } y>0,\\ \delta_1 - \int_x^{x_1} w\quad \text{if } y<0. \end{cases}\] Then \(\tilde{u}_{|\Sigma_i}=\delta_i\) by construction, and \(\tilde{u}\) is a solution of ?? in \(\Omega\setminus\{ y=0\}\). However, \(\tilde{u}\) and \(\partial_y \tilde{u}\) may have a jump across \(y=0\). As a consequence, \(\tilde{u}\) is not a solution of ?? in general. More precisely, \[\begin{align} & \tilde{u} \text{ is a solution of }\eqref{eq:shear} \\ \iff & [\tilde{u}]_{|y=0}= [\partial_y \tilde{u}]_{|y=0}=0\\ \iff & \delta_0(0)-\delta_1(0)+ \int_{x_0}^{x_1} w(x,0)\:dx= \delta_0'(0)-\delta_1'(0)+\int_{x_0}^{x_1}\partial_y w(x,0)\:dx=0. \end{align}\] Furthermore, if \([\tilde{u}]_{|y=0}= [\partial_y \tilde{u}]_{|y=0}=0\), then \(\tilde{u} \in Z^0\), and in this case \(\tilde{u}=u\).

Let us now say a few words about the more general case \((f,\delta_0, \delta_1)\in \mathcal{H}\). By definition of \(\mathcal{H}\), in this case \(\partial_x f \in L^2_x H^1_y\subset L^2\), and \(\Delta_i= \partial_y^2 \delta_i /y \in H^1(|y| \:\mathrm{d}y)\). Therefore, according to Proposition 1, there exists a unique solution \(w\in Z^0\) of 11 . Following the same argument as above, we define the linear forms \[\overline{\ell^j}:(f,\delta_0, \delta_1)\in \mathcal{H}\mapsto \int_{x_0}^{x_1}\partial_y^j w(x,0)\:dx + \partial_y^j (\delta_0-\delta_1)(0).\] Note that since \(w\in Z^0\subset {Q^{0}}\) (see 8 ), \(\partial_y w_{|y=0}\in H^{1/6}(x_0, x_1)\), and therefore the integral is well-defined.

We eventually obtain the following result:

For further purposes, it is useful to write the linear forms \(\overline{\ell^j}\) as scalar products. To that end, let us introduce the following definition:

The existence, uniqueness, and \(Z^0\) regularity of \(\overline{\Phi^j}\), \(j=0,1\), follow easily from the results of Fichera [14], Baouendi and Grisvard [15] and Pagani [16], [17] recalled above, and from a simple lifting argument. Furthermore, for any \(f \in H^1((x_0,x_1);L^2(-1,1))\), \(\delta_0, \delta_1 \in H^1((-1,1), |y| \:\mathrm{d}y)\) with \(\delta_0(1) = \delta_1(-1) = 0\) and \(\Delta_0, \Delta_1 \in H^1(-1,1, |y| \:\mathrm{d}y)\) with \(\Delta_0(1) = \Delta_1(-1)= 0\) (see 9 ), it can easily be proved that \[\int_{x_0}^{x_1}\partial_y^j w(x,0)\:dx = \int_\Omega\partial_x f \overline{\Phi^j} + \int_{\Sigma_0} y\Delta_0 \overline{\Phi^j} - \int_{\Sigma_1} y \Delta_1 \overline{\Phi^j} ,\] and therefore \[\overline{\ell^j}(f,\delta_0,\delta_1) = \partial_y^j \delta_0(0)-\partial_y^j\delta_1(0) + \int_\Omega\partial_x f \overline{\Phi^j} + \int_{\Sigma_0} y \Delta_0\overline{\Phi^j} - \int_{\Sigma_1} y \Delta_1 \overline{\Phi^j}.\] Additionally, when the data \((f,\delta_0, \delta_1)\) are smooth and satisfy the orthogonality conditions, we also deduce from the equation a gain of regularity in \(y\):

In order to conclude this section, let us now investigate the case when the orthogonality conditions are not satisfied. We first introduce some notation. For \(x\in (0, + \infty)\), \(y\in \mathbb{R}\), we introduce some “polar-like” variables \[\label{eq:r-t} r := (y^2 + x^{\frac{2}{3}})^{\frac{1}{2}} \quad \text{and} \quad t := y x^{-\frac{1}{3}}\tag{12}\] which are consistent with the scaling invariance of the operator \(y\partial_x - \partial_{yy}\).

