Abstract

We prove Hadamard ill-posedness for the non-autonomous linearised Prandtl equations around time-dependent shear-flow equilibria in function spaces up to Gevrey class 4. More precisely, we construct compactly supported smooth initial data, Gevrey class \(4\) in the tangential variable, for which the system admits no weak solution for any positive lifespan. In this regard, we improve previous results by showing that classical semigroup-type instabilities do not, by themselves, imply Hadamard ill-posedness in the non-autonomous case when the initial time is not a variable of the system.

Our argument is based on a family of exact unstable modes whose \(L^2\)-norms grow, at tangential frequency \(k\), like \(\exp(c \sqrt k t)\) up to times of order \(t \sim k^{-1/4}\). Their construction relies on an inner–outer gluing scheme, in the spirit of matched asymptotic expansions, which combines unstable inner solutions near a non-degenerate critical point with an exact outer solution and yields exponentially small matching errors for short times.

AMS Subject Classification: primary 35Q35; secondary 76D10, 35B30, 35B35, 35C20
Keywords: linearised Prandtl equations; Hadamard ill-posedness; non-solvability; boundary layers

section1 -3.5ex 1.5ex Introduction The Prandtl equations are a classical asymptotic model in fluid dynamics describing the leading-order dynamics of boundary layers in the vanishing viscosity limit of the Navier-Stokes equations. Among the various mathematical challenges of rigorously justifying their asymptotics [1]–[6], a central problem concerns the well-posedness of the Prandtl equations themselves.

The well-posedness theory reflects, in part, the physics of boundary layer separation since it falls into two categories. Under a monotonicity assumption on the initial velocity, the equations are well-posed in Sobolev spaces [7]–[11]; without monotonicity, well-posedness requires high tangential regularity, typically analytic or so-called Gevrey classes [12]–[15], with Gevrey-2 appearing as a borderline case [16], [17].

These regularity requirements are supported by the fact that the Prandtl equations, when linearised around suitable non-monotonic profiles, are ill-posed in Sobolev spaces [18], [19]. To the best of our knowledge, this is the first ill-posedness result in any Gevrey class for the linearised Prandtl equations. In this work, we establish ill-posedness for the linearised equations in Gevrey-4 spaces and provide some clarifications concerning the Sobolev instabilities.

We focus on the two-dimensional Prandtl model linearised around a time-dependent shear flow \((u,v) = (u_s,0)\). Here, the shear map \(u_s= u_s (t,y)\) is a smooth solution of the heat equation \[\label{eq:heat-equation} \partial_t u_s - \partial_y^2 u_s = 0,\qquad \qquad u_s |_{y = 0} = \lim\limits_{y \to + \infty} u_s = 0,\qquad \qquad u_s |_{t = 0} = U_s,\tag{1}\] with \(t>0\), \(y \in \mathbb{R}_+ = (0, \infty)\) and where \(U_s= U_s(y)\) is an appropriate non-monotonic profile (cf. 1). The shear flow \((u_s,0)\) is a smooth solution of the nonlinear Prandtl equations and the corresponding linearisation is non-autonomous, taking the form: \[\label{eq:lin-Prandtl} \begin{cases} \, \partial_t u + u_s \, \partial_x u + v \,\partial_y u_s - \partial_y^2 u = 0 \qquad &\!\!(t,x,y)\in(0,\delta) \times \mathbb{T} \times \mathbb{R}_+, \\ \, \partial_x u + \partial_y v = 0 \qquad & \phantom{(t,x,y) \in }(0,\delta) \times \mathbb{T} \times \mathbb{R}_+, \\ (u,v)\big|_{y = 0} =(0,0),\quad \lim\limits_{y \to + \infty} u = 0 \qquad & \phantom{(t,x,y) \in } (0,\delta) \times \mathbb{T}, \\ \,u\,|_{t = 0} = u_{\rm in} \qquad & \phantom{\,(t,x,y) \in (0,\delta) \times } \mathbb{T} \times \mathbb{R}_+, \end{cases}\tag{2}\] for a velocity field \((u,v)\) and an arbitrary (small) lifespan \(\delta >0\).

When \(u_s\) depends on time, [20] established a seminal result regarding the ill-posedness of 2 in Sobolev spaces: in general, System 2 does not generate a uniformly continuous two-parameter semigroup acting in Sobolev regularities. More specifically, under sufficient regularity and decay of \(u_s\), the equations are locally well-posed for \(u_{\rm in}\) with Fourier modes in \(x \in \mathbb{T}\) that are weighted Lebesgue/Sobolev in \(y \in \mathbb{R}_+\) and whose norms decay at an analytic rate (cf. Proposition 1 in [20]). If one denotes by \(T(\tau,t)(u_{\rm in}) = u(t, \cdot)\) the underlying solution operator with \(u|_{t = \tau} = u_{\rm in}\) imposed at a time \(\tau \in [0, \delta)\), Theorem 1-\((ii)\) in [20] shows that there exist non-monotonic profiles \(u_s(t,y)\) for which \(T\) does not extend uniformly as an operator in Sobolev regularities, since it satisfies: \[\label{eq:result-of-GV} \sup_{0 \leq \tau \leq t \leq \delta } \big\| \,e^{- \sigma(t-\tau) \sqrt{|\partial_x|}} T(\tau,t) \, \big\|_{\mathcal{L}(H^{m_1}, H^{m_2} )} = + \infty, \quad \forall m_1,\, m_2 \geq 0,\quad \; H^m = H^m (\mathbb{T}, L^\infty(\mathbb{R}_+, e^{\lambda y} dy )),\tag{3}\] for any \(\sigma\in [0, \sigma_0)\), with given \(\sigma_0>0\), and any \(\delta>0\), \(\lambda >0\). By this identity and the Banach–Steinhaus theorem, there exists a unit-norm initial datum \(u_{\rm in}\) in \(H^{m_1}\) such that \(T(\tau_n,t_n)(u_{\rm in})\) diverges in \(H^{m_2}\) for some sequence \((\tau_n, t_n)\to (0, 0)\) with \(0\leq \tau_n \leq t_n\). In other words, continuity at \((\tau,t) = (0, 0)\) fails in any Sobolev space, when the initial time \(\tau \in [0, \delta)\) is also considered a variable of the system.

Similar relations as in 3 were subsequently extended to three dimensions [21], MHD boundary layers [22], and to the case of lower decay of \(u_s\) as \(y \to \infty\) [23].

There are, however, several examples of non-autonomous PDEs that define a global-in-time, two parameter semigroup \(S = S(\tau,t)\) in Sobolev spaces, which is not uniformly continuous, yet satisfies the relations: \[\label{eq:intro-S-semigroup-relations} \sup_{0 \leq t < + \infty} \big\| \,S(0,t) \, \big\|_{\mathcal{L}(H^{m_1}, H^{m_1} )} \leq 1 ,\quad \qquad \sup_{0 \leq \tau \leq t \leq \delta } \big\| \,e^{- \sigma(t-\tau) \sqrt{|\partial_x|}} S(\tau,t) \, \big\|_{\mathcal{L}(H^{m_1}, H^{m_2} )} = + \infty,\tag{4}\] for any \(m_1,\, m_2\in \mathbb{R}\), \(\sigma \in [0,1)\) and \(\delta >0\). A simple toy-model is presented in [sec:sec:toy-model]. In particular, if the initial time is not treated as a variable, for instance \(\tau = 0\) is fixed, these PDEs are well-posed in any Sobolev spaces in the sense of Hadamard. As a natural question: does System 2 fall into this category, or does it instead exhibit Hadamard ill-posedness even when the initial time is fixed? We determine that the latter holds.

We emphasize that this issue is exclusive to the non-autonomous case of System 2 . If \(u_s(t,y) = U_s(y)\), indeed, Gérard-Varet and Nguyen [18] proved the existence of non-monotonic \(U_s\) and smooth, fast decaying initial data \(u_{\rm in}\) that do not generate weak solutions. However, as we discuss in 1, their approach does not extend directly to the non-autonomous setting of 2 .

0.1 Main Results↩︎

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Our approach combines a refined construction of unstable quasi-eigenmodes with a superposition argument adapted to Gevrey regularity. Inspired by [20] and [24], we revisit a special class of unstable quasi-eigenmodes \(u(t,x,y) = u_k(t,y)e^{i k x}\), at fixed tangential frequencies \(k \in \mathbb{N}\). Our first main result (cf. 1) establishes the existence of exact quasi-eigenmode solutions (as opposed to approximate ones as in [20]) to System 2 that exhibit growth of order \(e^{c \sqrt{k}\,t}\) persisting at least up to times of order \(t \sim k^{-\frac{1}{4}}\). By superposition, our second result exploits their maximal growth \(e^{c \sqrt[4]{k}}\) to construct smooth, compactly supported initial data which are Gevrey class \(4\) in \(x\in \mathbb{T}\), and for which no weak solution to 2 exists for any lifespan \(\delta >0\) (cf. 2 and 2).

Our results rely on structural assumptions on the initial profile \(u_s|_{t =0} = U_s\). We first introduce the weighted Sobolev spaces \(H^m_\lambda(\mathbb{R}_+)\), with \(\mathbb{R}_+ = (0, \infty)\), defined for \(m \in \mathbb{N}\) and \(\lambda >0\): \[\begin{align} H^{m}_\lambda(\mathbb{R}_+) := H^m(\mathbb{R}_+,\, e^{\lambda y} dy ) = \Big\{ \, f \in H^m(\mathbb{R}_+, \mathbb{R})\; \big|\; f^{(j) } \in L^2_\lambda(\mathbb{R}_+) := L^2(\mathbb{R}_+, e^{\lambda y} dy),\; \forall j = 0,\dots, m \Big\}, \end{align}\] endowed with its natural norm \[\| \, f \, \|_{H^{m}_\lambda} := \sum_{j = 0}^m\| \,f^{(j)} \,\|_{L^2_\lambda} = \sum_{j = 0}^m \Bigg( \int_{\mathbb{R}_+} |f^{(j)}(y)|^2 \,e^{2\lambda y} dy \Bigg)^\frac{1}{2}.\] The space \(H^m_\lambda(\mathbb{R}_+, \mathbb{C})\) for complex-valued functions is defined analogously.

Hypotheses 1. We assume the following conditions for the initial profile \(u_s|_{t = 0} = U_s\):

There exist \(m \in \mathbb{N}\setminus\{1,\,2\}\) and \(\lambda >0\) such that \(U_s\in H^{m+1}_\lambda(\mathbb{R}_+)\).
\(U_s(0) = U_s''(0) = \dots = U_s^{2 \lfloor \frac{m}{2} \rfloor }(0) = 0\).
There exist \(a_0>0\) and \(0< d < a_0\) such that \(U_s\) admits the following quadratic representation: \[U_s(y) = U_s(a_0) + \frac{U_s''(a_0)}{2} (y-a_0)^2,\qquad \text{for any}\quad y \in \left[ a_0 - d, a_0 + d\right],\] for some given \(U_s(a_0) \in \mathbb{R}\) and \(U_s''(a_0)<0\).

Under these assumptions, the unique solution \(u_s=u_s(t,y)\) to 1 is smooth and satisfies (cf. 8) \[u_s \in \mathcal{C}^\infty( \mathbb{R}_+ \times \mathbb{R}_+ ) \cap \mathcal{C}([0, \infty[, H^{m+1}_\lambda(\mathbb{R}_+)) \cap \mathcal{C}^1([0, \infty[, H^{m-1}_\lambda(\mathbb{R}_+)),\] which in particular implies exponential decay in \(y\) for each \(t \geq 0\). As a consequence, projecting 2 onto Fourier modes yields, for each \(k \in \mathbb{Z}\), the problem \[\label{eq:lin-Prandtl-projected-k} \begin{cases} \, \partial_t u_k + i k \, \big( u_s \, u_k - \partial_y u_s \int_0^y u_k \big) - \partial_y^2 u_k = 0 \qquad &\!\!(t,y)\in(0,\delta) \times \mathbb{R}_+, \\ \,u_k |_{y = 0} = \lim\limits_{y \to + \infty} u_k = 0 \qquad & \phantom{(t,y) \in } (0,\delta), \\ \,u_k\,|_{t = 0} = u_{{\rm in},k} \qquad & \phantom{\,(t,y) \in (0,\delta) \times } \mathbb{R}_+, \end{cases}\tag{5}\] For each fixed \(k\), this system can be treated as a linear perturbation of the heat equation on the half-line with Dirichlet boundary conditions, and for any \(u_{{\rm in},k} \in H^1_0(\mathbb{R}_+)\) it admits a unique solution \[\label{eq:fct-space-uk} u_k \in \mathcal{C}([0, \delta], H^1_0(\mathbb{R}_+, \mathbb{C})) \cap L^2(0, \delta ; H^2(\mathbb{R}_+, \mathbb{C}))\cap W^{1,2}(0, \delta; L^2(\mathbb{R}_+, \mathbb{C})),\tag{6}\] for any \(\delta>0\) (cf. the a-priori estimate ?? ). The key assumption for the existence of unstable \(u_k\) is (H3) in 1. It ensures the existence of a non-degenerate critical point at \(y = a_0>0\), i.e. \(U_s'(a_0) = 0\) and \(U_s''(a_0)\neq 0\). We stress, however, that the quadratic structure near \(y = a_0\) is essential for our construction, and not merely the existence of a non-degenerate critical point (see 0.2).

Let us introduce the explicit form of the unstable initial datum \(u_{\rm in, k} =\frac{d}{dy}\phi_{{\rm unst}, k}\) for System 5 . Its construction relies on an entire function arising in [24] in the analysis of the Prandtl equations around a quadratic shear flow.

Definition 1. Let \(\Upsilon:\mathbb{C} \to \mathbb{C}\) be defined by \[\Upsilon(\zeta):= \frac{1}{\sqrt{2\pi}}\,\zeta\, e^{-\frac{\zeta^2}{2}} + \frac{1}{2} \big(1+\zeta^2\big) \left(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}} \right) \right).\] Moreover, let \(H\) denote the Heaviside function and let \(\chi \in \mathcal{C}^\infty_c(\mathbb{R}_+)\) be a cutoff function satisfying \[\chi(y) = 1 \quad \text{for all } y\in \Big[a_0-\frac{d}{4}, a_0+\frac{d}{4}\Big] \qquad \text{and}\qquad \chi(y) = 0\quad \text{for all } y \in \mathbb{R}_+\setminus \Big[a_0-\frac{d}{2}, a_0+\frac{d}{2}\Big].\] For any \(k \in \mathbb{N}\), we define the unstable stream function \(\phi_{{\rm unst}, k}:\mathbb{R}_+ \to \mathbb{C}\) as follows: \[\label{eq:intro-def-phiinnk} \begin{align} \phi_{{\rm unst}, k}(y):= &\chi(y) \, \Upsilon \Bigg( (y-a_0) \, e^{-\frac{i\pi}{8}} \sqrt[4]{ k \frac{|U_s''(a_0)|}{2}} \,\Bigg) \sqrt{\frac{|U_s''(a_0)|}{2k}} \, e^{i\frac{5\pi}{4}} \,+ \\ +& \Big(1-\chi(y) \Big)\, H(y-a_0) \bigg(\, U_s(y) -U_s(a_0) + e^{i\frac{5\pi}{4}}\sqrt{\frac{|U_s''(a_0)|}{2k}} \; \bigg). \end{align}\qquad{(1)}\]

We defer further discussion of \(\phi_{{\rm unst}, k}\) to the next section. For now, we note that it matches two profiles via the cutoff \(\chi\): an inner layer based on \(\Upsilon\) near \(y = a_0\), and an outer layer at \(y > a_0 + d/2\), determined by \(U_s\). Moreover, \(\Upsilon\) can be associated with the so-called shear-layer profile introduced in [20] (cf. Appendix [sec:sec:comparison-with-shear-layer]).

The inner layer is the main source of instability (cf. 3), and the following theorem shows that the outer layer and the matching scheme do not suppress the resulting growth, at least for short times \(t \lesssim k^{-\frac{1}{4}}\).

Theorem 1. Let \(u_s= u_s(t,y)\) be solution of the heat equation 1 with \(u_s|_{t= 0} = U_s\) satisfying 1.

Consider the initial datum \(u_{{\rm in}, k}= \frac{d}{dy} \phi_{{\rm unst}, k}: \mathbb{R}_+ \to \mathbb{C}\) defined in 1. Then the following statements hold true:

One has \(u_{{\rm in}, k} \in H^{m}_\lambda(\mathbb{R}_+, \mathbb{C})\) and \(u_{{\rm in}, k}(y) = 0\) for any \(y\in [0, a_0-d/2]\). Moreover, there exists a constant \(\mathcal{D}_m = \mathcal{D}_m(U_s, d, \chi)>0\), depending only on \(m\), \(U_s\), \(d\) and \(\chi\), such that \[\| \, u_{{\rm in}, k} \,\|_{H^{m}_\lambda } \leq \mathcal{D}_m e^{2\lambda a_0} \, k^{\frac{2m-1}{4} }, \qquad \forall \;k \in \mathbb{N}.\]
Let \(u_k\) denote the unique classical solution of 5 in the space 6 . Then there exists \(K \in \mathbb{N}\), depending only on \(U_s\), \(d\) and \(\lambda\), such that for all \(k \geq K\) and all \[\begin{align} t \in \left[ 0,\, \frac{1}{\sqrt[4]{k}} \frac{d}{16(1 +\| U_s'' \|_{L^\infty} )} \right], \end{align}\] one has the lower bound \[\begin{align} \big\| \, u_k(t , \cdot )\, \big\|_{L^2(\mathbb{R}_+)} \geq \frac{1}{4} \bigg( \int_{a_0+\frac{d}{2}}^\infty | \, U_s'(y)\,|^2 dy \,\bigg)^\frac{1}{2} \exp \left( \frac{t}{2} \sqrt{k \frac{|U_s''(a_0)|}{2} }\right). \end{align}\]

Part \((i)\) shows that \(u_{{\rm in}, k}\) inherits the regularity of \(U_s' \in H^m_\lambda(\mathbb{R}_+)\) and satisfies the boundary conditions. Moreover, its norm grows at most polynomially in \(k\in \mathbb{N}\). An exact form of \(\mathcal{D}_m>0\) given in 2.

Part \((ii)\) presents the key aspect of the theorem. The lower bound already holds in the unweighted space \(L^2(\mathbb{R}_+)\)-norm (not weighted) and all constants are explicit.

Since \(U_s\) is quadratic in \([a_0-d, a_0+d]\), its derivative \(U_s'\) does not vanish identically in \((a_0 + d/2, +\infty)\) and the lower bound is nontrivial. We also observe that the exponential growth is proportional to \(|\,U_s''(a_0)\,|^{1/2}\) and degenerates when \(U_s''(a_0) = 0\), highlighting the role of the non-degeneracy as in [20].

In addition to this, Part \((ii)\) highlights the dependence of the maximal time of norm-inflation with respect to \(d > 0\). In our construction, the lifespan over which instability persists, shrinks as the interval in (H3) of 1 collapses to the critical point \(y = a_0\). As a consequence, our result cannot be extended to more general non-monotonic profiles by a simple approximation argument. Moreover, we infer that \(K=K(U_s, d, \lambda)\in \mathbb{N}\) diverges as \(d \to 0+\).