We first prove the following Lemma:

The function \(v_0\) is linked with a solution to ?? which has \(Z^0\) regularity, but does not belong to \(H^1_x H^1_y\). Similarly, for each \(k \geq 0\), \(v_k\) is linked with a solution \(u\) such that \(\partial^k_x u \in Z^0(\Omega)\) but \(u \notin H^{k+1}_x H^1_y\). More precisely, we now introduce singular profiles \(\bar{u}_{\mathrm{sing}}^i\), for \(i=0,1\), localized in the vicinity of \((x_i,0)\). Let \(\chi_i \in C^\infty(\overline{\Omega})\) be a cut-off function such that \(\chi_i \equiv 1\) in a neighborhood of \((x_i, 0)\), and \(\operatorname{supp}\chi \subset B((x_i, 0), \bar{R})\) for some \(\bar{R} < \min (1, x_1-x_0)/2\).

Sketch of proof By definition of \(\bar{u}_{\mathrm{sing}}^i\), the function \(f_i\) is supported in \(\operatorname{supp}\nabla \chi_i\). Furthermore there exists \(r_\pm>0\) such that \(\operatorname{supp}\nabla \chi_i \subset \{ r_-\leq r \leq r_+\}\). Hence the definition of \(\overline{f_i}\) follows from straightforward computations, and its regularity is a consequence of the smoothness of \(v_0\) away from the origin.

There remains to check that \(\bar{u}_{\mathrm{sing}}^0\), \(\partial_{yy} \bar{u}_{\mathrm{sing}}^0\) and \(z \partial_x \bar{u}_{\mathrm{sing}}^0\) are in \(L^2(\Omega)\) but \(\partial_x \partial_y \bar{u}_{\mathrm{sing}}^0 \notin L^2(\Omega)\). We use once again the formulas 13 14 , together with the observation that the jacobian of the change of variables \((x,z)\to (r,t)\) is \({(1+t^2)^2}/(3r^3)\). We infer that \[y\partial_x \bar{u}_{\mathrm{sing}}^i= O( r_i^{-3/2} (1+|t_i|)),\] and therefore \(y\partial_x \bar{u}_{\mathrm{sing}}^i \in L^2\). In a similar fashion \(\partial_{yy} \bar{u}_{\mathrm{sing}}^i \in L^2\). However \[\partial_x \bar{u}_{\mathrm{sing}}^i \in r_i^{-5/2} H_i(t) + L^2\] for some non-zero function \(H_i\), and therefore \(\bar{u}_{\mathrm{sing}}^i \notin H^1_x L^2_y\). 0◻

3 Extension to a broader class of degenerate elliptic equations↩︎

Let us now extend briefly the results of the previous section to equations of the type \[\label{eq:coeff-variables} \begin{cases} y \partial_x u + \gamma \partial_y u- \alpha \partial_{yy} u = f & \text{in } \Omega, \\ u_{\rvert \Sigma_i} = \delta_i, \\ u_{\rvert y = \pm 1} = 0. \end{cases}\tag{18}\] For the applications we have in mind, the coefficient \(\gamma\) will be small (in a norm which will be made precise shortly), and the coefficient \(\alpha\) is close to \(1\), and therefore keeps a positive sign within \(\Omega\). We follow the outline of the previous section: we first state an existence and uniqueness result for weak solutions in \(Z^0\). Then, we exhibit orthogonality conditions for higher regularity. Eventually, stepping on the analysis of the previous paragraph, we provide a decomposition of the solution associated with smooth data into a sum of singular profiles and a smooth remainder.

3.1 \(Z^0\) regularity for weak solutions↩︎

We start with the following result:

3.2 Orthogonality conditions for higher order regularity↩︎

We now address the higher regularity theory. In order to simplify the presentation, we focus on the case when the coefficients \(\alpha\) and \(\gamma\) are smooth.

We assume that \(f\in H^1_x L^2_y\), \(\delta_i \in H^2(\Sigma_i)\), and we set \[h_i :=\partial_x f + \alpha_x \partial_{zz}{\delta}_i -\gamma_x \partial_y {\delta}_i,\label{def:h95i}\tag{19}\] and \[\label{def:Delta95i} {\Delta}_i(y):=\frac{1}{y}\left(f(x_i,y) + \alpha(x_i,y) \partial_{yy} {\delta}_i(y) - \gamma(x_i,y) \partial_y {\delta}_i(y)\right).\tag{20}\]

Let \(u\in Z^0\) be the solution of 18 . Assume that \(\partial_x u \in Z^0\). Then, differentiating 18 with respect to \(x\) and setting \(\Omega_\pm:=\Omega\cap \{y\gtrless 0\}\), we observe that \(v=\partial_x u\) is a solution of \[\label{eq:V} \begin{cases} y \partial_x v + \gamma \partial_y v - \alpha \partial_{yy} v -\alpha_x \partial_{yy} \int_{x_0}^x v+ \gamma_x \partial_y\int_{x_0}^x v = h_0 & \text{in } \Omega_+,\\ y \partial_x v + \gamma \partial_y v - \alpha \partial_{yy} v +\alpha_x \partial_{yy} \int^{x_1}_x v - \gamma_x \partial_y\int^{x_1}_x v = h_1 & \text{in } \Omega_- , \\ [v]_{y=0}=[\partial_y v]_{y=0}=0 & \text{on } (x_0,x_1),\\ v(x_0,y) = {\Delta}_0 & \text{for } y\in (0,1), \\ v(x_1,y) = {\Delta}_1 & \text{for } y \in (-1,0), \\ v(x,\pm 1) = 0 & \text{for } x \in (x_0,x_1), \end{cases}\tag{21}\] Reciprocally, assuming that the above system has a unique weak solution \(v\in Z^0\). Then \(u_1\) defined by \[u_1 := \begin{cases} {\delta}_0 + \int_{x_0}^x v & \text{in } \Omega_+, \\ {\delta}_1 + \int_{x_1}^x v & \text{in } \Omega_- \end{cases}\] is a solution to 26 if and only if \(v\) satisfies \[\label{cond-V} \begin{align} \int_{x_0}^{x_1} v(x,0)\:\mathrm{d}x & = {\delta}_1(0)-{\delta}_0(0),\\ \int_{x_0}^{x_1} \partial_y v(x,0)\:\mathrm{d}x & = \partial_y{\delta}_1(0)-\partial_y {\delta}_0(0). \end{align}\tag{22}\] As in the previous section, we find that \(\partial_x u\in Z^0\) if and only if the data \((f,\delta_0,\delta_1)\) satisfy two orthogonality conditions. More precisely, we prove the following result:

Furthermore, the linear forms \(\ell^j_{\alpha,\gamma}\) depend smoothly on the coefficients:

4 The nonlinear scheme↩︎

We now turn towards the proof of Theorem 2. Uniqueness follows easily from simple energy estimates and is left to the reader. Therefore we focus on the existence part of the Theorem.

We construct an iterative sequence \((u_n)_{n\in \mathbb{N}}\) in the following way.

Let \(\chi \in C^\infty(\mathbb{R},[0,1])\), identically equal to one on \([-\frac{1}{3}, \frac{1}{3}]\) and compactly supported in \([-\frac{1}{2}, \frac{1}{2}]\). We define the initialization profile of our iterative scheme as \[\label{eq:u0} u_0(x,y) := \delta_0(y) \chi \left(\frac{x - x_0}{x_1-x_0}\right) + \delta_1(y) \chi \left(\frac{x_1 - x}{x_1-x_0}\right).\tag{23}\] Then, for any \(n\geq 0\), we define \(u_{n+1}\) as the solution of \[\label{eq:NL-un-un1} \begin{cases} (y + u_n) \partial_x u_{n+1} - \partial_{yy} u_{n+1} = f+ \nu^0_{n+1} f^0 + \nu^1_{n+1} f^1=:f_{n+1}, \\ (u_{n+1})_{\rvert \Sigma_i} = \delta_i + \nu^0_{n+1} \delta^0_i + \nu^1_{n+1} \delta^1_i=:\delta_{i,n+1}, \\ (u_{n+1})_{\rvert y = \pm 1} = 0, \end{cases}\tag{24}\] In the above equation \((f^j, \delta^j_0, \delta^j_1)\) are fixed triplets (independent of \(n\)) such that \[\label{def:Xi-j} \overline{\ell^i}(f^j, \delta^j_0, \delta^j_1)=\delta_{i,j}.\tag{25}\] The coefficients \(\nu^0_{n+1}\) and \(\nu^1_{n+1}\) are designed to ensure that the orthogonality conditions are satisfied at every step.