Building upon the quasi-eigenmodes constructed in 1, we obtain, by superposition, a smooth initial datum for the system 2 , for which no weak solution exists in the following sense:

Definition 2. Let \(u_{\rm in} \in L^2(\mathbb{T}\times \mathbb{R}_+)\). A weak solution to 2 is a function \[u \in L^\infty(0, \delta; L^2(\mathbb{T} \times \mathbb{R}_+)) \cap L^2(0,\delta; H^{0,1}_0(\mathbb{T} \times \mathbb{R}_+)),\quad \text{with}\quad H^{0,1}_0(\mathbb{T} \times \mathbb{R}_+) = \overline{\mathcal{C}_c^\infty( \mathbb{T} \times \mathbb{R}_+)}^{\| \cdot \|_{H^{0,1}}},\] and \(\| f \|_{H^{0,1}} =\| f \|_{L^2} + \| \partial_y f \|_{L^2}\), such that for all test functions \(\varphi \in \mathcal{C}^\infty_c([0, \delta) \times \mathbb{T} \times \mathbb{R}_+)\), one has \[\begin{align} \int_0^\delta \int_{\mathbb{T}} \int_{\mathbb{R}_+} &\bigg( u \, \partial_t \varphi + u_s \, u \, \partial_x \varphi - \partial_y u \, \partial_y \varphi \bigg) (t,x,y) \, dy dx dt \, + \\ &- \int_0^\delta \int_{\mathbb{T}} \int_{\mathbb{R}_+} u_s(t,y) \bigg( \int_0^y u (t,x,z) dz \bigg) \partial_x \varphi (t,x,y) dy dx dt = - \int_{\mathbb{T}} \int_{\mathbb{R}_+} u_{\rm in}(x,y) \varphi(0, x,y)dy dx. \end{align}\]

This corresponds to a distributional solution with natural energy regularity for the nonlinear Prandtl equations. We now construct an initial datum that does not admit any weak solution, under the further assumption that \(U_s \in \mathcal{C}^\infty_c (\mathbb{R}_+)\). In this case, conditions (H1) and (H2) in 1 are satisfied for all \(m \in \mathbb{N}\setminus\{1, 2\}\) and all \(\lambda > 0\), and the ill-posedness holds true regardless of their values.

Consider the Fourier series \[\label{eq:intro-Uincomplex-series} \mathbf{U}_{\rm in}(x,y) := \sum_{k = 1}^\infty e^{-\sigma_0 \sqrt[4]{k}} \frac{d}{dy} \phi_{{\rm unst,k}} (y) e^{ {i}k x} \in \mathbb{C},\quad (x,y) \in \mathbb{T} \times \mathbb{R}_+.\tag{7}\] where \(\sigma_0>0\) is sufficiently small constant. By Part \((i)\) of 1, the series converges in \(\mathcal{D}(\mathbb{T} \times \mathbb{R}_+, \mathbb{C})\). Moreover, \(\mathbf{U}_{\rm in}\) is Gevrey-class \(4\) in \(x\in \mathbb{T}\) and Sobolev in \(y\in \mathbb{R}_+\) in the following sense: a function \(f : \mathbb{T} \times \mathbb{R}_+ \to \mathbb{C}\) belongs to \(\mathcal{G}_\sigma^4(\mathbb{T}, H^m_\lambda(\mathbb{R}, \mathbb{C}))\) with \(\sigma>0\) and \(m\in \mathbb{N}_0\), if there exists a sequence \((f_k)_{k\in \mathbb{Z}} \subset H^m_\lambda(\mathbb{R}_+, \mathbb{C})\) such that \[\| \,f \,\|_{\mathcal{G}^4_\sigma H^m_\lambda }:= \sup_{k \in \mathbb{Z}} \Big( e^{\sigma \sqrt[4]{k}} \| f_k \|_{H^m_\lambda} \Big) < + \infty,\quad \text{and}\quad f(x,y) = \sum_{k \in \mathbb{Z}} f_k(y)e^{i k x} \quad\text{for a.e.}\quad (x,y) \in \mathbb{T} \times \mathbb{R}_+.\]

Theorem 2. Let \(u_s= u_s(t,y)\) be solution of 1 , with initial datum \(U_s \in \mathcal{C}^\infty_c(\mathbb{R}_+)\) satisfying (H3) of 1. Let \[0< \sigma_0 <\, \frac{d|U''(a_0)|^{1/2}}{32(1+ \left\| U''_s \right\|_{L^\infty} )}\] and let \(\mathbf{U}_{\rm in}\) be defined in 7 . Then \(\mathbf{U}_{\rm in} \in \mathcal{G}^4_\sigma(\mathbb{T}, H^m_\lambda(\mathbb{R}_+, \mathbb{C}))\), for all \(\sigma \in ( 0, \sigma_0)\), all \(m\in \mathbb{N}\) and all \(\lambda>0\). Moreover, for at least one of the initial data \[u_{\rm in} \in \Big\{ \,{\rm Re}(\mathbf{U}_{\rm in}),\, {\rm Im}(\mathbf{U}_{\rm in})\, \Big\} \subset \mathcal{C}^\infty_c ( \mathbb{T} \times \mathbb{R}_+) \cap \bigg( \bigcap_{m \in \mathbb{N}} \bigcap_{\lambda>0} \bigcap_{0 < \sigma < \sigma_0 } \mathcal{G}^4_\sigma\big(\mathbb{T}, H^m_\lambda(\mathbb{R}_+)\big) \bigg),\] the system 2 admits no a weak solution in the sense of 2.

In summary, 2 establishes Hadamard ill-posedness of the non-autonomous linearised Prandtl equations 2 both in Sobolev spaces and Gevrey classes of order \(4\). In contrast to [20], this ill-posedness persists even when the initial time is no variable of the system.

0.2 Structure of the proof↩︎

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The proof follows a perturbative strategy: we construct exponentially growing approximate solutions and show that the associated error can be controlled. We outline the main steps in the proof of 1. The proof of 2 is deferred to Section [sec:sec:proof-of-second-main-thm].

In [sec:sec:matched-asymptotics] and [sec:sec:estimates-forced-terms], we build a family of forced quasi-eigenmodes \(u_k^{\rm fr}\) satisfying estimates analogous to those in 1. In particular, there exist constants \(c_0,\, c_1>0\) such that \(\| \, u_k^{\rm fr}(t, \cdot) \,\|_{L^2} \geq c_0 \, e^{c_1 \sqrt{k} \, t}\). Moreover, \(u_k^{\rm fr}\) generates a forcing term \(f_k(t,y) e^{c_1 \sqrt{k} \, t}\) with \[\label{intro:behaviour-forcing-term} \| \, f_k(t, \cdot)\, \|_{L^2(\mathbb{R}_+)} =\mathcal{O}\Big( e^{-c_2/t} + e^{-c_3\sqrt{k}}\Big)\tag{8}\] for some constants \(c_2,\, c_3>0\) (see 1 and 2). The initial datum satisfies \(u_k^{\rm fr}(0, \cdot)\) coincides with \(\frac{d}{dy} \phi_{{\rm unst}, k}\) introduced in 1.
In [sec:sec:correcting-the-perturbed-solution], with 3, we estimate the correction \(\delta u_k= u^{\rm fr}_k- u_k\) between the approximate solution \(u^{\rm fr}_k\) and the exact solution \(u_k\) with \(\delta u_k (0, \cdot) = 0\). Let \(T_k(\tau, t)\in \mathcal{L}(L^2(\mathbb{R}_+), L^2(\mathbb{R}_+))\) denote the associated two-parameter semigroup on \(L^2(\mathbb{R}_+)\). Then \(u_k\) satisfies the Duhamel formula \[u_k(t,\cdot) = u_k^{\rm fr} (t, \cdot) - \delta u_k(t, \cdot),\quad \text{with}\quad \delta u_k(t,\cdot) = \int_0^t T_k(\tau, t) f_k(\tau, \cdot) e^{c_1 \sqrt{k} \, \tau }d\tau.\] To control this term, we establish the upper bound \[\label{intro:upper-bound-of-Tk} \|\, T_k(\tau, t) \,\|_{\mathcal{L}(L^2(\mathbb{R}_+), L^2(\mathbb{R}_+))} \leq e^{c_4 \sqrt{k} (t-\tau)},\tag{9}\] for constants \(c_4>c_1>0\), uniformly in \(k\in \mathbb{N}\). Combining these ingredients yields \[\begin{align} \|\, u_k(t, \cdot) \,\|_{L^2(\mathbb{R}_+)} &\geq \|\, u_k^{\rm fr}(t,\cdot)\, \|_{L^2(\mathbb{R}_+)} - \|\, \delta u_k (t, \cdot)\, \|_{L^2(\mathbb{R}_+)} \\ &\geq c_0 e^{c_1 \sqrt{k} \, t} - c_5 \int_0^t e^{c_4\sqrt{k}(t-\tau) + c_1 \tau} \big( e^{ - c_2/\tau } + e^{ - c_3 \,\sqrt{k}} \big)d\tau \\ &\geq c_0 e^{c_1 \sqrt{k} \, t} - c_5 \big( e^{ c_4 \sqrt{k} \, t - c_2/t } + e^{ c_4 \sqrt{k} \, t - c_3 \,\sqrt{k}} \big) t, \end{align}\] for some constant \(c_5>0\).
We conclude the proof of 1 in [sec:sec:proof-of-first-theorem] by choosing a time interval of length \(\sim k^{-\frac{1}{4}}\), for which the last negative term remains \(\mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{\scriptscriptstyle\mathcal{O}}} (1)\) as \(k \to \infty\), whereas the leading term \(c_0 e^{c_1 \sqrt{k} \, t}\) dominates. For instance, one may take \(t \leq (c_2/c_4)^{\frac{1}{2}} k^{-\frac{1}{4}}\) with \(k\gg 1\).

Tracking the dependence of \(c_0, \dots, c_5 > 0\) on \(U_s\) and \(d > 0\) yields the explicit bounds stated in 1.

We mention that we cannot reach an improved time \(k\)-scaling, since \(c_4 > c_1\).

The construction of \(u_k^{\rm fr}\) in Part-(a) is at the core of our approach. The main novelty is the asymptotics 8 of the forcing term: Gérard-Varet and Dormy [20] determined, for any \(N \in \mathbb{N}\), approximate solutions with \(f_k \in \mathcal{O}\big(t^N + k^{-\frac{N}{2}} \big)\), namely polynomially decaying both as \(k\to \infty\) and \(t \to 0\); We establish that, if \(u_s\) satisfies the heat equation, one can reach the asymptotics of 8 (the essential factor for Part (b) and Part (c)).

The function \(u_k^{\rm fr}\) is defined via a matched asymptotic expansion (e.g. [25]). We set in 3 \[u_k^{\rm fr}(t,y) = \partial_y \phi_{{\rm inn}, k}(t,y) + \partial_y \phi_{{\rm out}, k}(t,y),\qquad (t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+,\] for some \(T_{\rm max}>0\) and where \(\phi_{{\rm inn}, k}\) and \(\phi_{{\rm out}, k}\) are layers localised near and far from \(y = a_0\), respectively.

The inner profile \(\phi_{{\rm inn}, k}\) is supported in \(y \in [a_0 - \tfrac{d}{2}, a_0 + \tfrac{d}{2}]\) and relies on an instability ansatz of the equations in the region \(y \in [a_0 - \tfrac{d}{4}, a_0 + \tfrac{d}{4}]\), where \(U_s\) is quadratic. The outer layer \(\phi_{{\rm out}, k}\) is supported in \(y \geq a_0 + \tfrac{d}{4}\) and exhibits a similar instability to \(\phi_{{\rm inn}, k}\) for \(y \geq a_0 + \tfrac{d}{2}\). Below, we provide some heuristics and refer to [sec:sec:matched-asymptotics] for further details.

The construction of \(\phi_{{\rm out}, k}\) relies mainly on the following observation: for any \(\sigma_k \in \mathcal{C}([0,T_{\rm max}],\mathbb{C})\), then \[\phi_{{\rm out},k}(t,y) := \bigg( u_s(t,y) + \frac{\sigma_k(t)}{ik} \bigg) \exp \bigg( \int_0^t \sigma_k(\tau ) d \tau\bigg) \qquad (t,y) \in [0, T_{\rm max}] \times \Big[ a_0 + \frac{d}{2}, + \infty \Big[\] satisfies the identity: \[\begin{align} \Big(\, \partial_t + ik \, u_s - \partial_y^2 \,\Big) &\partial_y \phi_{{\rm out},k} - ik \, \partial_y u_s \, \phi_{{\rm out},k} = \\ &= \bigg( \partial_t \partial_y u_s + \sigma_k \, \partial_y u_s + i k \,u_s\, \partial_y u_s - \partial_y^3 u_s - i k \, \partial_y u_s \Big( u_s + \frac{\sigma_k }{ik} \Big) \bigg) e^{ \int_0^t \sigma_k } = 0. \end{align}\] Notably, \(\partial_t \partial_y u_s - \partial_y^3 u_s = 0\), since \(\partial_y u_s\) satisfies the heat equation. Due to the boundary conditions, these functions cannot define alone “unstable” quasi-eigenmodes on \(y>0\). On the other hand, by first building an “unstable” inner layer, we can incorporate the corresponding \(\sigma_k\) to the above expression. This is precisely possible because \(u_s\) satisfies the heat equation. By contrast (e.g. if \(u_s(t,y)=U_s(y)\)) an \(\mathcal{O}(1)\cdot e^{c_1 \sqrt{k} t}\) remainder would need to be corrected. In [20], the correction led to the mentioned \(\mathcal{O}(t^N + k^{-\frac{N}{2}})\cdot e^{c_1 \sqrt{k} t}\). If \(u_s\) satisfies the heat equation, no correction is needed.

The inner layer \(\phi_{{\rm inn}, k}\) is more involved; its construction is based on two aspects: first, \(u_s(0,y)= U_s(y)\) is quadratic in \(y \in [a_0 - \tfrac{d}{4},\, a_0 + \tfrac{d}{4}]\); second, for \(t>0\), the deviation of \(u_s(t,y)\) from its second-order Taylor expansion becomes in the same region exponentially small as \(t \to 0+\).

We recall from [24] that, for any constants \(\sigma_k \in \mathbb{C}\), \(a >0\), \(\alpha \in \mathbb{R}\) and \(\beta <0\), the spectral ODE arising from the autonomous linear Prandtl equations around a quadratic shear profile \[\bigg( \sigma_k + i k \,\Big( \alpha + \beta (y-a)^2 \Big) \bigg) \phi_k'(y) - 2 i k\, \beta (y-a) \phi_k(y) - \phi_k'''(y) = 0\] admits explicit general solutions in terms of hypergeometric functions. Among these solutions, we focus on the one characterised by \(\Upsilon\) of 1: \[\sigma_k = \sqrt{|\beta | k} \, e^{-\frac{{i}\pi}{4}} - {i}k \,\alpha \in \mathbb{C},\qquad \phi_k(y) = \Upsilon \Big( \,(y-a) \, e^{-\frac{i\pi}{8}} \sqrt[4]{ k |\beta|} \, \Big) \sqrt{\frac{|\beta|}{k}} e^{i \frac{5 \pi}{4}}.\] For simplicity, a concise proof of this assertion is provided in 3. As the real part of \(\sigma_k\) is strictly positive, the corresponding spectral mode exhibits exponential growth in time.

Our strategy in [sec:sec:matched-asymptotics] involves tracking the critical point \(\partial_y u_s(t,a(t)) = 0\) with \(a(0) = a_0\), for finite time \(t\in [0, T_{\rm max}]\) and define \(\phi_{{\rm inn}, k}\) in the region \((t,y) \in [0, T_{\rm max}] \times [a_0 - \tfrac{d}{4},\, a_0+\tfrac{d}{4}]\) via the above \(\Upsilon\)-profile: \[\begin{align} \alpha(t) &:= u_s(t,a(t)),\qquad \beta(t) := \partial_y^2u_s(t,a(t)),\qquad \sigma_k(t) := \sqrt{|\beta(t) | k} \, e^{-\frac{{i}\pi}{4}} - {i}k \,\alpha(t),\\ \phi_{{\rm inn}, k}(t,y) &:= \Upsilon \Big( \,(y-a(t)) \, e^{-\frac{i\pi}{8}} \sqrt[4]{ k |\beta(t)|} \, \Big) \sqrt{\frac{|\beta(t)|}{k}} e^{i \frac{5 \pi}{4}} \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg). \end{align}\] As explained, this construction determines simultaneously \(\phi_{{\rm out}, k}\). Unlike the outer layer, however, \(\phi_{{\rm inn}, k}\) does not solve System 2 in the region \((t,y) \in ]0, T_{\rm max}[ \times [a_0-\frac{d}{4},\, a_0+\frac{d}{4}]\), since \(u_s(t,y)\) ceases to be quadratic at any \(t>0\). Consequently, nontrivial remainders appear near \(y = a_0\) at any \(t>0\).

From the properties of the function \(\Upsilon\), however, we establish that these remainders mainly depend on the deviation of \(u_s\) from its second Taylor approximation (cf. 1-Part (a)). Thanks to the Green’s formula, they depend roughly on \[u_s(t,y) -\alpha(t) - \frac{\beta(t) }{2}(y-a(t))^2 = \frac{(y-a(t))^3}{6} \frac{1}{\sqrt{4\pi t}} \int_0^\infty \Big( e^{- \frac{|\xi(t,y) -w|^2}{4t}} - e^{- \frac{|\xi(t,y) + w|^2}{4t}} \Big) U_s^{(3)}(\omega)d\omega,\] where \(\xi(t,y)\) lies between \(y\) and \(a(t)\). The third derivative of \(U_s\) is zero in \([a_0-d, a_0+d]\) and the last integral is therefore \(\mathcal{O}(e^{-c_2/t})\) for some constant \(c_2 > 0\) proportional to \(d > 0\). Consequently, the remainders of \(\phi_{{\rm inn}, k}\) near \(y = a_0\) are estimated by the first asymptotics of 8 . Here the quadratic assumption becomes crucial: if \(U^{(3)}_s \not\equiv 0\) near \(y = a_0\), the integral would be \(\mathcal{O}(t)\) at best.

Once the inner and outer layers are defined on \(y \in [a_0 - \tfrac{d}{4}, a_0 + \tfrac{d}{4}]\) and \(y \geq a_0 + \tfrac{d}{2}\), respectively, we match them in the remaining regions. The transition is handled via the cutoff \(\chi\) of 1.

Multiplying the \(\Upsilon\)-profile by \(\chi\), the inner layer \(\phi_{{\rm inn}, k}\) extends to \((t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+\) as follows: \[\phi_{{\rm inn}, k}(t,y) = \chi(y)\Upsilon \Big( \,(y-a(t)) \, e^{-\frac{i\pi}{8}} \sqrt[4]{ k |\beta(t)|} \, \Big) \sqrt{\frac{|\beta(t)|}{k}} e^{i \frac{5 \pi}{4}} \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg).\] It satisfies \(\phi_{{\rm inn}, k}(t, 0) = \partial_y \phi_{{\rm inn}, k}(t, 0) = 0\). Additionally, due to the exponential decay \(\Upsilon(\zeta) \sim e^{-\zeta^2/2}\) in the sector \(\arg(\zeta)\in ]\frac{3\pi}{4}, \frac{5\pi}{4}[\), the remainders generated by \(\phi_{{\rm inn}, k}\) in the region \(y \in [0, a_0 - \tfrac{d}{4}]\) are indeed of order \(\mathcal{O}(e^{-c_3\sqrt{k}}) \cdot e^{c_1 \sqrt{k} t}\) as in 8 .

The transition in the region \(y \in [a_0+\tfrac{d}{4},\, a_0 + \tfrac{d}{2}]\) is more subtle. We globally extend \(\phi_{\text{out},k}\) via the cutoff function: \[\phi_{{\rm out},k}(t,y) := (1-\chi(y) )H(y-a(t)) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{ik} \bigg) \exp \bigg( \int_0^t \sigma_k(\tau ) d \tau\bigg).\] Both \(\phi_{\text{inn},k}\) and \(\phi_{\text{out},k}\) are \(\mathcal{O}(1) \cdot e^{c_1\sqrt{k} t}\) in \((t,y) \in [0, T_{\rm max}] \times [a_0+\tfrac{d}{4}, a_0 + \tfrac{d}{2}]\). However, from the properties of \(\Upsilon\) in 1, and the fact that \[\, \mathrm{erf}(\zeta) \to 1,\qquad \text{as}\qquad \zeta\to \infty\qquad\text{with}\qquad \arg(\zeta) = -\frac{\pi}{8},\] the asymptotics of \(\phi_{\text{inn},k}(t,y)\) in the region \((t,y) \in [0, T_{\rm max}] \times [a_0+\tfrac{d}{4}, a_0 + \tfrac{d}{2}]\) as \(k\to +\infty\) is given by \[\begin{align} \phi_{\text{inn},k}(t,y) &\sim \chi(y) \left( 1+ \sqrt{|\beta(t)|k}\,(y-a(t))^2e^{-\frac{{i}\pi}{4}} \right) \frac{1}{2} \left( 1 + 1 \right) e^{{i}\frac{5\pi}{4}} \sqrt{\frac{|\beta(t)|}{2k}} \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg) \\ &\sim \chi(y)\left( \frac{\sigma_k(t)}{ik} + u_s(t,a(t)) + \frac{\partial_y^2 u_s(t, a(t))}{2}\,(y-a(t))^2 \right) \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg). \end{align}\] This term is balanced by the corresponding \(\chi\)-term in the outer layer \(\phi_{\text{out},k}\): in the region \((t,y) \in [0, T_{\rm max}] \times [a_0+\tfrac{d}{4}, a_0 + \tfrac{d}{2}]\), where the main transition occurs, the leading-order term of the remainders still depends on \[\chi(y) \Big( u_s(t,a(t)) + \frac{\partial_y^2 u_s(t, a(t))}{2}\,(y-a(t))^2 -u_s(t,y) \Big).\] Once more, under the quadratic assumption and the fact that \(u_s\) satisfies the heat equation, the order of these remainders is \(\mathcal{O}(e^{-c_2/t})\).

We conclude this introduction with a final comparison between our approach and a prior result.

Remark 1. Our result extends that of [18]. For the autonomous case \(u_s(t,y) = U_s(y)\) of System 2 , Gérard-Varet and Nguyen identified initial data \(u_{\rm in} \in e^{-y} H^\infty(\mathbb{T} \times \mathbb{R}_+)\) that, for any \(T>0\), fail to generate distributional solutions with \(u \in L^\infty(0, T; L^2(\mathbb{T} \times \mathbb{R}))\) and \(\partial_y u \in L^2((0,T) \times \mathbb{T} \times \mathbb{R}))\). In this remark, we highlight why 2 is not a simple adaptation.

The proof in [18] relies on a contradiction, assuming that 2 defines a continuous linear map \[\mathcal{T} : e^{-y} H^\infty(\mathbb{T}\times \mathbb{R}_+) \to L^\infty(0, T; L^2(\mathbb{T} \times \mathbb{R})) \times L^2((0,T) \times \mathbb{T} \times \mathbb{R}),\quad \text{with}\quad \mathcal{T}(u_{\rm in}) = (u, \, \partial_y u).\] Namely, any \(u_{\rm in} \in e^{-y} H^\infty(\mathbb{T} \times \mathbb{R})\) defines a unique \(u\), and there are \(K \in \mathbb{N}\) and \(\mathcal{C} > 0\) (independent of \(u_{\rm in}\)) with \[\sup_{t \in [0, T]} \| u(t) \|_{L^2_{x,y}} + \| \partial_y u \|_{L^2_{t, x, y}} \leq \mathcal{C} \| e^y u_{\rm in} \|_{H^{K}_{x,y}}.\] We refer to page 5 and 6 of [18]. The associated autonomous semigroup \(S(t)(u_{\rm in}) = (\mathcal{T}(u_{\rm in}))_1(t) \in L^2_{x,y}\) satisfies for any \(0 \leq s \leq t \leq T\): \[\| S(t-s) \|_{\mathcal{L}(H^{K}_{x,y}, L^2_{x,y})} \leq \mathcal{C} \quad \Leftrightarrow \quad \max_{0 \leq t \leq T}\| S(t) \|_{\mathcal{L}(H^{K}_{x,y}, L^2_{x,y})} \leq \mathcal{C}<+\infty.\] This uniform bound is fundamental: at page 6 of [18], with \(\varepsilon= 1/\sqrt{k}\), they leverage the unstable approximate solutions \(u_\varepsilon^n\) of [20] with remainders \(r_\varepsilon^n\) to estimate a correction \(v(t) = u - u_\varepsilon^n\) via the Duhamel relation \[\| v(t) \|_{L^2_{x,y}} = \bigg\| \int_0^t S(t-s) r_\varepsilon^n(s) ds \bigg\|_{L^2_{x,y}} \leq \mathcal{C} \int_0^t \| e^y r_\varepsilon^n (s) \|_{H^{K}_{x,y}} ds.\] Their analysis leads eventually to an estimate that fails to hold for small \(\varepsilon> 0\) and \(t>0\). In the non-autonomous case, however, the corresponding inequality would be \[\| v(t) \|_{L^2_{x,y}} \leq \int_0^t \| S(s,t) r_\varepsilon^n(s) \|_{L^2_{x,y}} ds \leq \mathcal{C} \int_0^t \| e^y r_\varepsilon^n (s) \|_{H^{K}_{x,y}} ds,\] which holds only under the further assumption that \(S\) is uniformly continuous: \(\mathcal{C}>0\) and \(K\in \mathbb{N}\) do not change if the initial time for \(\mathcal{T}\) is replaced by \(t =s\geq 0\). Hence, any eventual contradiction based uniquely on this approach would not distinguish whether is from non-existence of solutions or from the fact that \(S\) is simply not uniformly continuous. Therefore, the second case would not improve the result of [20].

section1 -3.5ex 1.5ex Ill-posedness induced by the initial time

We present a toy model for a two parameter semigroup \(S = S(\tau, t)\) satisfying the relations introduced in 4 . For simplicity, we restrict our attention to a linear PDE in the variables \((t,x) \in \mathbb{R}_+ \times \mathbb{T}\).

We introduce an operator \(\mathcal{A}\in L^\infty(\mathbb{R}_+, \mathcal{L}(H^m(\mathbb{T}), H^{m-\frac{1}{2}}(\mathbb{T}) ))\), for any \(m \in \mathbb{R}\). Given a function \(\phi \in H^m(\mathbb{T})\) \[\phi(x) = \sum_{k\in \mathbb{Z}} \phi_{k} e^{ikx}, \qquad \sum_{k \in \mathbb{Z}} (1+k^2)^m|\phi_k|^2<+\infty,\] \([\mathcal{A}(t)](\phi) \in H^{m-\frac{1}{2}}(\mathbb{T})\) acts on \(\phi\) via the following Fourier multipliers at any time \(t\geq 0\): \[[\mathcal{A}(t) \phi](x) := \sum_{k\in \mathbb{Z}\setminus\{0\}} \sqrt{|k|} A_k(t) \phi_k e^{i kx} \quad \text{with} \quad A_k(t) := - \mathbf{1}_{\left[0, \frac{1}{2\sqrt[4]{|k|}}\right)}(t) + \mathbf{1}_{\left[\frac{1}{2\sqrt[4]{|k|}}, \frac{1}{\sqrt[4]{|k|}}\right)}(t).\] Each \(A_k\) belongs to \(L^\infty(\mathbb{R})\) and \(\| \sqrt{|k|} A_k \|_{L^\infty(\mathbb{R}_+)}\leq \sqrt{|k|}\), so that \([\mathcal{A(\cdot)}](\phi) \in L^\infty(\mathbb{R}_+,H^{m-\frac{1}{2}}(\mathbb{T}))\).

Next, given an arbitrary initial time \(\tau \geq 0\) and initial datum \(\phi_{in} \in H^m(\mathbb{T})\), we address the Cauchy problem: \[\label{eq:evol-eq} \begin{cases} \phi_\tau'(t) = \mathcal{A}(t) \phi_\tau(t)\qquad \qquad t \in [\tau , \infty) , \\ \phi_\tau(\tau ) = \phi_{in}. \end{cases}\tag{10}\] As main property: when \(\tau = 0\), each Fourier multiplier \(\sqrt{|k|}A_k(t)\) produces a short-time dissipation \(e^{-\sqrt[4]{|k|}t}\) on the mode \(\phi_{0,k}(t)\) for \(t \in [0, \,\tfrac{1}{2}|k|^{-1/4})\), followed by an opposite inflation phase in \(t \in [\tfrac{1}{2}|k|^{-1/4}, \, |k|^{-1/4})\) that restores the initial amplitude; thereafter, the solution remains constant.

The following result states that 10 is weakly well-posed in any Sobolev space for any initial time \(\tau \geq 0\).

Lemma 1. For any \(m \in \mathbb{R}\), any \(\tau \geq 0\) and any \(\phi_{in} \in H^m(\mathbb{T})\), 10 admits a unique solution \[\label{fct-space-phi} \phi_\tau \in \mathcal{C}( [\tau, +\infty), H^m(\mathbb{T}))\cap W^{1, \infty}( [\tau, +\infty), H^{m-\frac{1}{2}}(\mathbb{T})).\qquad{(2)}\] Additionally, given the constants \(C_0 := 1\) and \(C_\tau := e^{1/\tau}\), for any \(\tau >0\), the following inequality holds true: \[\label{eq:cont-wrt-initial-data} \max_{\tau \leq t \leq +\infty} \| \phi_\tau(t) \|_{H^m(\mathbb{T})} \leq C_\tau \| \phi_{\rm in} \|_{H^m(\mathbb{T})}.\qquad{(3)}\]

Thus, by setting \(S(\tau, t)(\phi_{\rm in}) := \phi_\tau(t)\), we define a (non-uniform) strongly continuous evolution family of linear bounded operator on \(H^m(\mathbb{T})\). Moreover, since \(C_0 =1\), then \(S(0, \cdot)\) satisfies \[\sup_{0 \leq t < + \infty} \big\| \,S(0,t) \, \big\|_{\mathcal{L}(H^{m}(\mathbb{T}), H^{m}\mathbb{T}))} \leq 1,\qquad \text{for any }m \in \mathbb{R},\] which is the first inequality of 4 . The next lemma establishes the second relation.

Lemma 2. For any \(\delta >0\), any \(0 \leq \sigma < 1\) and any \(m_1,\, m_2 \in \mathbb{R}\) the following identity holds true: \[\label{eq:impostor-identity} \sup_{0 \leq \tau \leq t \leq \delta} \left\| e^{-\sigma (t-\tau) \sqrt{|\partial_x|} } S(\tau,t) \right\|_{\mathcal{L}( H^{m_1}(\mathbb{T}), H^{m_2}(\mathbb{T})\big)} = + \infty.\qquad{(4)}\]

Proof of 1. We address 10 at each frequency \(k\in \mathbb{Z}\setminus\{0\}\) (the \(0\)-mode remains constant). We denote by \(\phi_{\tau, k} \in \mathcal{C}([\tau, +\infty)) \cap W^{1,\infty}(\tau, +\infty)\) the unique weak solution of the linear ODE: \[\label{eq:ODE-projected} \phi_{\tau,k}'(t) = \sqrt{|k|} A_k(t)\phi_{\tau,k}(t)\quad t \in [\tau, \infty), \qquad \qquad \phi_{\tau,k}(\tau) = \phi_{in, k},\tag{11}\] where \(\phi_{in, k} \in \mathbb{C}\) is the \(k\)-mode of \(\phi_{in }\). \(\phi_{\tau,k}\) is explicitly determined by \[\label{eq:def-Tk} \phi_{\tau,k}(t) = S_k(\tau, t) \phi_{in, k}, \quad \text{with} \quad S_k(\tau, t) := \exp \left( \sqrt{|k|} \int_\tau^t A_k(\omega) d\omega \right)\in \mathbb{R}.\tag{12}\] We next consider the time-dependent Fourier series \(\phi_\tau(t,x) := \sum_{k \in \mathbb{Z}} \phi_{\tau,k}(t) e^{i k x}\) and show that it converges in \(\mathcal{C}([\tau, \infty), H^m(\mathbb{T}))\). We first address \(\tau = 0\) and remark that, from the definition of \(S_k\) and \(A_k\), \[\begin{alignedat}{4} &t \in \left[0, \frac{1}{2\sqrt[4]{k}}\right) \quad &&\Rightarrow \quad 0 \leq S_k(0,t) = \exp \left( - t \sqrt{|k|} \right) \leq 1, \\ &t \in \left[\frac{1}{2\sqrt[4]{k}},\frac{1}{\sqrt[4]{k}}\right) \quad &&\Rightarrow \quad 0 \leq S_k(0,t) = \exp \left( \sqrt{|k|} \left( - \frac{1}{2\sqrt[4]{k}} + t - \frac{1}{2\sqrt[4]{k}} \right) \right) \leq 1, \\ &t \in \left[\frac{1}{\sqrt[4]{k}}, +\infty\right) \quad &&\Rightarrow \quad 0 \leq S_k(0,t) = 1. \end{alignedat}\] Thus \(0\leq S_k(0, t)\leq 1\) for any \(k \in \mathbb{Z}\setminus\{0 \}\) and any \(t\geq 0\). This implies in particular \[\| \phi_0(t) \|_{H^m(\mathbb{T})}^2 = \sum_{k \in \mathbb{Z}}(1+ k^2)^\frac{m}{2} \left| S_k(0,t)\phi_{k,in} \right|^2 \leq \| \phi_{in} \|_{H^m(\mathbb{T})}^2 < \infty.\] Since \(|S_k(0,t)\phi_{k,in} | \leq |\phi_{k,in}|\), the function \(\phi_0\) belongs to \(\mathcal{C}([0, \infty), H^m(\mathbb{T}))\) thanks to the dominated convergence theorem for series and the continuity of \(S_k(0,\cdot)\). We have thus established the constant \(C_0 = 1\) in ?? .

Next, we handle \(\tau>0\) and split the Fourier series \(\phi_\tau\) into two components: \[\phi_\tau (t,x ) = \phi^1_\tau (t,x ) + \phi^2_\tau (t,x ) := \sum_{|k| \leq \frac{1}{\tau^4}} S_k(\tau ,t)\phi_{k,in}e^{i k x} + \sum_{|k| > \frac{1}{\tau^4}}S_k(\tau ,t)\phi_{k,in}e^{i k x}.\] The second series \(\phi^2_\tau\) handles frequencies satisfying \(1/\sqrt[4]{|k|}<\tau\). Hence, for any \(t \geq \tau\), we obtain the identity \(S_k(\tau, t) = \exp( \sqrt{|k|}\int_\tau^t A_k(\omega) d\omega)= \exp( \sqrt{|k|}\int_\tau^t 0 \,d \omega )= 1\). Therefore \(\phi^2_\tau(t,x)\) is constant in time and satisfies \(\| \phi^2_\tau(t,\cdot) \|_{H^m(\mathbb{T})} \leq \| \phi_{in} \|_{H^m(\mathbb{T})}\).

The first sum \(\phi^1_\tau\) is finite and therefore in \(\mathcal{C}([\tau, \infty), H^m(\mathbb{T}))\). Moreover, by definition, \(A_k(t) \leq \mathbf{1}_{[0,1/\sqrt[4]{|k|})}(t)\) for any \(t \geq 0\). Thus, for any \(|k| \leq 1/\tau^{4}\), we gather that \[0 \leq S_k(\tau,t) = \exp\bigg( \sqrt{|k|} \int_\tau^tA_k(\omega) d \omega \bigg) \leq \exp\left( \sqrt{|k|} \frac{1}{\sqrt[4]{|k|}} \right) = e^{ \sqrt[4]{|k|} } \leq e^{1/\tau}.\] We have established therefore that \(\phi_\tau \in \mathcal{C}([\tau, \infty), H^m(\mathbb{T}))\) satisfies ?? .

We conclude observing that \(\partial_t \phi_\tau = [\mathcal{A}(\cdot)](\phi_\tau(\cdot)) \in L^\infty(\tau, \infty; H^{m-1/2}(\mathbb{T}))\) and that the uniqueness follows from the uniqueness of the ODE 11 for each Fourier mode. ◻

Proof of [{thm:impostor-identity}]. Let \(\delta>0\) and \(\sigma\in[0,1)\). We consider a general \(k\in\mathbb{N}\) satisfying \(\frac{1}{\sqrt[4]{k}}\leq \delta\). Then, by setting \(0 < \tau_k =\tfrac{1}{2\sqrt[4]{k}} < t_k = \tfrac{1}{\sqrt[4]{k}} \leq \delta\), we obtain \[\begin{align} \sup_{0 \leq \tau \leq t \leq \delta} \left\| e^{-\sigma (t-s) \sqrt{|\partial_x|} } S(\tau ,t) \right\|_{\mathcal{L}( H^{m_1}(\mathbb{T}), H^{m_2}(\mathbb{T})\big)} &\geq e^{-\sigma (t_k-\tau_k) \sqrt{k} } S(\tau_k,t_k)\left( 1+ k^2 \right)^\frac{m_2-m_1}{2} \\ &\geq \exp \left( -\sigma \frac{\sqrt[4]{k}}{2}\right) S_k\left( \frac{1}{2\sqrt[4]{k}}, \frac{1}{\sqrt[4]{k}} \right) \left( 1+ k^2 \right)^\frac{m_2-m_1}{2}. \end{align}\] Next, since \(A_k \equiv 1\) on the interval \([\tau_k, t_k] = \big[\frac{1}{2\sqrt[4]{|k|}},\frac{1}{\sqrt[4]{|k|}}\big]\), we obtain \[\begin{align} S_k\left( \frac{1}{2\sqrt[4]{k}}, \frac{1}{\sqrt[4]{k}} \right) = \exp \left( \sqrt{k} \int_{\frac{1}{2\sqrt[4]{k}}}^\frac{1}{\sqrt[4]{k}} A_k(\omega) d\omega \right) = \exp \left( \frac{1}{2}\sqrt[4]{k} \right). \end{align}\] Summarising, for any \(k\in\mathbb{N}\) with \(\frac{1}{\sqrt[4]{|k|}}\le \delta\), the following lower bound holds true: \[\sup_{0 \leq \tau \leq t \leq \delta} \left\| e^{-\sigma (t-\tau) \sqrt{|\partial_x|} } S(\tau,t) \right\|_{\mathcal{L}( H^{m_1}(\mathbb{T}), H^{m_2}(\mathbb{T})\big)} \geq \exp \left( \frac{1-\sigma}{2}\sqrt[4]{k} \right) \left( 1+ k^2 \right)^\frac{m_2-m_1}{2}\] Since \(\sigma\in [0, 1)\), by sending \(k\to\infty\), we obtain the relation ?? , which concludes the proof of 2. ◻

section1 -3.5ex 1.5ex The matched asymptotic expansion

We begin with the proof of 1 and 2 and devote this section to the construction of the approximate solution \(u_k^{\rm fr} = \partial_y \phi_{{\rm inn}, k}+ \partial_y \phi_{{\rm out}, k}\) as described in Part-(a) of 0.2. The first building block for the inner layer \(\phi_{{\rm inn}, k}\), is the entire function \(\Upsilon\) of 1. For simplicity, we recall its definition: \[\label{def:Upsilon-fct} \Upsilon(\zeta):= \frac{1}{\sqrt{2\pi}}\,\zeta\, e^{-\frac{\zeta^2}{2}} + \frac{1}{2} \big(1+\zeta^2\big) \left(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}} \right) \right),\tag{13}\] where \(\, \mathrm{erf}: \mathbb{C} \to \mathbb{C}\) satisfies \(\, \mathrm{erf}(0) = 0\) and \(\, \mathrm{erf}'(\zeta) = \frac{2}{\sqrt{\pi}} e^{-\zeta^2}\). In the next lemma, we establish via \(\Upsilon\) a solution of the spectral ODE for the linear Prandtl equations around a quadratic shear flow.