The main steps of the analysis are the following:

• First, we perform a change of variables in order to transform 24 into an equation of the form 18 with coefficients \(\alpha_n,\gamma_n\) depending on \(u_n\). This will allow us to define rigorously the coefficients \(\nu^j_n\), and therefore the sequence \((u_n)_{n\in \mathbb{N}}\).

• Then, we derive uniform estimates in \({Q^1}\) on the sequence \((u_n)_{n\in \mathbb{N}}\). This step is fairly easy and is a straightforward consequence of the regularity analysis of solutions of equation 18 .

• Eventually, we prove that \((u_n)_{n\in \mathbb{N}}\) is a Cauchy sequence in a suitable functional space. This is the most involved step of the convergence proof. Indeed, the sequence \((u_n)_{n\in \mathbb{N}}\) is not sufficiently smooth in order to obtain a Cauchy bound in \({Q^1}\). Hence we must work in a space with lower regularity. However, the Lipschitz estimate on the linear forms \(\ell^j_{\alpha_n,\gamma_n}\) from Lemma 5 involves a regularity which is strictly bigger than \({Q^{0}}\). This prompts us to work in a fractional Sobolev space.

4.1 Change of variables and uniform \({Q^1}\) estimate↩︎

Let \(n\in \mathbb{N}\), and assume that \(u_n\in {Q^1}\) is such that \(\|u_n\|_{{Q^1}}\ll 1\). Define \(Y_n=Y_n(x,z)\) by the implicit formula \[Y_n(x,z) + u_n(x, Y_n(x,z))=z\quad \forall (x,z)\in \overline{\Omega}.\] We then look for \(u_{n+1}\) under the form \[u_{n+1}(x,y) = U_{n+1}(x, y+ u_n(x,y)),\] so that \(U_{n+1}=U_{n+1}(x,z)\) solves \[\label{eq:U} \begin{cases} z \partial_x U_{n+1} + \gamma_n \partial_z U_{n+1} - \alpha_n \partial_{zz} U_{n+1} = g_{n+1} & \text{in } \Omega, \\ U_{n+1\rvert \Sigma_i} = \widetilde{\delta}_{i,n+1}, \\ U_{n+1\rvert y = \pm 1} = 0, \end{cases}\tag{26}\] where \[\begin{align} \tag{27} \alpha_n(x,z) & := (1 + \partial_y u_n)^2(x,Y_n(x,z)), \\ \tag{28} \gamma_n(x,z) & := (z \partial_x u_n- \partial_{yy} u_n)(x,Y_n(x,z)) \\ \tag{29} g_{n+1}(x,z) & := f_{n+1}(x, Y_n(x,z)), \\ \tag{30} \widetilde{\delta}_{i,n+1}(z) & := \delta_{i,n+1} (Y_n(x_i,z)). \end{align}\] It follows from Lemma 4 that equation 26 has a unique solution for every couple \((\nu^0_{n+1}, \nu^1_{n+1})\in \mathbb{R}^2\). Furthermore, it can be checked that \(\alpha_n, \gamma_n\) satisfy the assumptions of [prop:U-recap]. With a slight abuse of notation, let us denote by \(\ell^j_n\) the linear form \(\ell^j_{\alpha_n,\gamma_n}\). We then choose the coefficients \((\nu^0_{n+1}, \nu^1_{n+1})\in \mathbb{R}^2\) so that \[\ell^j_n (g_{n+1}, \widetilde{\delta}_{0,n+1},\widetilde{\delta}_{1,n+1})=0\quad \text{for }j=0,1.\] Using Lemma 5, we prove that the matrix \[M_n:=\left( \ell^j_n ( f^i(x, Y_n), \delta^i_0(Y_n(x_0)), \delta^i_1(Y_n(x_1)))\right)_{0\leq i, j \leq 1}\] is close to identity, and thus invertible. The coefficients \((\nu^0_{n+1}, \nu^1_{n+1})\) are therefore uniquely defined.