Lemma 3. Let \(a>0\), \(\alpha \in \mathbb{R}\), \(\beta<0\) and \(k \in \mathbb{N}\). Introduce \(\sigma_k\in \mathbb{C}\) and the change of variable \(z : \mathbb{C} \to \mathbb{C}\) \[\sigma_k:= \sqrt{|\beta | k} \, e^{-\frac{{i}\pi}{4}} - {i}k \,\alpha \in \mathbb{C} \qquad\qquad z(y) := \sqrt[4]{|\beta|}(y-a)e^{-\frac{{i}\pi}{8}} \in \mathbb{C}.\] Then, the entire function \(\phi_k(y) =\Upsilon( \sqrt[4]{k}\, z(y) ) \sqrt{\frac{|\beta|}{k}} e^{i\frac{5\pi}{4}}\) satisfies the following ODE in a classical sense: \[\label{eq:lemma-Upsilon-identity} \Big( \sigma_k + i k \,\Big( \alpha + \beta (y-a)^2 \Big) \Big) \phi_k'(y) - 2 i k\, \beta (y-a) \phi_k(y) - \phi_k'''(y) = 0,\qquad y \in \mathbb{C}.\qquad{(5)}\]

Proof. We compute the first three derivatives of \(\Upsilon\): \[\Upsilon'(\zeta) = \sqrt{\frac{2}{\pi}}e^{-\frac{\zeta^2}{2}} + \zeta\bigg(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}}\right) \bigg), \qquad \Upsilon''(\zeta) = 1+ \, \mathrm{erf}\Big( \frac{\zeta}{\sqrt{2}}\Big),\qquad \Upsilon'''(\zeta) = \sqrt{\frac{2}{\pi}} e^{-\frac{\zeta^2}{2}}.\] Hence, we remark that \[\begin{align} -\big( 1 + \zeta^2 \big) \Upsilon'(\zeta) + 2 \zeta \Upsilon(\zeta) + \Upsilon'''(\zeta) = &-\big( 1 + \zeta^2 \big)\sqrt{\frac{2}{\pi}} e^{-\frac{\zeta^2}{2}} - \big( 1 + \zeta^2 \big)\zeta \bigg(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}}\right) \bigg) + \\ &+\sqrt{\frac{2}{\pi} }\zeta^2 \,e^{-\frac{\zeta^2}{2}} + \zeta (1+\zeta^2) \bigg(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}}\right) \bigg) + \sqrt{\frac{2}{\pi}} e^{-\frac{\zeta^2}{2}} = 0. \end{align}\] By setting the substitution \(\zeta = \zeta(y) = \sqrt[4]{k}\,z(y) =\sqrt[4]{-\beta k} (y-a)e^{-\frac{i\pi}{8}}\), we recast each \(\zeta\)-derivative into: \[\frac{d}{d\zeta} = \frac{e^\frac{\rm i\pi}{8}}{\sqrt[4]{-\beta k}} \frac{d}{dy}, \qquad \frac{d^2}{d\zeta^2} = \frac{e^\frac{\rm i\pi}{4}}{\sqrt{-\beta k}} \frac{d^2}{dy^2}, \qquad \frac{d^3}{d\zeta^3} = \frac{e^{{i}\frac{\rm 3\pi}{8}}}{(-\beta k)^\frac{3}{4}} \frac{d^3}{dy^3}.\] Thus, we obtain \[\bigg( - \Big( 1 + \sqrt{-\beta k}(y-a)^2e^{-\frac{{i}\pi}{4}} \Big) \frac{e^\frac{\rm {i}\pi}{8}}{\sqrt[4]{-\beta k}} \frac{d}{dy} + 2\sqrt[4]{-\beta k}(y-a)e^{-\frac{{i}\pi}{8}} + \frac{e^{{i}\frac{\rm 3\pi}{8}}}{(-\beta k)^\frac{3}{4}} \frac{d^3}{dy^3} \bigg)\Big[ \Upsilon( \sqrt[4]{k}\, z(y) )\Big] = 0.\] Multiplying this identity by \(-(-\beta k)^{3/4} e^{-\frac{3 {i}\pi}{8}}\): \[\bigg( \Big( \sqrt{-\beta k} e^{-\frac{{i}\pi}{4}} + (-\beta k) (y-a)^2e^{-\frac{{i}\pi}{2}} \Big) \frac{d}{dy} - (-\beta k)(y-a)e^{-\frac{{i}\pi}{2}} - \frac{d^3}{dy^3} \bigg) \Big[ \Upsilon( \sqrt[4]{k}\, z(y) )\Big] = 0.\] Recalling that \(\sigma_k = \sqrt{-\beta k} e^{-\frac{{i}\pi}{4}} - {i}k \alpha\), we finally obtain \[\label{eq:lemma-Upsilon-identity-proof-sigma} \bigg( \Big( \sigma_k + {i}k\,\Big( \alpha + \beta (y-a)^2 \Big) \Big) \frac{d}{dy} - 2 {i}k\, \beta (y-a) - \frac{d^3}{dy^3}\bigg) \Big[ \Upsilon( \sqrt[4]{k}\, z(y) )\Big] = 0.\tag{14}\] Multiplying this identity by \(\sqrt{\frac{|\beta|}{k}} e^{\frac{5\pi}{4}i}\) concludes the proof of the lemma. ◻

Building upon this \(\Upsilon\)-profile, we next define the inner and outer layers \(\phi_{{\rm inn}, k}\) and \(\phi_{{\rm out}, k}\). We invoke the quadratic assumption (H3) of 1 for \(u_{s|t = 0}= U_s\): \[U_s(y) = U_s(a_0) + \frac{U_s''(a_0)}{2} (y-a_0)^2,\qquad\text{with}\quad U_s''(a_0) <0,\qquad \text{for any }y \in \left[ a_0 - d, a_0 + d\right].\] Similarly as in [20], recalling that \(u_s\) satisfies the heat equation 1 , we track the evolution of the critical point \(y = a_0\) by solving the nonlinear ODE: \[\label{def:a40t41} \partial_y^3 u_s(t,a(t)) \,+\,a'(t) \,\partial_y^2 u_s(t,a(t)) = 0,\qquad a(0) = a_0,\qquad t \in [0,T_{\rm max}].\tag{15}\] The Cauchy–Lipschitz theorem guaranties the existence of a local-in-time smooth solution \(a \in \mathcal{C}^\infty([0,T_{\text{max}}])\). Without loss of generality, we can restrict \(T_{\text{max}} > 0\) to be finite and sufficiently small to ensure: \[\label{eq:assumptions-for-Tmax} a(t) \in \left[ a_0 - \frac{d}{8},\, a_0 + \frac{d}{8} \right] \quad \text{and} \quad \partial_y^2 u_s(t,a(t)) \in \Big[2U_s''(a) ,\, \frac{U_s''(a_0)}{2}\Big], \quad \text{for any}\quad t \in [0, T_{\rm max}] .\tag{16}\] Moreover, to simplify some estimates, we assume from the continuity in time of \(\| \partial_y u_s(t, \cdot) \|_{L^2(a_0+\frac{d}{2}, \infty)}\) that \[\label{eq:assumption-for-Tmax-and-partialyus} \| \partial_y u_s(t, \cdot )\|_{L^2(a_0+\frac{d}{2}, \infty)} \geq \frac{1}{2} \| U_s'\|_{L^2(a_0+\frac{d}{2}, \infty)}>0,\qquad \text{for any }t \in [0, T_{\rm max}].\tag{17}\] We recall from 1 that \(\chi \in \mathcal{C}^\infty_c(\mathbb{R}_+)\) is a cutoff function satisfying the relations: \[\chi(y) = 1 \quad \text{for any }y \in \left[a_0-\frac{d}{4},\, a_0+\frac{d}{4}\right], \qquad \chi(y) = 0 \quad \text{for any }y \in \mathbb{R}_+\setminus \left]a_0-\frac{d}{2},\, a_0+\frac{d}{2} \right[.\]

Definition 3. For each frequency \(k \in \mathbb{N}\), define \(z \in \mathcal{C}^\infty([0, T_{\rm max}] \times \mathbb{R}_+ , \mathbb{C})\) and \(\sigma_k \in \mathcal{C}^\infty([0, T_{\rm max}], \mathbb{C})\): \[\label{eq:z-sigmak-in-def-of-phi-inn-out} z(t,y) :=(y-a(t)) e^{-\frac{{i}\pi}{8}} \sqrt[4]{\frac{|\partial_y^2u_s(t,a(t))|}{2}} \in \mathbb{C}, \quad \quad \sigma_k(t):= \sqrt{k} \,e^{-\frac{{i}\pi}{4}} \sqrt{\frac{|\partial_y^2u_s(t,a(t))|}{2}} - {i}k \, u_s( t,a(t))\in \mathbb{C},\qquad{(6)}\] We define the inner and outer layers at any \((t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+\) as \[\label{eq:phi-inn-out} \begin{align} (i)\quad &\phi_{{\rm inn}, k}(t,y) := \chi(y) \Upsilon( \sqrt[4]{k} \, z(t,y) ) \, e^{{i}\frac{5\pi}{4}} \sqrt{\frac{|\partial_y^2u_s(t,a(t))|}{2k}} \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg) ,\\ (ii)\quad &\phi_{{\rm out}, k}(t,y) := \big(1 - \chi(y)\big) H\big(y-a(t)\big) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg) , \end{align}\qquad{(7)}\] where \(H\) denotes the Heaviside function.

Remark 2. Under (H1)-(H2) of 1, we recall that \(u_s \in \mathcal{C}([0, \infty[, H^{m+1}_\lambda(\mathbb{R}_+))\) with \(m\geq 3\). Hence \[\label{def:phi-inn-out-regularities} \begin{align} &\phi_{{\rm inn}, k} \in \mathcal{C}^\infty_c\big(\,[0, T_{\rm max}] \times ]0, \infty[\,\,\big),\quad \phi_{\rm out, k}\in \mathcal{C}^\infty(\,]0, T_{\rm max}] \times [0, \infty[\,)\cap \mathcal{C}([0, T_{\rm max}], H_\lambda^{m+1}(\mathbb{R}_+) \oplus \mathbb{C}), \end{align}\qquad{(8)}\] Furthermore, the associated \(u^{\rm fr}_k := \partial_y ( \phi_{{\rm inn}, k} + \phi_{{\rm out}, k} ) \in \mathcal{C}^\infty(\,]0, T_{\rm max}] \times [0, \infty[\,)\) belongs to \[\label{eq:ukfr-vkfr-def-and-spaces} u^{\rm fr}_k \in \mathcal{C}([0, T_{\rm max}], H_\lambda^m(\mathbb{R}_+)) \cap L^2(0, T_{\rm max}; H^{m+1}_\lambda(\mathbb{R}_+)) \cap W^{1,2}(0, T_{\rm max}; H^{m-1}_\lambda(\mathbb{R}_+)).\qquad{(9)}\] Notably, \((1-\chi(y))\) is zero around \(y = a(t)\), eliminating any discontinuities introduced by the Heaviside function. The boundary conditions \(u_k^{\rm fr}(t,0) = \phi_{{\rm inn}, k}(t, 0)= \phi_{{\rm out}, k}(t, 0)= 0\), for any \(t \in [0, T_{\rm max}]\), are also satisfied.

The next proposition determines the forcing term generated by \(u_k^{\text{fr}}\) when inserted into System 5 . For the sake of abbreviation, we introduce the short notation \(\alpha,\, \beta,\,\gamma \in \mathcal{C}^\infty([0, T_{\rm max}])\) with \[\label{eq:notation-alpha-beta-gamma} \alpha(t):= u_s(t,a(t)),\qquad \beta(t) := \frac{1}{2}\partial_y^2 u_s(t,a(t)),\qquad \gamma(t) := \sqrt[4]{-\beta(t)} = \sqrt[4]{\frac{|\partial_y^2 u_s(t,a(t))|}{2}}>0,\tag{18}\] which implies \(z(t,y) = \gamma(t) (y-a(t))e^{-\frac{i\pi}{8}}\), thanks to ?? .

Proposition 1. For any frequency \(k \in \mathbb{N}\), \(u_k^{\rm fr}\) satisfies the following system in a classical sense: \[\begin{align} \left\{ \begin{alignedat}{4} &\Big( \partial_t + {i}k\, u_s(t,y) - \partial_y^2 \Big) u_k^{\rm fr}(t,y) - {i}k\, \partial_y u_s(t,y) \int_0^y u_k^{\rm fr}(t, \omega)d \omega = f_k(t,y) e^{ \int_0^t \sigma_k(\tau) d \tau} ,\qquad && (t,y) \in \,]0, T_{\rm max}[ \times \mathbb{]}0, \infty[, \\ &\,u_k^{\rm fr}(t,0) = \lim_{y\to +\infty} u_k^{\rm fr}(t,y) = 0 &&t \in [0, T_{\rm max}], \\ &\,u_k^{\rm fr}(0,y) = \partial_y \phi_{\rm inn, k}(0,y) + \partial_y \phi_{\rm out, k}(0,y) &&y \in ]0, \infty[, \end{alignedat} \right. \end{align}\] where \(f_k \in \mathcal{C}^\infty_c([0, T_{\rm max}] \times \mathbb{R}_+, \mathbb{C})\) is the sum \[f_k(t,y) := \sum_{j = 1}^7\mathcal{R}^1_{j,k}(t,y) + \sum_{j = 1}^3\mathcal{R}^2_{j,k}(t,y) ,\qquad (t,y) \in [0, T_{\rm max}] \times [0, \infty[\] of two family of remainders:

The remainders \[\label{eq:first-family-remainders} \begin{align} \mathcal{R}^1_{1,k}(t,y) &= \bigg[\, u_s(t,y) - \alpha(t) - \beta(t) (y-a(t))^2 \,\bigg] \chi (y) \Upsilon'(\sqrt[4]{k}\,z(t,y) ) \gamma(t)^3 e^{{i}\frac{13\pi}{8}} k^\frac{3}{4} , \\ \mathcal{R}^1_{2,k}(t,y) &= \bigg[ u_s(t,y) - \alpha(t) - \beta(t) (y-a(t))^2 \bigg] \Big( \chi'''(y) - \Big( \sigma_k(t) + {i}k \, u_s(t,y) \Big) \chi'(y) \Big) H(y-a(t)), \\ \mathcal{R}^1_{3,k}(t,y) &= \bigg[ \partial_y u_s(t,y) - 2\beta(t)(y-a(t)) \bigg] \chi(y) \gamma(t)^2 \Upsilon( \sqrt[4]{k} \, z(t,y)) e^{{i}\frac{3\pi}{4}} \sqrt{k}, \\ \mathcal{R}^1_{4,k}(t,y) &= \Big[ \partial_y u_s(t,y) - 2\beta(t)(y-a(t)) \Big] 3 \chi''(y) H(y-a(t)), \\ \mathcal{R}^1_{5,k}(t,y) &= \bigg[\partial_y^2 u_s(t,y) -2\beta(t) \bigg] 2\chi'(y) H(y-a(t)), \\ \mathcal{R}^1_{6,k}(t,y) &=\gamma'(t) \bigg( \chi'(y) \frac{2\gamma(t)e^{{i}\frac{5\pi}{4}}}{\sqrt{k}} \Big( \Upsilon(\sqrt[4]{k} \,z(t,y)) - H(y-a(t)) \Big) + \chi(y) \frac{3\gamma(t)^2e^{{i}\frac{9\pi}{8}}}{\sqrt[4]{k}} \Upsilon'(\sqrt[4]{k} \,z(t,y)) \bigg), \\ \mathcal{R}^1_{7,k}(t,y) &= \Big[ \gamma'(t)(y-a(t)) - \gamma(t) a'(t) \Big] \bigg( \chi'(y) \frac{e^{{i}\frac{9\pi}{8}}\gamma(t)^2}{\sqrt[4]{k}} \Upsilon'(\sqrt[4]{k} \,z(t,y)) - \chi(y) \gamma(t)^3 \Upsilon''(\sqrt[4]{k} \,z(t,y)) \bigg). \end{align}\qquad{(10)}\]
The remainders \[\label{eq:second-family-remainders} \begin{align} \mathcal{R}^2_{1,k}(t,y) &= \bigg[ \, \Upsilon(\sqrt[4]{k}\,z(t,y)) - \big( 1 + \sqrt{k}\, z(t,y)^2\, \big) H(y-a(t))\, \bigg] \Big( \big( \sigma_k(t) + {i}k\, u_s(t,y) \big)\chi'(y) - \chi'''(y) \Big) \frac{e^{{i}\frac{5\pi}{4}}\gamma(t)^2 }{\sqrt{k}} , \\ \mathcal{R}^2_{2,k}(t,y) &= \bigg[\, \Upsilon'(\sqrt[4]{k}\,z(t,y)) - 2 \sqrt[4]{k} \,z(t,y) H(y-a(t)) \,\bigg] \big( - 3 \chi''(y) \big) \frac{e^{{i}\frac{9\pi}{8}}\gamma(t)^3 }{\sqrt[4]{k}} , \\ \mathcal{R}^2_{3,k} (t,y) &= \bigg[\, \Upsilon''(\sqrt[4]{k}\,z(t,y))-2 H(y-a(t)) \,\bigg] \big( -3 \chi'(y) \big)\beta(t) . \end{align}\qquad{(11)}\]

The leading terms are isolated in the first (squared) brackets. We note that each remainder depends on the cutoff function \(\chi\) or its derivatives, supported in \([0, T_{\text{max}}] \times [a_0 - \tfrac{d}{2}, a_0 + \tfrac{d}{2}]\). Furthermore, since \(u_s\) is smooth in this region (\(U_s\) is quadratic), it follows that \(f \in \mathcal{C}_c^\infty([0, T_{\text{max}}] \times \,]0, \infty[)\).

The family \(\mathcal{R}_{1,k}^1, \dots, \mathcal{R}_{7,k}^1\) depends mainly on the deviation of \(u_s\) from its second-order Taylor expansion centered at \(y = a(t)\). As discussed in 0.2, these are therefore the \(\mathcal{O}(e^{-c_2/t})\)-remainders.

The second family \(\mathcal{R}_{1,2}^2\), \(\mathcal{R}_{1,2}^2\) and \(\mathcal{R}_{1,2}^3\) has leading-order terms governed by the interaction between the function \(\Upsilon\) and the Heaviside function \(H\). Eventually, these yield estimates of the form \[\bigg| \, \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \, \bigg| \lesssim e^{ -\sqrt{k} \,(y-a(t))^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}\] Since \(\chi' \equiv 0\) on the interval \([a_0 - \tfrac{d}{4}, a_0 + \tfrac{d}{4}]\) and given our assumption that \(a(t) \in [a_0 - \tfrac{d}{8}, a_0 + \tfrac{d}{8}]\), we conclude that these are the \(\mathcal{O}(e^{-c_3\sqrt{k}})\)-remainders anticipated in 0.2. We formalise these estimates in 2.