Furthermore, using Proposition 3 and deriving additional regularity estimates in the spirit of Lemma 1, we prove by induction that \[\| u_n\|_{{Q^1}} + | \nu^0_n | + |\nu^1_n| \lesssim \|(f,\delta_0, \delta_1)\|_{\mathcal{H}} \quad \forall n\in \mathbb{N}.\]

4.2 Geometric estimate in \({Q^{1/2}}\)↩︎

The next step is to prove a bound of the type \[\label{in-geom} \|u_{n+1}- u_n \|_X \lesssim \eta \| u_n - u_{n-1}\|_X\tag{31}\] for some \(0<\eta \ll 1\), in a functional space \(X\) to be determined. To that end, we observe that \(w_n:=u_{n+1}- u_n\) satisfies \[\label{eq:wn} \begin{cases} (y + u_n) \partial_x w_n - \partial_{yy} w_n = - w_{n-1} \partial_x u_n + (\nu^0_{n+1}-\nu^0_{n}) f^0 + (\nu^1_{n+1}-\nu^1_{n}) f^1, \\ (w_n)_{\rvert \Sigma_i} = (\nu^0_{n+1} - \nu^0_n) \delta^0_i + (\nu^1_{n+1} - \nu^1_n) \delta^1_i, \\ (w_n)_{\rvert y = \pm 1} = 0. \end{cases}\tag{32}\] We already know that the solution \(w_n\) belongs to \({Q^1}\), as the difference between two \({Q^1}\) functions. Hence, there is no need to check that the orthogonality conditions are satisfied. However, because of the term \(-w_{n-1} \partial_x u_n\) in the right-hand side, the source term does not belong to \(H^1_x L^2_y\). Note that the right-hand side also involves \(\nu^j_{n+1} - \nu^j_n\). From the definition of \(\nu^j_n\) and Lemma 5, we have \[\begin{align} |\nu^j_{n+1} - \nu^j_n| \lesssim &\| \ell^0_{n}- \ell^0_{n-1}\|_{\mathcal{L}(\mathcal{H})} + \| \ell^1_{n}- \ell^1_{n-1}\|_{\mathcal{L}(\mathcal{H})} \\ \lesssim & \|\alpha_n-\alpha_{n-1}\|_{L^\infty_z(H^{7/12}_x)} + \|\gamma_{1,n}-\gamma_{1, n-1} \|_{L^\infty_z(L^2_x)} \\&+ \|\gamma_{2, n} -\gamma_{2, n-1}\|_{H^{1/2}_x L^2_z}. \end{align}\] Looking at the definition of the coefficients \(\alpha_n, \gamma_n\), one can prove that the right-hand side of the above inequality is bounded by \(\| u_n - u_{n-1}\|_{{Q^{1/2}}}\). This prompts us to take \[X=Q^{1/2}:= H^{7/6}_x L^2_y \cap L^2_x H^{7/2}_y(\Omega).\] in 31 .

As a consequence, we need to derive estimates in \({Q^{1/2}}\) for the solutions of 18 , when the data satisfies an orthogonality condition. We choose to argue by interpolation. This leads to a subtle matter: omitting the boundary data to simplify the discussion, one needs to interpolate between \(L^2(\Omega)\) (which is the regularity of the source term for \({Q^{0}}\) solutions) and the space \(\{f\in H^1_x L^2_z \cap L^2_x H^3_z\;\ell^j_{\alpha,\gamma} (f,0,0)=0\}\). This requires a rather precise description of the linear forms \(\ell^j\), involving in particular their representation in terms of the dual profiles \(\Phi^j\), and a decomposition of the latter into singular profiles and a smooth and/or explicit remainder, in the spirit of Corollary 1. We refer to [1] for all details. Eventually, we conclude that \[\|w_n \|_{{Q^{1/2}}} \lesssim \eta \|w_{n-1}\|_{Q^{1/2}}+ \eta^2 \|w_{n-2}\|_{Q^{1/2}},\] where \(\eta= \|(f,\delta_0, \delta_1)\|_{\mathcal{H}}\).