Proof of 1. We begin with the following identity form Lemma 3 (cf. 14 ): \[\Bigg[ \bigg( \sigma_k(t) + {i}k \Big( \alpha(t) + \beta(t) (y-a(t))^2 \Big) \bigg) \partial_y -2 i k \,\beta(t) (y-a(t)) - \partial_y^3 \Bigg] \Big( \Upsilon(\sqrt[4]{k}\,z(t,y)) \Big) = 0,\] for any \((t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+\). Next, we replace the quadratic terms with the shear flow \(u_s\) and move any remaining terms to the right-hand side: \[\label{eq:Upsilon-eq-in-us-with-remainders} \begin{align} \bigg[ \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \partial_y -{i}k \,\partial_y u_s(t,y) &- \partial_y^3 \bigg] \Big( \Upsilon(\sqrt[4]{k}\,z(t,y)) \Big) = \\ = {i}k \, \Big( u_s(t,y) - \alpha(t) -\beta(t)(y&-a(t))^2 \Big) \Upsilon'(\sqrt[4]{k}\,z(t,y))\sqrt[4]{k} \,\gamma(t) e^{-\frac{{i}\pi}{8}} +\\ &- {i}k \, \Big( \partial_y u_s(t,y) - 2\beta(t)(y-a(t)) \Big) \Upsilon(\sqrt[4]{k}\,z(t,y)), \end{align}\tag{19}\] using \(z(t,y) = \gamma(t)(y-a(t))e^{-\frac{{i}\pi}{8}}\), which implies \(\partial_y z(t,y) = \gamma(t) e^{-\frac{{i}\pi}{8}}\) Recall from ?? that the inner layer is \[\phi_{\text{inn}, k}(t,y) = \chi(y) \Upsilon(\sqrt[4]{k}\,z(t,y)) \frac{e^{{i}\frac{5\pi}{4}} \gamma(t)^2}{\sqrt{k}} \exp \bigg( \int_0^t \sigma_k(\tau) d \tau\bigg), \qquad (t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+.\] Thus, multiplying the identity 19 by \(\chi(y) \frac{e^{{i}\frac{5\pi}{4}}\gamma(t)^2}{\sqrt{k}} e^{\int_0^t \sigma_k(\tau)d\tau}\) and applying the product rule, we obtain: \[\begin{align} \bigg[ \Big( \sigma_k(t) + {i}k\, u_s(t,y) &\Big) \partial_y -{i}k \,\partial_y u_s(t,y) - \partial_y^3 \bigg] \phi_{\text{inn}, k}(t,y) = \\ = e^{\int_0^t \sigma_k(\tau)d\tau } \Bigg\{ & \underbrace{ \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) - \chi'''(y) \bigg) \frac{e^{{i}\frac{5\pi}{4}} \gamma(t)^2}{\sqrt{k}} \Upsilon(\sqrt[4]{k}\,z(t,y)) }_{=:\mathcal{I}_1} \,+ \\ & \underbrace{ - 3 \chi''(y) \frac{e^{{i}\frac{9\pi}{8}}\gamma(t)^3}{\sqrt[4]{k}}\Upsilon'(\sqrt[4]{k}\,z(t,y)) - 3 \chi'(y) \beta(t) \Upsilon''(\sqrt[4]{k}\,z(t,y)) }_{=:\mathcal{I}_2}\,+ \\ & + \underbrace{ {i}k \, \Big( u_s(t,y) - \alpha(t) -\beta(t)(y-a(t))^2 \Big) \chi(y) \frac{e^{{i}\frac{9\pi}{8}}\gamma(t)^3}{\sqrt[4]{k}} \Upsilon'(\sqrt[4]{k}\,z(t,y)) }_{= \mathcal{R}_{1,k}^1(t,y)}\,+\\ & \underbrace{- {i}k \, \Big( \partial_y u_s(t,y) - 2\beta(t)(y-a(t)) \Big) \chi(y) \frac{e^{{i}\frac{5\pi}{4}}\gamma(t)^2}{\sqrt{k}} \Upsilon(\sqrt[4]{k}\,z(t,y)) }_{= \mathcal{R}_{3,k}^1(t,y)} \Bigg\}. \end{align}\] In the third line with \(\mathcal{I}_2\), we used \(e^{{i}\frac{5\pi}{4}}\gamma(t)^4e^{-\frac{{i}\pi}{4}} = - \gamma(t)^4 = \beta(t)\). Additionally, we identified the remainder terms \(\mathcal{R}_{1,k}^1(t,y)\) and \(\mathcal{R}_{3,k}^1(t,y)\) in the last two lines, thanks to the identity \({i}k \frac{e^{{i}\frac{9\pi}{8}}}{\sqrt[4]{k}} =e^{{i}\frac{13\pi}{8}} k^\frac{3}{4}\), as well as \(- {i}k \frac{e^{{i}\frac{5\pi}{4}}}{\sqrt{k}} = e^{{i}\frac{3\pi}{4}}\sqrt{k}\). Using further the notation introduced in ?? and ?? , we decompose \(\mathcal{I}_1\) and \(\mathcal{I}_2\) as: \[\begin{align} \mathcal{I}_1 &= \mathcal{R}_{1, k}^2(t,y) + \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) - \chi'''(y) \bigg) \bigg( \frac{ \gamma(t)^2 e^{{i}\frac{5\pi}{4}} }{\sqrt{k}} + \beta(t)(y-a(t))^2 \bigg) H(y-a(t)), \\ \mathcal{I}_2 &= \mathcal{R}_{2, k}^2(t,y) + \mathcal{R}_{3, k}^2(t,y) - 6\chi'(y) \beta(t)H(y-a(t)) - 6\chi''(y) \beta(t) (y-a(t)) H(y-a(t)) , \end{align}\] where we have also used the fact that \(\gamma(t)^4 = - \beta(t)\) and \(z(t,y) = \gamma(t)(y-a(t))e^{-\frac{i\pi}{8}}\), so that \[\begin{align} \big( 1+ \sqrt{k} \, z(t,y)^2 \big) \frac{e^{i \frac{5 \pi}{4}}\gamma(t)^2}{\sqrt{k}}H(y-a(t)) &= \bigg( \frac{e^{i \frac{5 \pi}{4}}\gamma(t)^2}{\sqrt{k}} + \frac{e^{i \frac{5 \pi}{4}}\gamma(t)^2}{\sqrt{k}} \sqrt{k} \, e^{-\frac{i\pi}{4} }\gamma(t)^2 (y-a(t))^2 \bigg) H(y-a(t))\\ &= \bigg( \frac{e^{i \frac{5 \pi}{4}}\gamma(t)^2}{\sqrt{k}} + \beta(t) (y-a(t))^2 \bigg) H(y-a(t)) \end{align}\] and similarly \[\begin{align} 2 \sqrt[4]{k} \,z(t,y) H(y-a(t)) \frac{e^{i \frac{9\pi}{8}}\gamma(t)^3}{\sqrt[4]{k}} =2 \gamma(t)^4 (y-a(t)) e^{-\frac{i \pi}{8}}e^{i \frac{9\pi}{8}}H(y-a(t)). \end{align}\] This yields the identity: \[\begin{align} \bigg[ \Big( \sigma_k(t) + i k\, u_s(t,y) \Big) \partial_y -i k \,\partial_y u_s(t,y) - \partial_y^3 \bigg] \phi_{\text{inn}, k}(t,y) = e^{\int_0^t \sigma_k(\tau)d\tau } \Bigg\{ \mathcal{R}_{1, k}^1(t,y) + \mathcal{R}_{3, k}^1(t,y) + \mathcal{R}_{1, k}^2(t,y) + \\ + \mathcal{R}_{2, k}^2(t,y) + \mathcal{R}_{3, k}^2(t,y) \Bigg\} + \Bigg\{ \bigg( \Big( \sigma_k(t) + i k\, u_s(t,y) \Big) \chi'(y) - \chi'''(y) \bigg) \bigg( \frac{ \gamma(t)^2 e^{i\frac{5\pi}{4}} }{\sqrt{k}} + \beta(t)(y-a(t))^2 \bigg) +\\ - 6\chi''(y) \beta(t) (y-a(t)) - 6\chi'(y) \beta(t) \Bigg\} H(y-a(t))e^{\int_0^t \sigma_k(\tau)d\tau }. \end{align}\] Finally, we replace the multiplication by \(\sigma_k(t)\) on the left-hand side with the time derivative and move any additional term to the right-hand side. Observing that \[\begin{align} \partial_t \partial_y\phi_{\text{inn}, k}(t,y) &- \sigma_k(t) \partial_y\phi_{\text{inn}, k}(t,y) = \partial_t \partial_y \bigg( \chi(y) \Upsilon(\sqrt[4]{k}\,z(t,y)) \frac{e^{{i}\frac{5\pi}{4}} \gamma(t)^2}{\sqrt{k}} \bigg) e^{\int_0^t \sigma_k(\tau)d\tau } \\ &= \partial_t \bigg( \chi'(y) \Upsilon(\sqrt[4]{k}\,z(t,y)) \frac{e^{{i}\frac{5\pi}{4}} \gamma(t)^2}{\sqrt{k}} + \chi(y) \Upsilon'(\sqrt[4]{k}\,z(t,y)) \frac{e^{{i}\frac{9\pi}{8}} \gamma(t)^3}{\sqrt[4]{k}} \bigg) e^{\int_0^t \sigma_k(\tau)d\tau } \\ &= \gamma'(t) \bigg( \chi'(y) \frac{e^{{i}\frac{5\pi}{4}} 2\gamma(t)}{\sqrt{k}} \Upsilon(\sqrt[4]{k}\,z(t,y)) + \chi(y) \Upsilon'(\sqrt[4]{k}\,z(t,y)) \frac{e^{{i}\frac{9\pi}{8}} 3\gamma(t)^2}{\sqrt[4]{k}} \bigg) e^{\int_0^t \sigma_k(\tau)d\tau } +\\ & + \partial_t z(t,y) e^{\frac{{i}\pi}{8}} \bigg( \chi'(y) \frac{e^{{i}\frac{9\pi}{8}} \gamma(t)^2}{\sqrt[4]{k}} \Upsilon'(\sqrt[4]{k}\,z(t,y)) - \chi(y) \gamma(t)^3 \Upsilon''(\sqrt[4]{k}\,z(t,y)) \bigg) e^{\int_0^t \sigma_k(\tau)d\tau } \\ &= \bigg( \mathcal{R}_{6, k}^1(t,y) + 2\gamma'(t)\frac{e^{{i}\frac{5\pi}{4}}\gamma(t) }{\sqrt{k}}H(y-a(t)) + \mathcal{R}_{7, k}^1(t,y) \bigg) e^{\int_0^t \sigma_k(\tau)d\tau }, \end{align}\] we gather finally \[\label{eq:proof-eq-of-phiinn} \begin{align} \bigg[ \Big( \partial_t + {i}k\, u_s \Big) \partial_y -{i}k \,\partial_y u_s - \partial_y^3 \bigg] \phi_{\text{inn}, k}(t,y) = e^{\int_0^t \sigma_k(\tau)d\tau } \Bigg\{ \mathcal{R}_{1, k}^1(t,y)+ \mathcal{R}_{3, k}^1(t,y)+ \mathcal{R}_{6, k}^1(t,y)+ \mathcal{R}_{7, k}^1(t,y)+\\ + \mathcal{R}_{1, k}^2(t,y)+ \mathcal{R}_{2, k}^2(t,y)+ \mathcal{R}_{3, k}^2(t,y) \bigg\} + F(t,y) e^{\int_0^t \sigma_k(\tau)d\tau }, \end{align}\tag{20}\] where the term \(F\) will cancel out with an analogous contribution of the outer profile and it is given by: \[\label{eq:F-proof-prop} \begin{align} F(t,y) := \Bigg\{ \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) &- \chi'''(y) \bigg) \bigg( \frac{ \gamma(t)^2 e^{{i}\frac{5\pi}{4}} }{\sqrt{k}} + \beta(t)(y-a(t))^2 \bigg) \,+\\ & - 6\chi''(y) \beta(t) (y-a(t)) - 6\chi'(y) \beta(t) + 2\gamma'(t)\frac{e^{{i}\frac{5\pi}{4}}\gamma(t) }{\sqrt{k}} \Bigg\} H(y-a(t)). \end{align}\tag{21}\] We now focus on the outer layer \(\phi_{{\rm out}, k}\). Our starting point is the identity \[\label{eq:identity-for-exact-sol-large-y-prop} \bigg[ \Big( \partial_t + {i}k \, u_s(t,y) \Big)\partial_y - {i}k\, \partial_y u_s(t,y) - \partial_y^3 \bigg] \left[ \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) e^{ \int_0^t \sigma_k(\tau) d \tau} \right] = 0,\qquad (t,y) \in [0, T_{\rm max}]\times \mathbb{R}_+.\tag{22}\] Recall from ?? that \(\phi_{{\rm out}, k}\) is defined as \[\phi_{{\rm out}, k}(t,y) = \big(1 - \chi(y)\big) H\big(y-a(t)\big) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) e^{ \int_0^t \sigma_k(\tau) d \tau}.\] We hence apply the following product rule \[\begin{align} \big(1 - \chi(y)\big) H\big(y-a(t)\big) & \bigg[ \Big( \partial_t + {i}k \, u_s(t,y) \Big)\partial_y \bigg] \bigg[ \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg)e^{ \int_0^t \sigma_k(\tau) d \tau} \bigg] = \\ &= \Big( \partial_t + {i}k \, u_s(t,y) \Big) \bigg[ \big(1 - \chi(y)\big) H\big(y-a(t)\big) \partial_y \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg)e^{ \int_0^t \sigma_k(\tau) d \tau} \bigg], \end{align}\] since \(\partial_t [(1-\chi(y))H(y-a(t))] = 0\) for any \((t,y) \in [0, T_{\rm max}]\times \mathbb{R}_+\). Additionally, by bringing the \(\partial_y\) derivative outside the brackets, we can rewrite the last expression as: \[\begin{align} \big(1 &- \chi(y)\big) H\big(y-a(t)\big) \bigg[ \Big( \partial_t + {i}k \, u_s(t,y) \Big)\partial_y \bigg] \bigg[ \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg)e^{ \int_0^t \sigma_k(\tau) d \tau} \bigg] = \\ & = \Big( \partial_t + {i}k \, u_s(t,y) \Big)\partial_y \phi_{\text{out}, k}(t,y) + \Big( \partial_t + {i}k \, u_s(t,y) \Big) \bigg[ \chi'(y) H\big(y-a(t)\big) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg)e^{ \int_0^t \sigma_k(\tau) d \tau} \bigg]. \end{align}\] We multiply 22 by the factor \((1-\chi(y))H(y-a(t))\), commuting this factor with the third-order derivative \(\partial_y^3\) \[\begin{align} \bigg[ \Big( \partial_t &+ {i}k \, u_s(t,y) \Big)\partial_y - {i}k\, \partial_y u_s(t,y) - \partial_y^3 \bigg] \phi_{\text{out,k}}(t,y) = \\ &= - \Big(\partial_t + {i}k u_s(t,y) \Big) \bigg[ \chi'(y) H(y-a(t)) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) e^{\int_0^t \sigma_k(\tau)d\tau } \bigg] + \\ & + e^{\int_0^t \sigma_k(\tau)d\tau } H\big(y-a(t)\big) \Bigg\{ \chi'''(y) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) +3 \chi'(y) \partial_y^2 u_s(t,y) +3 \chi''(y) \partial_y u_s(t,y) \Bigg\}. \end{align}\] We can further rearrange this identity to isolate the time derivative: \[\label{eq:id1-for-phi-out} \begin{align} \bigg[ \Big( \partial_t + {i}k \, u_s (t,y) \Big)\partial_y - {i}k\, \partial_y u_s (t,y) - \partial_y^3 \bigg] \phi_{\text{out,k}}(t,y) = e^{\int_0^t \sigma_k(\tau)d\tau } H(y-a(t)) \Bigg\{ - \chi'(y) \partial_t\bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) + \\ -\, \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) -\chi'''(y) \bigg) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) +3 \chi'(y) \partial_y^2 u_s(t,y) +3 \chi''(y) \partial_y u_s(t,y) \Bigg\}. \end{align}\tag{23}\] To develop the first term on the right-hand side, we expand \(\tfrac{\sigma_k(t)}{{i}k} =\tfrac{e^{{i}\frac{5\pi}{4}}\gamma(t)^2}{\sqrt{k}} -\alpha(t) = \tfrac{e^{{i}\frac{5\pi}{4}} \gamma(t)^2}{\sqrt{k}} -u_s(t, a(t))\). Furthermore, since \(u_s\) is a solution to the heat equation, this allows us to express the time derivative as follows: \[\begin{align} \partial_t \Big( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \Big) &= \partial_t u_s(t,y) + \frac{d}{dt}\left[\frac{e^{i\frac{5\pi}{4}}\gamma(t)^2}{\sqrt{k}} - u_s(t,a(t)) \right] \\ &= \partial_y^2 u_s(t,y) + \frac{e^{{i}\frac{5\pi}{4}}2\gamma(t)\gamma'(t)}{\sqrt{k}} - (\partial_t u_s)(t,a(t)) - \underbrace{\partial_y u_s(t,a(t))}_{ = 0}a'(t), \\ & = \partial_y^2 u_s(t,y) - \partial_y^2 u_s(t,a(t)) + \frac{e^{{i}\frac{5\pi}{4}}2\gamma(t)\gamma'(t)}{\sqrt{k}} \\ & = \partial_y^2 u_s(t,y) - 2\beta(t) + \frac{e^{{i}\frac{5\pi}{4}}2\gamma(t)\gamma'(t)}{\sqrt{k}}. \end{align}\] Substituting this result into the identity 23 , we reorder the right-hand side of the equation as follows: \[\begin{align} \bigg[ \Big( \partial_t + {i}k \, u_s (t,y) \Big)\partial_y - {i}k\, \partial_y u_s (t,y) - \partial_y^3 \bigg] \phi_{\text{out,k}}(t,y) = e^{\int_0^t \sigma_k(\tau)d\tau } H(y-a(t)) \Bigg\{ - \chi'(y) \Big( \partial_y^2 u_s(t,y) - 2\beta(t) \Big) + \\ - \chi'(y) \frac{e^{{i}\frac{5\pi}{4}}2\gamma(t)\gamma'(t)}{\sqrt{k}} - \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) -\chi'''(y) \bigg) \bigg( u_s(t,y) -\alpha(t) -\beta(t) (y-a(t))^2 \bigg) \,+ \\ - \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) -\chi'''(y) \bigg) \bigg( \frac{\gamma(t)^2 e^{{i}\frac{5\pi}{4}}}{\sqrt{k}} + \beta(t) (y-a(t))^2 \bigg) + 3 \chi'(y) \Big( \partial_y^2 u_s(t,y) -2\beta(t) \Big) + \\ +6 \chi'(y) \beta(t) + \Big( \partial_y u_s(t,y) - 2\beta(t) (y-a(t)) \Big) 3 \chi''(y) + 6 \chi''(y) \beta(t) (y-a(t)) \Bigg\}. \end{align}\] We invoke once more the notation for the remainder terms as defined in ?? and ?? : \[\begin{align} \bigg[ \Big( \partial_t + {i}k \, u_s (t,y) \Big)\partial_y - {i}k\, \partial_y u_s (t,y) - \partial_y^3 \bigg] \phi_{\text{out,k}}(t,y) = e^{\int_0^t \sigma_k(\tau)d\tau } \Bigg\{ \mathcal{R}_{5, k}^1(t,y) - \chi'(y) \frac{e^{{i}\frac{5\pi}{4}}2\gamma(t)\gamma'(t)}{\sqrt{k}} + \mathcal{R}_{2,k}^1(t,y) \\ - \bigg( \Big( \sigma_k(t) + {i}k\, u_s(t,y) \Big) \chi'(y) -\chi'''(y) \bigg) \bigg( \frac{\gamma(t)^2 e^{{i}\frac{5\pi}{4}}}{\sqrt{k}} + \beta(t) (y-a(t))^2 \bigg) H(y-a(t)) + 6\chi'(y)\beta(t)H(y-a(t)) \\ + \mathcal{R}_{4, k}^1(t,y) + 6 \chi''(y) \beta(t) (y-a(t))H(y-a(t)) \Bigg\}. \end{align}\] From the definition of \(F\) in 21 , this leads us to the following simplified form: \[\label{eq:proof-eq-of-phiout} \begin{align} \bigg[ \Big( \partial_t + {i}k \, u_s (t,y) \Big)\partial_y &- {i}k\, \partial_y u_s (t,y) - \partial_y^3 \bigg] \phi_{\text{out,k}}(t,y) = \\ &= e^{\int_0^t \sigma_k(\tau)d\tau } \Bigg\{ \mathcal{R}_{2, k}^1(t,y)+ \mathcal{R}_{4, k}^1(t,y)+ \mathcal{R}_{5, k}^1(t,y) \bigg\} -F(t,y)e^{\int_0^t \sigma_k(\tau)d\tau }. \end{align}\tag{24}\] Finally, we sum Equation 20 for the inner layer and 24 for the outer layer. This yields \[\begin{align} \bigg[ \Big( \partial_t + {i}k \, u_s (t,y) \Big)\partial_y - {i}k\, \partial_y u_s (t,y) - \partial_y^3 \bigg] \Big( \phi_{\text{inn,k}}(t,y) + \phi_{\text{out,k}}(t,y) \Big) = e^{\int_0^t \sigma_k(\tau)d\tau } \bigg( \sum_{j = 1}^7 \mathcal{R}_{j, k}^1 + \sum_{j = 1}^3 \mathcal{R}_{j, k}^2 \bigg), \end{align}\] which concludes the proof of the proposition. ◻

section1 -3.5ex 1.5ex Estimates of the remainders In the next proposition, we establish some estimates for the forced solution \(u_k^{\rm fr}\) and its associated remainders along those described by Part-(a) of 0.2.

Proposition 2. Consider \(u^{\rm fr}_k = \partial_y (\phi_{{\rm inn}, k} +\phi_{{\rm out}, k} )\) of 3 and the two families of remainders \(\mathcal{R}_{1,k}^1,\dots,\mathcal{R}_{7,k}^1\) and \(\mathcal{R}_{1,k}^2, \mathcal{R}_{2,k}^2, \mathcal{R}_{3,k}^2\) introduced in 1. Define the positive constants \[\begin{align} {4} C_{\mathcal{R}} &= C_{\mathcal{R}}\big( U_s,\, d,\,\chi \big) &&:= 10^2 \bigg( 1 + \frac{2}{|U_s''(a_0)|} \bigg) \|\chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4, \\ \mathcal{D}_m &= \mathcal{D}_m \big( U_s,\, d,\,\chi \big) &&:= m! \,2^{m+2} ( 1 + d )^{m+1} \Big( 1 + \|U_s\|_{H^m_\lambda}+\|U_s\|_{W^{3,1}} \Big)^m \Big( 1 + \|\chi \|_{W^{m+1,\infty}} \Big). \end{align}\] Then the following results hold true:

For any \(t\in [0, T_{\rm max}]\) and any \(k\in \mathbb{N}\) \[\label{eq:EstimateRemainderFirstKind} \sum_{j = 1}^7 \big\|\, \mathcal{R}_{j,k}^1(t, \cdot) \,\big\|_{L^2(\mathbb{R}_+)} \leq C_{\mathcal{R}}\, k \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}}.\qquad{(12)}\]
For any \(t\in [0, T_{\rm max}]\) and any \(k\in \mathbb{N}\) \[\label{eq:prop-remainders-exp-decay-in-k} \sum_{j = 1}^3 \big\|\,\mathcal{R}_{j,k}^2(t, \cdot) \,\big\|_{L^2(\mathbb{R}_+)} \leq C_{\mathcal{R}} k^\frac{3}{4} \frac{1}{d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}.\qquad{(13)}\]
For any time \(t\in [0, T_{\rm max}]\) and any \(k\in \mathbb{N}\) \[\label{eq:ForcedSolutionLowerBound} \| u_k^{\rm fr}(t, \cdot) \|_{L^2(\mathbb{R}_+)} \geq \frac{1}{2} \| U_s' \|_{L^2(a_0+\frac{d}{2}, \infty)} e^{t \sqrt{k} \frac{|U_s''(a_0)|^{1/2}}{2\sqrt{2}}}.\qquad{(14)}\]
For any \(k\in \mathbb{N}\), \(m \in \mathbb{N}\setminus\{1, 2\}\) and \(\lambda>0\), the initial data \(u_k^{\rm fr}(0, \cdot)\) satisfies \[\label{eq:InitialDataUpperBound} \| u_k^{\rm fr}(0, \cdot ) \|_{H^m_\lambda(\mathbb{R}_+)} \leq \mathcal{D}_m \,e^{2 \lambda a_0} k^{\frac{2m-1}{4} }.\qquad{(15)}\]

0.3 The upper bound on the first family of remainders↩︎

We begin with the proof of part \((i)\) of 2 on the first family of remainders \(\mathcal{R}_{j,k}^1\). The argument relies on the following estimates about the deviation of \(u_s\) from its second-order Taylor polynomial at \(y = a(t)\), the derivatives \(a'(t)\) and \(\gamma'(t)\), and the growth properties of \(\Upsilon\):

For any time \(t \in [0, T_{\rm max}]\) and any index \(n \in \{0,1,2\}\) \[\label{eq:ineq-in-lemma-us} \begin{align} &\max_{a_0-\frac{d}{2}\leq y \leq a_0+ \frac{d}{2}} \bigg| \partial_y^n \Big( u_s(t,y) - \alpha(t) - \beta(t) (y-a(t))^2 \Big) \bigg| \leq d^{3-n} \big\| U_s^{(3)}\big\|_{L^1(\mathbb{R}_+)} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}} . \end{align}\tag{25}\]
For any time \(t \in [0, T_{\rm max}]\) \[\label{eq:ineq-in-lemma-us-a39-gamma39} \begin{align} |a'(t)| &\leq \frac{2}{|U_s''(a_0)|} \| U_s^{(3)} \|_{L^1(\mathbb{R}_+)} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}}, \qquad |\gamma'(t)| \leq \frac{2}{|U_s''(a_0)|^\frac{3}{4}} \| U_s^{(3)} \|_{W^{1,1}(\mathbb{R}_+)} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}} . \end{align}\tag{26}\]
For any frequency \(k \in \mathbb{N}\), any time \(t \in [0, T_{\rm max}]\) and any index \(n\in \{ 0, 1, 2\}\) \[\label{eq:ineq-in-lemma-us-Upsilon} \max_{a_0-\frac{d}{2}\leq y \leq a_0+ \frac{d}{2}} \Big| \Upsilon^{(n)}\big(\sqrt[4]{k}\,z (t,y) \big) \Big| \leq 4 \big( 1+ |U_s''(a_0)|^{\frac{2-n}{4}} d^{2-n} \big) k^\frac{2-n}{4}.\tag{27}\]

In this subsection, we focus on the application of 25 , 26 and 27 to the remainders \(\mathcal{R}_{j,k}^1\), postponing their proofs to 5 in the Appendix, where they are formalised using the Taylor theorem and the Green’s formula for the heat kernel with Dirichlet boundary conditions.

Proof of 2, Part \((i)\). To shorten the notation, we set the interval \(I = [a_0 - \tfrac{d}{2}, a_0 + \tfrac{d}{2}]\). Additionally, we repeatedly use the following upper bounds for \(|U_s''(a_0)|\) and \(\gamma(t)\) at any time \(t\in [0, T_{\rm max}]\): \[\label{eq:124Us393940a41124and124gamma3940t41124} |U_s''(a_0)|= \bigg| \int_{a_0}^{+\infty} U_s^{(3)}(\omega) d\omega \bigg| \leq \| U_s^{(3)} \|_{L^1} ,\qquad |\gamma(t)| = \sqrt[4]{\frac{|\partial_y^2 u_s(t,a(t))|}{2}} \leq \sqrt[4]{\frac{2|U_s''(a_0)|}{2}} = \| U_s^{(3)} \|_{L^1}^\frac{1}{4},\tag{28}\] which are satisfied thanks to our assumption in 16 . We begin by estimating the first five remainders, proving that they satisfy the bound \[\label{eq:ineq-for-first-five-remainders-first-family} \| \mathcal{R}_{j,k}^1 (t) \|_{L^2} \leq k\frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}}\, \,\frac{4}{\sqrt{\pi}} \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4,\qquad j \in \{1, \dots, 5\}.\tag{29}\] The first remainder \(\mathcal{R}_{1, k}^1\) in ?? has \(L^2\)-norm equal to \[\| \mathcal{R}_{1,k}^1 (t) \|_{L^2} = k^\frac{3}{4} \gamma(t)^3 \bigg( \int_0^\infty \bigg| \Big( u_s(t, y) -\alpha(t) - \beta(t) (y -a(t))^2 \Big) \chi(y) \Upsilon'\big(\sqrt[4]{k}\, z(t, y) \big) \bigg|^2 dy \bigg)^\frac{1}{2}.\] We hence recall that \(\chi\) is supported on \(I = [a_0 - \tfrac{d}{2}, a_0 + \tfrac{d}{2}]\), therefore, thanks to 28 and the Hölder inequality: \[\begin{align} \| \mathcal{R}_{1,k}^1 &(t ) \|_{L^2} \leq k^\frac{3}{4} \| U_s^{(3)} \|_{L^1}^\frac{3}{4} \left\| u_s(t, \cdot ) -\alpha(t) - \beta(t) (\cdot -a(t))^2 \right\|_{L^\infty(I)} \| \chi \|_{L^2} \left\| \Upsilon'\big(\sqrt[4]{k}\, z(t, \cdot) \right\|_{L^\infty(I)} \end{align}\] where we denoted \(L^p = L^p(\mathbb{R}_+)\) for any \(p \geq 1\). Next, we invoke 25 with \(n = 0\) and 26 with \(n = 1\), to obtain \[\begin{align} \| \mathcal{R}_{1,k}^1 (t) \|_{L^2} &\leq k^\frac{3}{4} \| U_s^{(3)} \|_{L^1}^\frac{3}{4} \; \big\| U_s^{(3)} \big\|_{L^1} d^3\, \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \; \| \chi \|_{L^2} \; 4 \big( 1 + \big\| U_s^{(3)} \big\|_{L^1}^\frac{1}{4} d \big) \sqrt[4]{k} \\ &\leq 4 k \big( 1 + \| U_s^{(3)} \|_{L^1} \big)^\frac{7}{4} \big( 1 + d \big)^3 \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \; \| \chi \|_{H^3} \; \big( 1 + \| U_s^{(3)} \|_{L^1} \big)^\frac{1}{4} \big( 1 + d \big) \\ & \leq k\frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}}\, \,\frac{4}{\sqrt{\pi}} \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4, \end{align}\] which is 29 for \(j = 1\). Next, we address the second remainder \(\mathcal{R}_{2,k}^1\) in ?? , whose \(L^2\)-norm is equal to \[\| \mathcal{R}_{2,k}^1 (t) \|_{L^2} = \bigg( \int_{a(t)}^{\infty} \bigg| \Big( u_s(t,y) - \alpha(t) - \beta(t) (y-a(t))^2 \Big) \Big( \chi^{(3)}(y) - \Big( \sigma_k(t) + {i}k \, u_s(t,y) \Big) \chi'(y) \Big) \bigg|^2 dy \bigg)^\frac{1}{2}.\] We hence remark that the function \(\sigma_k(t)= e^{-\frac{i\pi}{4}}\gamma(t)^2 \sqrt{k} - i k \,u_s(t,a(t))\) defined in ?? satisfies \[\label{eq:est-of-sigmak40t41} |\sigma_k(t)| \leq \gamma(t)^2 \sqrt{k} + k \| u_s \|_{L^\infty(\mathbb{R}_+ \times \mathbb{R}_+) } \leq \sqrt{k} \, | U_s''(a_0) |^\frac{1}{2} + k \| U_s \|_{L^\infty(\mathbb{R}_+)} \leq 2 k \big( 1 + \| U_s \|_{W^{4,1}}\big),\tag{30}\] thanks to 28 and the maximum principle of the heat equation. Thus applying the Hölder inequality and the relation 25 with \(n = 0\), we obtain \[\begin{align} \| \mathcal{R}_{2,k}^1 (t) \|_{L^2} &\leq \left\| u_s(t, \cdot ) -\alpha(t) - \beta(t) (\cdot -a(t))^2 \right\|_{L^\infty(I)} \Big( \| \chi^{(3)} \|_{L^2} + \big( |\sigma_k(t)| + k \| u_s(t) \|_{L^\infty} \big) \| \chi' \|_{L^2} \Big) \\ &\leq \big\| U_s^{(3)} \big\|_{L^1} d^3\, \frac{e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \| \chi \|_{H^3} \Big( 1 +2 k \big( 1 + \| U_s \|_{W^{4,1}}\big) + k \| U_s \|_{L^\infty} \Big) \\ &\leq k \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}} \frac{4}{\sqrt{\pi}} \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4, \end{align}\] which is 29 for \(j = 2\). The third remainder \(\mathcal{R}_{3,k}^1\) in ?? is estimated using 25 with \(n=1\) and 27 with \(n=0\): \[\begin{align} \| \mathcal{R}_{3,k}^1 (t) \|_{L^2} &= \sqrt{k}\, \gamma(t)^2 \bigg( \int_0^\infty \bigg| \Big( \partial_y u_s(t, y) - 2 \beta(t) (y -a(t)) \Big) \chi(y) \Upsilon\big(\sqrt[4]{k}\, z(t, y) \big) \bigg|^2 dy \bigg)^\frac{1}{2} \\ &\leq \sqrt{k}\, \|U_s^{(3)}\|_{L^1}^\frac{1}{2} \big\| \partial_y u_s(t, \cdot ) - 2 \beta(t) (\cdot -a(t)) \big\|_{L^\infty(I)} \| \chi \|_{L^2} \| \Upsilon\big(\sqrt[4]{k}\, z(t, \cdot ) \|_{L^\infty(I)} \\ &\leq \sqrt{k}\, \|U_s^{(3)}\|_{L^1}^\frac{1}{2} d^2 \| U_s^{(3)} \|_{L^1} \frac{e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \| \chi \|_{H^3} 4 \big( 1+ \| U_s^{(3)} \|_{L^1}^\frac{1}{2} d^2 \big) \sqrt{k} \\ & \leq k\frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}}\frac{4}{\sqrt{\pi}} \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4, \end{align}\] namely 29 for \(j = 3\). The fourth and fifth remainders \(\mathcal{R}_{4,k}^1\), \(\mathcal{R}_{5,k}^1\) in ?? can be tackled together using 25 once more: \[\begin{align} \max \Big\{ \| \mathcal{R}_{4,k}^1 (t)\|_{L^2},\, \| \mathcal{R}_{5,k}^1 (t) \|_{L^2} \Big\} &\leq \max \Big\{ \| \partial u_s (t, \cdot ) - 2\beta(t)(\cdot -a(t)) \|_{L^\infty(I)},\, \| \partial^2 u_s (t, \cdot ) - 2\beta(t) \|_{L^\infty(I)} \Big\} 4 \| \chi \|_{H^3} \\ &\leq \max\{ d^2,\, d\} \| U_s^{(3)} \|_{L^1}\frac{e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \leq 4\, k\, \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4 \; \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}}, \end{align}\] which is 29 for \(j = 4,\,5\). Since \(20/\sqrt{\pi} \leq 12\), we have proven therefore that \[\label{eq:sum-first-five-remainders-first-family} \sum_{j = 1}^5 \| \mathcal{R}_{j,k}^1 (t) \|_{L^2} \leq k\frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}}\, \,12 \| \chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4.\tag{31}\] It remains to estimate \(\mathcal{R}_{6, k}^1\) and \(\mathcal{R}_{7, k}^1\) in ?? . For \(\mathcal{R}_{6, k}^1\), we first apply 25 and 27 , to gather \[\begin{align} \| \mathcal{R}_{6,k}^1 (t)\|_{L^2} &\leq \frac{1}{\sqrt{k}} \gamma'(t) \|\chi \|_{H^1} \Big( 2 \gamma(t) + 2 \gamma(t) \| \Upsilon\big(\sqrt[4]{k}\, z(t, \cdot ) \|_{L^\infty(I)} + 3 \gamma(t)^2 \| \Upsilon'\big(\sqrt[4]{k}\, z(t, \cdot ) \|_{L^\infty(I)} \Big) \\ &\leq \frac{1}{\sqrt{k}} \gamma'(t) \|\chi \|_{H^1} \bigg( 2 \| U_s^{(3)} \|_{L^1}^\frac{1}{4} + 8 \| U_s^{(3)} \|_{L^1}^\frac{1}{4} ( 1 + \| U_s^{(3)} \|_{L^1}^\frac{1}{2} d^2 ) \sqrt{k} + 12\| U_s^{(3)} \|_{L^1}^\frac{1}{2} ( 1 + \| U_s^{(3)} \|_{L^1}^\frac{1}{4} d ) \sqrt[4]{k} \bigg) \\ & \leq 22\, \gamma'(t) \|\chi \|_{H^3} \big( 1+ d\big)^2 \big( 1 + \| U_s^{(3)} \|_{L^1} \big). \end{align}\] Next, thanks to 26 and the fact that \(22/\sqrt{\pi} \leq 13\), we further obtain \[\| \mathcal{R}_{6,k}^1 (t)\|_{L^2} \leq k \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}} \, 13 \frac{2}{|U_s''(a_0)|} \|\chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4.\] Finally \[\begin{align} \| \mathcal{R}_{7,k}^1 (t)\|_{L^2} &\leq \Big( |\gamma'(t)| d + \gamma(t)|a'(t)|\Big) \bigg( \| \chi' \|_{L^2} \frac{\gamma(t)^2}{\sqrt[4]{k}} \| \Upsilon'\big(\sqrt[4]{k}\, z(t, \cdot ) \|_{L^\infty(I)} + \| \chi \|_{L^2} \gamma(t)^3 \| \Upsilon''\big(\sqrt[4]{k}\, z(t, \cdot ) \|_{L^\infty(I)} \bigg) \\ &\leq \| U_s^{(3)} \|_{W^{1,1}} \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{\pi t}} \bigg( \frac{2}{|U_s''(a_0)|^\frac{3}{4}} d + \frac{2}{|U_s''(a_0)|^\frac{1}{4}} \bigg) \| \chi \|_{H^3} 4 \bigg( \| U_s^{(3)} \|_{L^1} \big( 1 + \| U_s^{(3)} \|_{L^1}^\frac{1}{4}d \big) + \| U_s^{(3)} \|_{L^1}^\frac{3}{4} \bigg) \\ &\leq k \frac{ e^{ - \frac{d^2}{16 t} } }{\sqrt{t}} \, 5 \bigg( 1 + \frac{2}{|U_s''(a_0)|} \bigg) \|\chi \|_{H^3} \big( 1 + \| U_s \|_{W^{4,1}}\big)^2 \big(1+d\,\big)^4. \end{align}\] Taking the sum between 31 and the last two relations concludes the proof of Part \((i)\) of 2. ◻

0.4 An upper bound on the second family of remainders↩︎

We now deal with Part (ii) of 2 about the second family of remainders \(\mathcal{R}_{1,k}^2\), \(\mathcal{R}_{2,k}^2\), \(\mathcal{R}_{3,k}^2\) as defined in ?? . We prove that their \(L^2\)-norms exhibit exponential decay in the frequencies \(k \in \mathbb{N}\), as described by ?? .

These terms depend mainly on combinations between the function \(\Upsilon\) of 13 and the Heaviside function \(H(y-a(t))\). In particular, we will repeatedly rely on the following inequality about the \(\, \mathrm{erf}\)-term of \(\Upsilon\) and \(H\): \[\label{eq:erf-H-estimate} \bigg| \, \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \, \bigg| \leq \frac{13}{|U_s''(a_0)|^\frac{1}{4}} \frac{1}{\sqrt[4]{k}\,d} \, e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}\tag{32}\] for any \(y \in \mathbb{R}_+ \setminus [a_0 - \tfrac{d}{4}, a_0 + \tfrac{d}{4}]\). Roughly speaking, this inequality holds true, since \(z(t,y) = \gamma(t)(y-a(t))e^{-\frac{i\pi}{8}}\) is in the sector \(|\arg(z) |< \tfrac{\pi}{4}\) if \(y >a_0 +\tfrac{d}{2}>a(t)\) and \(|\arg(-z) |< \tfrac{\pi}{4}\) if \(y<a_0 -\tfrac{d}{2}<a(t)\), so that the \(\, \mathrm{erf}\) function converges exponentially to \(\pm 1\) with same sign to the one of \(y-a(t)\). We postpone the formal proof of 32 to 6, in the Appendix, and focus on its application to \(\mathcal{R}_{1,k}^2\), \(\mathcal{R}_{2,k}^2\), and \(\mathcal{R}_{3,k}^2\).