As a consequence, \((u_n)_{n\in \mathbb{N}}\) is a Cauchy sequence in \({Q^{1/2}}\). Passing to the limit as \(n\to \infty\), we infer that there exists \(u\in {Q^1}\), \(\|u\|_{{Q^1}}\lesssim \eta\), and \((\nu^0, \nu^1)\in \mathbb{R}^2\), \(| \nu^j|\lesssim \eta\), such that \[\begin{cases} (y+u)\partial_x u - \partial_{yy} u =f + \nu^0 f^0 + \nu^1 f^1,\\ u_{\rvert \Sigma_i} = \delta_i + \nu^0 \delta^0_i + \nu^1 \delta^1_i, \\ u_{\rvert y = \pm 1} = 0. \end{cases}\] Let \(B_\eta\) be the open ball of radius \(\eta\) in \(\mathcal{H}\). Note that it follows from the above construction that the maps \[(f,\delta_0, \delta_1)\in B_\eta\mapsto \nu^j (f,\delta_0, \delta_1)\in \mathbb{R}\] are Lipschitz continuous. We also denote by \(\mathcal{U}_\perp\) the map \(\mathcal{U}_\perp:(f,\delta_0, \delta_1)\in B_\eta\mapsto u\in {Q^1}\). The above argument shows that \(\mathcal{U}_\perp\) is Lipschitz continuous from \(B_\eta\) to \({Q^{1/2}}\).

4.3 Definition of the manifold \(\mathcal{M}\)↩︎

For the sake of brevity, let us denote by \(\Xi=(f,\delta_0,\delta_1)\) the elements of \(\mathcal{H}\). We set \(\Xi^j= (f^j, \delta_0^j, \delta_1^j)\) (see 25 ), and \(\mathcal{H}^\perp_{\operatorname{sg}}:=\{\Xi\in \mathcal{H}, \;\langle \Xi, \Xi^j\rangle_{\mathcal{H}} =0\;j=0,1 \}\). For every \(\Xi \in \mathcal{H}\), one has the decomposition \[\Xi = \Xi^\perp + \langle \Xi^0 ; \Xi \rangle_{\mathcal{H}} \Xi^0 + \langle \Xi^1 ; \Xi \rangle_{\mathcal{H}} \Xi^1,\] where \(\Xi^\perp \in \mathcal{H}^\perp_{\operatorname{sg}}\) and the linear maps \(\Xi \mapsto \Xi^\perp\) and \(\Xi \mapsto \langle \Xi^k ; \Xi \rangle\) are continuous.

Let us now define \[\label{eq:M} \mathcal{M} := \left\{ \Xi \in \mathcal{H}; \enskip \| \Xi \|_{\mathcal{H}} < \eta \text{ and } \langle \Xi^k ; \Xi \rangle_{\mathcal{H}} = \nu^k(\Xi^\perp) \text{ for } k = 0,1 \right\}\tag{33}\] where \(\nu^0\) and \(\nu^1\) are the maps constructed in the previous paragraph. We set, for \(\Xi \in \mathcal{M}\), \[\mathcal{U}(\Xi) := \mathcal{U}_\perp(\Xi^\perp).\] It can be easily checked that for all \(\Xi\in \mathcal{M}\), \(\mathcal{U}(\Xi)\) is a solution of 1 . It is also clear from the definition that \(\mathcal{M}\) is a Lipschitz manifold modeled on \(\mathcal{H}^\perp_{\operatorname{sg}}\) since \(\nu^0\) and \(\nu^1\) are Lipschitz maps. It contains \(0\) since \(\nu^j(0)=0\). Furthermore, \(\mathcal{H}^\perp_{\operatorname{sg}}\) is tangent to \(\mathcal{M}\) at \(0\) in the following weak senses:

• For \(\Xi \in \mathcal{M}\), \(\operatorname{d} ( \Xi, \mathcal{H}^\perp_{\operatorname{sg}}) \lesssim \| \Xi \|^2_{\mathcal{H}}\).

• For every \(\Xi^\bot \in \mathcal{H}^\perp_{\operatorname{sg}}\), for \(\varepsilon\in \mathbb{R}\) small enough, \(\operatorname{d} (\varepsilon\Xi^\bot, \mathcal{M}) \lesssim \varepsilon^2\).

These two properties are left to the reader. Theorem 2 follows.

Acknowledgements↩︎

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC “Mathematical Study of Boundary Layers in Oceanic Motion”. This work was supported by the SingFlows project, grants ANR-18-CE40-0027 of the French National Research Agency (ANR). A.-L.D.acknowledges the support of the Institut Universitaire de France. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while A.-L.D.participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester.

References↩︎