Proof of 2, Part \((ii)\). All remainders depend on some derivative of the cutoff function \(\chi\), which is supported in \([a_0 - \tfrac{d}{2}, a_0 + \tfrac{d}{2}]\) and equal to \(1\) on \([a_0 - \tfrac{d}{4}, a_0 + \tfrac{d}{4}]\). Consequently, \(\mathcal{R}_{1,k}^2\), \(\mathcal{R}_{2,k}^2\), and \(\mathcal{R}_{3,k}^2\) have support in \([0, T_{\rm max}] \times K\), with \(K = [a_0 - \tfrac{d}{2}, a_0 - \tfrac{d}{4}] \cup [a_0 + \tfrac{d}{4}, a_0 + \tfrac{d}{2}]\). We will use in particular that \[\label{eq:estimate-y-a40t41-propositionest-2nd-part} \frac{d}{8} < |\, y-a(t) \, | \leq d, \qquad \text{for any}\quad (t,y) \in [0, T_{\rm max}] \times K,\tag{33}\] which follows from our assumption \(a(t) \in [a_0-\tfrac{d}{8}, a_0+\tfrac{d}{8}]\), introduced in 16 .

Our goal is to establish that, for each \(j \in \{1,2,3\}\) and any \(t \in [0, T_{\rm max}]\), the following upper bound holds: \[\label{eq:ineq-for-remainders-second-family} \| \mathcal{R}_{j,k}^2(t) \|_{L^2} \leq \frac{k^\frac{3}{4} }{d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} 24 \| \chi \|_{H^3} \Big( 1 + \frac{2}{|U_s''(a_0)|} \Big) \big( 1 + \| U_s \|_{W^{4,1}} \big)^2(1 +d\,)^4 .\tag{34}\] The desired inequality ?? and the conclusion of the proof follows then by summing over \(j = 1, 2, 3\). We begin with the estimate of \(\mathcal{R}_{1,k}^2\) in ?? , whose \(L^2\)-norm satisfies at any \(t\in [0, T_{\rm max}]\): \[\begin{align} \| \mathcal{R}_{1,k}^2&(t) \|_{L^2} = \\ & \frac{\gamma(t)^2 }{\sqrt{k}} \bigg( \int_K \bigg| \Big[ \Upsilon(\sqrt[4]{k}\,z(t,y)) - H(y-a(t)) \big( 1 + \sqrt{k}\, z(t,y)^2 \big) \Big] \Big( \big( \sigma_k(t) + {i}k\, u_s(t,y) \big)\chi'(y) - \chi'''(y) \Big) \bigg|^2 dy \bigg)^\frac{1}{2}. \end{align}\] By applying the Hölder inequality, we obtain \[\| \mathcal{R}_{1,k}^2(t) \|_{L^2} \leq \frac{\gamma(t)^2 }{\sqrt{k}} \| \chi \|_{H^3} \Big( 1 + |\sigma_k(t)| + k \| u_s(t,\cdot) \|_{L^\infty} \Big) \Big\| \Upsilon(\sqrt[4]{k}\,z(t,\cdot )) - H(\cdot -a(t)) \Big( 1 + \sqrt{k}\, z(t,\cdot)^2 \Big) \Big\|_{L^\infty(K)}.\] We hence recall that \(\gamma(t)\leq \| U_s^{(3)} \|_{L^1}^{1/4}\) from 28 and \(|\sigma_k(t)| \leq 2k ( 1 + \| U_s\|_{W^{4,1}})\) from 30 . Thanks to the maximum principle of the heat equation \(\| u_s \|_{L^\infty(\mathbb{R}_+ \times \mathbb{R}_+)} \leq \| U_s \|_{L^\infty(\mathbb{R}_+)}\leq \| U_s \|_{W^{4,1}}\), we deduce therefore that \[\label{eq:first-ineq-of-R1k2} \| \mathcal{R}_{1,k}^2(t) \|_{L^2} \leq 3 \sqrt{k} \, \| \chi \|_{H^3} \big( 1 + \| U_s\|_{W^{4,1}} \big) \big\| \, \Upsilon(\sqrt[4]{k}\,z(t,\cdot ))-\big(1 + \sqrt{k}\, z(t,\cdot)^2 \big)H(\cdot -a(t)) \,\big\|_{L^\infty(K)}.\tag{35}\] We develop the last \(L^\infty\)-norm using the definition of \(\Upsilon\), provided in 13 . More precisely, for any \((t,y) \in [0, T_{\rm max}]\times K\), \[\label{eq:Ups-H-formula} \begin{align} \Upsilon(\sqrt[4]{k}\,z(t,y))-\Big(1 &+ \sqrt{k}\, z(t,y)^2 \Big)H(y -a(t)) = \\ &= \frac{ \sqrt[4]{k}\, z(t,y)}{\sqrt{2\pi}}e^{ - \sqrt{k} \frac{z(t,y)^2}{2}} + \big( 1 + \sqrt{k}\,z(t,y)^2 \big) \frac{1}{2} \bigg( \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \bigg). \end{align}\tag{36}\] We recall that \(z(t,y) = \gamma(t) (y-a(t)) e^{-\frac{i\pi}{8}}\), by definition in ?? , and observe that \[\frac{|U_s''(a_0)|^{1/4}}{\sqrt{2}} \leq \gamma(t)\leq |U_s''(a_0)|^{1/4}\leq \| U_s^{(3)} \|_{L^1}^{1/4}\qquad\] from the assumptions in 16 . For any \((t,y) \in [0, T_{\rm max}] \times K\), the first element of the sum in 36 is bounded by \[\begin{align} \bigg| \frac{ \sqrt[4]{k}\, z(t,y)}{\sqrt{2\pi}} e^{ - \sqrt{k} \frac{z(t,y)^2}{2}} \bigg| &= \frac{\sqrt[4]{k}\,\gamma(t) |\,y-a(t)\,| }{\sqrt{2\pi}} e^{ -\sqrt{k} \, \frac{\gamma(t)^2(y-a(t))^2}{2\sqrt{2}}}\\ &\leq \frac{\sqrt[4]{k}\| U_s^{(3)} \|_{L^1}^{1/4} \,d}{\sqrt{2\pi}} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \\ &\leq \sqrt[4]{k} \, \big( 1 + \| U_s \|_{W^{4,1}} \big)(1 +d\,)^4 \Big( 1 + \frac{2}{|U_s''(a_0)|} \Big) \frac{1}{d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}, \end{align}\] where we used 33 . Furthermore, applying 32 , we obtain \[\begin{align} \frac{1}{2} \Big| \,\Big( 1 + \sqrt{k}\,z(t,y)^2 \Big) & \Big( \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \Big) \Big| \leq \\ &\leq \frac{1}{2} \Big(\, 1 + \sqrt{k} \, \gamma(t) (y-a(t))^2 \Big) \frac{13}{|U_s''(a_0)|^\frac{1}{4}} \frac{1}{\sqrt[4]{k}\,d} \, e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \\ &\leq 7 \sqrt[4]{k} \, \big(\, 1 + \| U_s \|_{W^{4,1}} \big)( 1 + d \,)^4 \Big( 1 + \frac{2}{|U_s''(a_0)|} \Big) \frac{1}{d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \end{align}\] Coupling the last two relations together with 35 yields 34 for \(j = 1\). We next address the second remainder \(\mathcal{R}_{2, k}^2\) of ?? , whose \(L^2\)-norm satisfies at any time \(t\in [0, T_{\rm max}]\): \[\begin{align} \| \mathcal{R}_{2, k}^2(t) \|_{L^2} & = 3\frac{\gamma(t)^3}{\sqrt[4]{k}} \bigg( \int_K \Big| \Big[\, \Upsilon'(\sqrt[4]{k}\,z(t,y)) - 2 \sqrt{k} \,z(t,y) H(y-a(t)) \,\Big] \chi''(y) \Big|^2 dy \bigg)^\frac{1}{2} \\ &\leq \frac{3 \| U_s^{(3)} \|_{L^1}^\frac{3}{4}}{\sqrt[4]{k}} \| \chi \|_{H^3} \big\| \, \Upsilon'(\sqrt[4]{k}\,z(t,\cdot)) - 2 \sqrt{k} \,z(t,,\cdot) H(\cdot-a(t)) \,\big\|_{L^\infty(K)} \end{align}\] From the derivative of \(\Upsilon'\) we obtain with a similar argument as before \[\begin{align} \Big| \Upsilon'(\sqrt[4]{k}\,z(t,y)) &- 2 \sqrt{k} \,z(t,y) H(y-a(t)) \Big| = \bigg| \sqrt{\frac{2}{\pi}}e^{ - \sqrt{k} \frac{z(t,y)^2}{2}} + \sqrt[4]{k}\, z(t,y) \Big( \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \Big) \bigg| \\ &\leq \sqrt{\frac{2}{\pi}} e^{ -\sqrt{k} \, \frac{\gamma(t)^2(y-a(t))^2}{2\sqrt{2}}} + \sqrt[4]{k}\,\gamma(t) |\, y-a(t) | \Big| \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \Big| \\ &\leq \sqrt{\frac{2}{\pi}} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} + \sqrt[4]{k}\,|U_s''(a) | d \frac{13}{|U_s''(a_0)|^\frac{1}{4}} \frac{1}{\sqrt[4]{k}\,d} \, e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \\ & \leq 13 \Big( 1 + \| U_s^{(3)} \|_{L^1} \Big) e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}, \end{align}\] which implies in particular the inequality of 34 for \(j = 2\). Finally, for \(j = 3\), we use that \(\Upsilon''(\zeta) = 1+\, \mathrm{erf}( \zeta/ \sqrt{2})\) so that we gather \[\begin{align} \| \mathcal{R}_{3,k}^2(t) \|_{L^2} &= 3 \beta(t) \bigg( \int_0^\infty \Big|\, \big( \Upsilon''(\sqrt[4]{k}\,z(t,y))-2 H(y-a(t)) \Big) \chi'(y) \,\big|^2 dy \bigg)^\frac{1}{2} \\ &\leq 3 \sqrt{\frac{|\partial_y u_s(t,a(t))|}{2}} \big\| \Upsilon''(\sqrt[4]{k}\,z(t,\cdot ))-2 H(\cdot -a(t)) \big\|_{L^\infty(K)} \| \chi' \|_{L^2} \\ &\leq 3 \sqrt{|U_s''(a_0)|}\, \max_{y \in K } \bigg| \, \, \mathrm{erf}\Big( \frac{\sqrt[4]{k} \, z(t,y)}{\sqrt{2}} \Big) - H(y-a(t)) \, \bigg|. \end{align}\] We hence apply the inequality in 32 to obtain \[\begin{align} \| \mathcal{R}_{3,k}^2(t) \|_{L^2} &\leq 39 |U_s''(a_0)|^\frac{3}{4} \frac{1}{\sqrt[4]{k}\,d} \, e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \\ &\leq 20\Big( 1+ \frac{2}{|U_s''(a_0)|}\Big) |U_s''(a_0)|^\frac{7}{4} \frac{1}{d} \, e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \\ &\leq \frac{k^\frac{3}{4}}{d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} 24\| \chi \|_{H^3} \Big( 1+ \frac{2}{|U_s''(a_0)|}\Big) \Big( 1+ \|U_s\|_{W^{4,1}} \Big)^2 (1+d)^4, \end{align}\] which is indeed 34 for \(j = 3\). This concludes the proof of 2, Part \((ii)\). ◻

0.5 A lower bound for the initial profile.↩︎

We now address 2, Part (iii), which provides a lower bound for the \(L^2\)-norm of the forced solution \(u_k^{\rm fr}\). By construction, the cutoff function \(\chi\) is identically zero on the interval \([a_0 + \tfrac{d}{2}, +\infty)\). Consequently, the inner profile \(\phi_{\text{inn}, k}\), as defined in 3, is also zero identically in this region.

As a result, for all \((t,y) \in [0, T_{\rm max}] \times [a_0 + \tfrac{d}{2}, +\infty)\), \[u_k^{\rm fr}(t,y) = \partial_y \phi_{\text{out}, k}(t,y) = \partial_y u_s(t,y) e^{\int_0^t \sigma_k(\tau) d\tau},\] and the following \(L^2\)-norm lower bound holds: \[\| u_k^{\rm fr}(t) \|_{L^2(a_0+\frac{d}{2}, +\infty)} \geq \frac{1}{2} \| U_s' \|_{L^2(a_0+\frac{d}{2}, +\infty)} \exp\bigg( \int_0^t {\rm Re} \,\sigma_k(\tau) d\tau \bigg),\] where we have exploited the assumption on \(T_{\rm max} > 0\) given by 17 . The result follows by the definition of \(\sigma_k(t)\) always in 3, which yields \[\begin{align} {\rm Re}\, \sigma_k(t) &= {\rm Re}\, \bigg( \sqrt{k} \,e^{-\frac{{i}\pi}{4}} \sqrt{\frac{|\partial_y^2u_s(t,a(t))|}{2}} - {i}k \, u_s( t,a(t)) \bigg) \\ &= \sqrt{k} \,\frac{\sqrt{|\partial_y^2u_s(t,a(t))|}}{2} \geq \sqrt{k} \, \frac{|U_s''(a_0)|^\frac{1}{2}}{2\sqrt{2}}, \end{align}\] as assumed in 16 . This completes the proof of 2, Part (iii).

0.6 An upper bound for the initial profile.↩︎

We conclude the proof of 2 with Part \((iv)\), concerning the upper bound of the \(H^m_\lambda\)-norm of \(u_k^{\rm fr}(0,y) = \partial_y \phi_{\text{inn}, k}(0,y) + \partial_y \phi_{\text{out}, k}(0,y)\) in \(y\in \mathbb{R}_+\).

The component that grows polynomially in \(k\in \mathbb{N}\) is \(\partial_y \phi_{\text{inn}, k}\) and most of its norm can be estimated using the following inequality for the \(n\)-th derivative of \(\Upsilon\): \[\label{eq:ineq-Ups-n-derivative-prop} \max_{a_0-\frac{d}{2}\leq y \leq a_0 + \frac{d}{2}} \big|\, \Upsilon^{(n)}(\sqrt[4]{k}\, z(t,y) )\, \big| \leq (n-3)! \, 2\big(\, 1 + |\,U_s''(a_0)\,|^\frac{n-3}{4} d^{n-3}\,\big) k^\frac{n-3}{4},\tag{37}\] for any \(n \in \mathbb{N}\) with \(n \geq 3\) and any \(t\in [0, T_{\rm max}]\). Indeed, the third derivative of \(\Upsilon\) is the Gaussian function \(\Upsilon^{(3)}(\zeta) = \sqrt{\tfrac{2}{\pi}}e^{-\frac{\zeta^2}{2}}\). Consequently, any higher-order derivative \(\Upsilon^{(n)}(\zeta)\) with \(n> 3\) can be bounded (locally in \(\zeta \in \mathbb{C}\)) using the Hermite polynomial \({\rm He}_{n-3}(\zeta)\) of degree \(n-3\). The proof of 37 is postponed to 7 in the Appendix, while we focus on its application to \(\partial_y \phi_{\text{inn}, k}\) at time \(t = 0\).

Using the 3, we apply the Leibniz rule for the \((j+1)\)-th derivative of the product with \(j \in \{0, \dots, m\}\): \[\begin{align} \partial_y^{j+1} \phi_{\text{inn}, k}(0,y) &= \partial_y^{j+1} \bigg[ \chi(y) \Upsilon \big( \sqrt[4]{k}\,z(0,y) \big) \bigg] \\ &= \sum_{n = 0}^{j+1} \binom{j+1}{n} \chi^{(j+1-n)}(y) \, \partial_y^{n} \bigg[ \Upsilon\Big(\, \sqrt[4]{k}\,z(0,y) \,\Big) \bigg] \\ &= \sum_{n = 0}^{j+1} \binom{j+1}{n} \chi^{(j+1-n)}(y) \, \Upsilon^{(n)}\big( \sqrt[4]{k}\,z(0,y) \big) \, k^\frac{n}{4} \bigg(\frac{|U_s''(a_0)|}{2}\bigg)^\frac{n}{4} e^{-\frac{i n \pi}{8}} \end{align}\] where we used \(z(0,y) = \gamma(0)\, (y-a(0))\,e^{-\frac{i\pi}{8}}= ( |U_s''(a_0)|/2)^{1/4}(y-a_0)\,e^{-\frac{i\pi}{8}}\). For the \(n\)-th derivatives with \(n \leq 2\), we apply the inequality 27 , while for \(n\geq 3\) we use 37 : \[\begin{align} \big| \partial_y^{j+1} \phi_{\text{inn}, k}(0,y) \big| \leq \sum_{n = 0}^{\min\{ 2, j+1\}} \binom{j+1}{n} \big| \chi^{(j+1-n)}(y)\big| 4 \Big( 1 + |U_s''(a_0)|^{\frac{2-n}{4}} d^{2-n} \Big) k^{\frac{2-n}{4}}k^\frac{n}{4}\bigg(\frac{|U_s''(a_0)|}{2}\bigg)^\frac{n}{4} + \\ + \sum_{n = 3}^{j+1} \binom{j+1}{n} |\chi^{(j+1-n)}(y)| (n-3)! \, 2\big(\, 1 + |\,U_s''(a_0)\,|^\frac{n-3}{4} d^{n-3}\,\big) k^\frac{n-3}{4} k^\frac{n}{4}\bigg(\frac{|U_s''(a_0)|}{2}\bigg)^\frac{n}{4}, \end{align}\] where the second sum is simply zero, if \(j \leq 1\). Hence, for any \(j \in \{0, \dots, m\}\) and any \(y \in \mathbb{R}\), we obtain \[e^{\lambda y} \big| \partial_y^{j+1} \phi_{\text{inn}, k}(0,y) \big| \leq k^{\frac{2m-1}{4} } e^{2 \lambda a_0} (m-2)! \,2 ( 1 + d )^m \Big( 1 + |U_s''(a_0)| \Big)^\frac{m}{4} \sum_{n =0}^{j+1} \binom{j+1}{n} | \chi^{(j+1-n)}(y)|,\] where we used that \(\chi\) has support in \([a_0-\tfrac{d}{2}, \, a_0 + \tfrac{d}{2}]\). This implies that the \(H^m_\lambda\)-norm of \(\partial_y \phi_{\text{inn}, k}\) satisfies \[\begin{align} \| \partial_y \phi_{\text{inn}, k} (0)\|_{H^m_\lambda} &= \sum_{j = 0}^m \| e^{\lambda y} \partial_y^{j+1} \phi_{\text{inn}, k}(0) \|_{L^2(\mathbb{R}_+)} \\ &\leq k^{\frac{2m-1}{4} } e^{2 \lambda a_0} (m-2)! \,2 ( 1 + d )^m \Big( 1 + |U_s''(a_0)| \Big)^\frac{m}{4} \| \chi \|_{H^m} \sum_{j = 0}^m 2^{j+1} \\ &\leq k^{\frac{2m-1}{4} } e^{2 \lambda a_0} (m-1)! \, ( 1 + d )^m \Big( 1 + \|U_s\|_{W^{3,\infty}} \Big)^\frac{m}{4} \| \chi \|_{W^{m+1,\infty}}\,d\, 2^{m+2} , \end{align}\] which yields \[\label{eq:estimate-from-above-phiinn-in-Hm} \| \partial_y \phi_{\text{inn}, k} (0)\|_{H^m_\lambda} \leq k^{\frac{2m-1}{4} } e^{2 \lambda a_0} (m-1)! \,2^{m+2} ( 1 + d )^{m+1} \Big( 1 + \|U_s\|_{H^m_\lambda}+\|U_s\|_{W^{3,1}} \Big)^m \Big( 1 + \|\chi \|_{W^{m+1,\infty}} \Big)\tag{38}\] We now deal with the \(H^m_\lambda\)-norm of the outer layer \(\partial_y \phi_{\text{out}, k}\) at \(t =0\), which is indeed \(\mathcal{O}(1)\) in the frequencies \(k\in \mathbb{N}\). Indeed, from 3, we remark that \[\begin{align} \partial_y^{j+1} \phi_{\text{inn}, k} (0, y) = \partial_y^{j+1} \Bigg[ \big( 1- \chi(y) \big) H(y-a_0) \bigg( U_s(y)-U_s(a_0) + e^{i\frac{5\pi}{4}}\sqrt{\frac{|U_s''(a_0)|}{2k}} \bigg) \Bigg] \\ = (1-\chi(y)) H(y-a_0) U_s^{(j+1)}(y) - \sum_{n=1}^{j} \binom{j+1}{n} \chi^{(j+1-n)}(y) H(y-a_0) U_s^{(n)}(y) \,+\\ -\, \chi^{(j+1)}(y)H(y-a_0) \bigg( U_s(y)-U_s(a_0) + e^{i\frac{5\pi}{4}}\sqrt{\frac{|U_s''(a_0)|}{2k}} \bigg). \end{align}\] This implies that for any \(j\in \{ 0, \dots, m\}\) \[\begin{align} \| e^{\lambda y} \partial_y^{j+1} \phi_{\text{out}, k}(0) \|_{L^2(\mathbb{R}_+)} \leq \Big( 1 + \| \chi \|_{W^{m+1,\infty}}\Big) \Big( \| U_s \|_{H^m_\lambda} + \| U_s' \|_{L^1} + \| U_s^{(3)} \|_{L^1}^\frac{1}{2}\Big) \sum_{n=0}^{j+1} \binom{j+1}{n}, \end{align}\] which yields the estimate \[\| e^{\lambda y} \partial_y^{j+1} \phi_{\text{out}, k} (0) \|_{L^2} \leq 2^{j+1} \Big( 1 + \| \chi \|_{W^{m+1, \infty}} \Big) \Big( 1 + \| U_s \|_{H^{m+1}_\lambda} + \|U_s \|_{W^{3,1}} \Big)^m.\] Taking the sum \[\begin{align} \| \partial_y \phi_{\text{inn}, k} (0) \|_{H^m_\lambda} &\leq 2^{m+2} \Big( 1 + \| U_s \|_{H^{m+1}_\lambda} + \|U_s \|_{W^{3,1}} \Big)^m \Big( 1 + \| \chi \|_{W^{m+1, \infty}} \Big) \\ &\leq k^{\frac{2m-1}{4} } e^{2 \lambda a_0} \,2^{m+2} ( 1 + d )^{m+1} \Big( 1 + \|U_s\|_{H^m_\lambda}+\|U_s\|_{W^{3,1}} \Big)^m \Big( 1 + \|\chi \|_{W^{m+1,\infty}} \Big). \end{align}\] Summing the last inequality together with 38 concludes the proof of 2.

section1 -3.5ex 1.5ex Correcting the perturbed solution

Combining 3, 1 and 2, we have constructed a forced quasi-eigenmode solution \(u(t,x,y) = u_k^{\rm fr}(t,y)e^{i kx}\) to the linearised Prandtl system 2 , which grows as \(e^{\sigma_0 \sqrt{k} \,t}\) (cf. ?? ). To extend this growth to an exact solution, we need to subtract a corrector \(\delta u_k(t,y)e^{i k x}\) with same forcing term of \(u_k^{\rm fr}\) but zero initial data. Naturally, we shall also establish some appropriate upper bounds of \(\delta u_k\), as detailed in the next proposition. We recall that, by 1, the forcing term \(f_k\) belongs to \(\mathcal{C}^\infty_c([0, T_{\rm max}]\times \mathbb{R}_+, \mathbb{C})\).

Proposition 3. For any \(k\in \mathbb{N}\), there exists a unique \(\delta u_k : [0, T_{\rm max}] \times \mathbb{R}_+ \to \mathbb{C}\) in the function space \[\label{eq:function-space-for-deltauk} \delta u_k \in \mathcal{C}([0, T_{\rm max}], H^1_0(\mathbb{R}_+, \mathbb{C})), \quad \text{with}\quad \partial_t \delta u_k,\; \partial_y^2 \delta u_k \in L^2((0, T_{\rm max}) \times \mathbb{R}_+, \mathbb{C})\qquad{(16)}\] that satisfies both \(\delta u_{k }|_{t= 0 }\equiv 0\) as well as, for a.e. \((t,y) \in (0, T_{\rm max}) \times \mathbb{R}_+\), the pointwise identity \[\label{eq:equation-for-deltauk} \partial_t \delta u_k (t,y)+ i k\, u_s(t,y) \delta u_k (t,y) - \partial_y^2 \delta u_k (t,y) - i k\, \partial_y u_s(t,y) \int_0^y \delta u_k (t,\omega) d\omega = f_k(t,y) \, e^{\int_0^t \sigma_k(\tau) d\tau }.\qquad{(17)}\] Moreover, the \(L^2(\mathbb{R}_+, \mathbb{C})\)-norm of \(\delta u_k(t,\cdot)\) is bounded at any time \(t\in [0, T_{\rm max}]\) by \[\label{eq:ineq-deltauk} \|\, \delta u_k(t) \,\|_{L^2} \leq 2 \bigg( 1 + k \,\frac{e^{2\lambda^2 t}}{ \lambda } \| \,U_s' \,\|_{L^2_\lambda } \bigg) \int_0^ t \big\| \,f_k(\tau ) e^{\int_0^\tau \sigma_k(l) dl} \big\|_{L^2} e^{ (t-\tau) \sqrt{k} (1+2\|U_s''\|_{L^\infty} )} \,d\tau.\qquad{(18)}\]

Remark 3. The equation ?? for \(\delta u_k\) is linear and non-homogeneous, with the heat equation as its leading term under Dirichlet boundary conditions. Existence and uniqueness of a solution \(\delta u_k\) in ?? then follow from a standard fixed-point argument, relying also on the a-priori estimates: \[\label{eq:control-of-int0y-deltauk} \begin{align} \int_0^\infty \Big| {i}k \,\partial_y u_s(t,y) \int_0^y \delta u_k(t, \omega ) d \omega \Big|^2 dy &\leq k^2 \int_0^\infty \big|\,\partial_y u_s(t,y) \,y \,\big|^2 dy \, \|\delta u_k(t) \|_{L^2}^2 \\ &\leq \frac{k^2}{\lambda^2 } \int_0^\infty \big|\,\partial_y u_s(t,y) \big|^2 e^{2\lambda y} dy \, \| \delta u_k (t ) \|_{L^2}^2 \\ &\leq \frac{k^2 }{\lambda^2 } e^{4\lambda^2 t}\| U_s' \|_{L^2_\lambda}^2 \| \delta u_k (t ) \|_{L^2}^2. \end{align}\qquad{(19)}\] A similar a-priori estimate holds also for the term \(i k \,u_s(t,\cdot ) \delta u_k(t,\cdot )\) in \(L^2(\mathbb{R}_+, \mathbb{C})\). We shall therefore focus on the less-trivial inequality ?? .

Proof of inequality ?? . We proceed with a similar ansatz as in [16] and introduce some auxiliary functions for our estimates. We first consider the unique function \[\psi_k \in \mathcal{C}([0, T_{\rm max}], H^1(\mathbb{R}_+, \mathbb{C})),\quad \text{with}\quad \partial_y \psi_k \in L^2(0, T_{\rm max}, H^1_0(\mathbb{R}_+, \mathbb{C})) \quad \text{and}\quad \partial_t \psi_k \in L^2( (0, T_{\rm max}) \times \mathbb{R}_+, \mathbb{C}),\] that satisfies both \(\psi_{k}|_{t = 0} \equiv 0\) and the pointwise identity, for a.e. \((t,y) \in (0, T_{\rm max}) \times \mathbb{R}_+\), \[\partial_t \psi_k(t,y) + i k\, u_s(t,y)\, \psi_k(t,y) - \partial_y^2 \psi_k(t,y) + ik \,\partial_y u_s(t,y) \int_0^y \psi_k(t,\omega) d\omega = \delta u_k(t,y).\] Since \(\delta u_k \in L^2((0, T_{\rm max}) \times \mathbb{R}_+, \mathbb{C})\), the existence and uniqueness of \(\psi_k\) follow once more from a standard fixed-point argument, now with the heat equation with homogeneous Neumann boundary conditions as the leading operator. We remark in particular that the following identity is satisfied pointwise for a.e. \((t,y) \in (0, T_{\rm max}) \times \mathbb{R}_+\): \[\begin{align} \Big( \partial_t + i k \, u_s(t,y) - \partial_y^2 \Big) \int_0^y \psi_k(t,\omega) d \omega = \int_0^y \partial_t \psi_k(t, \omega) d\omega + ik \,u_s(t,y) \int_0^y \psi_k(t,\omega) d \omega - \partial_y \psi_k(t,y) \\ = \int_0^y \big( \partial_t -\partial_\omega^2 \big) \psi_k (t, \omega) d\omega + i k \int_0^y \partial_\omega \Big( u_s(t, \omega) \int_0^\omega \psi_k(t, l) dl \Big) d\omega = 0 . \end{align}\] Next, we consider a second auxiliary function \(\Psi_k\in \mathcal{C}([0, T_{\rm max}], H^1_0(\mathbb{R}_+, \mathbb{C}))\), with weak derivatives \(\partial_t \Psi_k, \partial_y^2 \Psi_k\) in \(L^2((0, T_{\rm max}) \times \mathbb{R}_+, \mathbb{C})\), defined by \[\label{eq:def-Psik} \Psi_k (t,y) := \delta u_k(t,y) - i k\, \partial_y u_s(t,y) \int_0^y \psi_k(t, \omega ) d\omega,\qquad \text{for a.e.}\quad (t,y) \in (0, T_{\rm max}) \times \mathbb{R}_+.\tag{39}\] Indeed, \(\delta u_k\) belongs to the same function space as \(\Psi_k\), while for the second term of the sum, we exploit the fast decay of \(u_s \in \mathcal{C}([0, T_{\rm max}], H^{m+1}_\lambda(\mathbb{R}_+))\), with \(m \geq 3\), which also satisfies the heat equation. Moreover, we note that \(\Psi_k |_{ t= 0} \equiv 0\) and \(\Psi_k\) satisfies the following pointwise identity for a.e. \((t,y)\in (0, T_{\rm max}) \times \mathbb{R}_+\): \[\begin{align} \Big( \partial_t &+ i k \, u_s(t,y) - \partial_y^2 \Big) \Psi_k (t, y) = f_k(t,y) e^{\int_0^t \sigma_k(\tau) d\tau } + i k\, \partial_y u_s(t,y) \int_0^y \delta u_k (t,\omega) d\omega \, + \\ &- i k \underbrace{ \big( \partial_t - \partial_y^2 \big) u_s(t,y)}_{=0} \int_0^y \psi_k(t, \omega ) d\omega - \underbrace{ \Big( \partial_t + i k \, u_s(t,y) - \partial_y^2 \Big)\int_0^y \psi_k(t, \omega ) d\omega}_{=0} - 2 i k \,\partial_y u_s(t,y) \psi_k(t,y). \end{align}\] In sum, the couple \((\Psi_k, \sqrt{k} \, \psi_k)\) satisfies pointwise, for a.e. in \((0, T_{\rm max})\times \mathbb{R}_+\), the following linear system: \[\label{eq:system-of-psik-and-Psik} \Big( \partial_t + i k \, u_s(t,y) - \partial_y^2 \Big) \begin{pmatrix} \Psi_k(t,y) \\ \sqrt{k} \, \psi_k (t,y) \end{pmatrix} = \sqrt{k} \begin{pmatrix} 0 & 2i \, \partial_y^2 u_s(t,y) \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \Psi_k (t,y)\\ \sqrt{k} \, \psi_k (t,y) \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix} f_k(t,y) e^{\int_0^t \sigma_k(\tau) d\tau }.\tag{40}\] Each summand and derivative in the above identity belongs at least to \(L^2((0, T_{\rm max}) \times \mathbb{R}_+)\). We remark, moreover, that for any complex vector \((\xi_1, \xi_2) \in \mathbb{C}^2\) and any \((t,y) \in (0, T_{\rm max})\times \mathbb{R}_+\) \[\label{eq:est-for-Mxi1xi2} \mathcal{M}(t,y) := \begin{pmatrix} 0 & 2i \, \partial_y^2 u_s(t,y) \\ 1 & 0 \end{pmatrix} \quad \Longrightarrow \quad \bigg| \begin{pmatrix} \overline{\xi_1} \\ \overline{\xi_2} \end{pmatrix} \cdot \mathcal{M}(t,y) \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix} \bigg| \leq \Big( 1 + 2 \| \,U_s ''\, \|_{L^\infty} \Big) \bigg( \frac{|\xi_1|^2}{2} + \frac{|\xi_2|^2}{2} \bigg).\tag{41}\] We hence introduce the absolutely continuous energy functional \(\mathcal{E}_k \in AC([0, T_{\rm max}])\) defined by: \[\begin{align} {4} \mathcal{E}_k(t) &:= \frac{1}{2}\|\, \Psi_k(t ) \,\|_{L^2 }^2 + \frac{\sqrt{k}}{2} \| \,\psi_k(t ) \,\|_{L^2 }^2 ,\qquad &&\text{for any } t \in [0, T_{\rm max}],\\ \mathcal{E}_k'(t) &= \langle \, \Psi_k(t) ,\, \partial_t \Psi_k(t) \,\rangle_{L^2 } + \sqrt{k} \langle \,\psi_k(t) ,\, \partial_t \psi_k(t) \,\rangle_{L^2 },\qquad &&\text{for a.e.~} t \in [0, T_{\rm max}]. \end{align}\] Thanks to the \(L^2\)-regularity of both derivatives \(\partial_t \psi_k, \, \partial_y^2 \psi_k\) and \(\partial_t \Psi_k,\, \partial_y^2 \Psi_k\), multiplying the identity 40 by \((\overline{\Psi_k}, \sqrt{k} \, \overline{\psi_k})^T\), integrating over \((0, t) \times \mathbb{R}_+\) and taking the real part yields the energy identity: \[\begin{align} \mathcal{E}_k&(t) - \mathcal{E}_k(0) + \int_0^t \|\partial_y \Psi_k(s) \|_{L^2 }^2 ds + \sqrt{k} \int_0^t \| \partial_y \psi_k(s) \,\|_{L^2 }^2 ds = \\ &= \sqrt{k} \, \mathfrak{Re} \bigg( \int_0^t \int_{\mathbb{R}_+} \begin{pmatrix} \overline{\Psi_k} \\ \sqrt{k} \, \overline{\psi_k} \end{pmatrix} \cdot \mathcal{M} \begin{pmatrix} \Psi_k \\ \sqrt{k} \, \psi_k \end{pmatrix} (s,y) dyds \bigg) + \mathfrak{Re} \bigg( \int_0^t \int_0^\infty \overline{\Psi_k } (s,y)f_k(s,y) e^{\int_0^s \sigma_k(\tau) d \tau} dyds \bigg), \end{align}\] which is satisfied for any \(t\in [0, T_{\rm max}]\). In particular, thanks to 41 and the fact that \(\mathcal{E}_k(0) = 0\) by construction, we further obtain \[\max_{0 \leq \tau \leq t } \mathcal{E}_k(\tau) \leq \sqrt{k} \,(1 + 2 \| \,U_s'' \,\|_{L^\infty}) \int_0^t \mathcal{E}_k(\tau) d\tau + \sqrt{2} \, \int_0^t \big\| \, f_k(\tau) e^{\int_0^\tau \sigma_k (l) d l } \,\big\|_{L^2} \sqrt{\mathcal{E}_k(\tau)} d\tau.\] for any \(t\in [0, T_{\rm max}]\). We shall now remark that \(\max\limits_{0 \leq s \leq t}\mathcal{E}_k(s) >0\) for any \(t>0\). Indeed, if zero, this would imply that \(\Psi_k\) and \(\psi_k\) vanish identically on \([0, t] \times \mathbb{R}_+\), which in turn would force both \(\delta u_k\) and the forcing term \(f_k\) to be identically zero in the same domain (which is a contradiction for \(f_k\)). For any \(t\in ]0, T_{\rm max}]\) we can divide the last inequality by \(\max\limits_{0 \leq \tau \leq t}\sqrt{\mathcal{E}_k(t)} >0\), to obtain \[\max\limits_{0 \leq \tau \leq t}\sqrt{\mathcal{E}_k(\tau)} \leq \sqrt{k} \Big( \,1 + 2 \| \,U_s''\, \|_{L^\infty} \Big) \int_0^t \max_{ 0 \leq l \leq \tau} \sqrt{\mathcal{E}_k(l)} d\tau + \sqrt{2} \int_0^t \big\| \, f_k(\tau) e^{\int_0^\tau \sigma_k (l) d l } \, \big\|_{L^2} d\tau.\] This last relation is satisfied also at \(t = 0\), given that \(\mathcal{E}_k(0) = 0\). We apply the Gronwall’s inequality to obtain the estimate \[\label{eq:est-final-EEk40t41} \sqrt{\mathcal{E}_k(t)} \leq \sqrt{2} \int_0^t \big\| \,f_k(\tau,\cdot ) e^{\int_0^\tau \sigma_k (l) d l } \,\big\|_{L^2} e^{ (t-\tau) \sqrt{k} \big(1 + 2\|U_s''\|_{L^\infty} \big)} d\tau .\tag{42}\] for any \(t \in [0, T_{\rm max}]\). Finally, recalling the relation 39 between \(\Psi_k\), \(\psi_k\) and \(\delta u_k\), we remark that \[\| \,\delta u_k(t) \, \|_{L^2} \leq \| \Psi_k(t) \|_{L^2} + \bigg( \int_0^\infty \Big| i k \,\partial_y u_s(t,y) \int_0^y \psi_k(t, \omega ) d \omega \Big|^2 dy \bigg)^\frac{1}{2}.\] A similar argument as the one used in ?? implies finally that for any \(t \in [0, T_{\rm max}]\): \[\label{eq:final-estimate-for-deltauk-in-the-proof} \| \,\delta u_k(t) \, \|_{L^2} \leq \| \Psi_k(t) \|_{L^2} + k\frac{e^{2 \lambda^2 t}}{\lambda} \| U_s' \|_{L^2_\lambda} \| \psi_k (t) \|_{L^2} \leq \bigg( 1 + k\frac{e^{2 \lambda^2 t}}{\lambda} \| U_s' \|_{L^2_\lambda} \bigg) \sqrt{2} \sqrt{\mathcal{E}_k(t)} .\tag{43}\] By combining 42 and 43 , we obtain ?? , thus concluding the proof of the proposition. ◻

section1 -3.5ex 1.5ex Proof of Theorem 1 With respect to the previous sections, we are in the position to prove Theorem 1. The strategy consists of the following: ansatz ?? gives rise to a forced solution of the linearised Prandtl equations as described in Proposition 1. In order to control the exact solution with initial data \(u_k(0,\cdot )\), we make use of Proposition 2 and ?? , which estimates the solution arising from the forcing terms with vanishing initial data \(\delta u_k\).

The proof will be a consequence of the following simple lemma:

Lemma 4. Let \(b_1,b_2>0\) as well as \(c_i, ~ i=1,...,4\) be positive constants with \(c_1<c_2\) and \(h=h_{m,k}: [0,1] \to \mathbb{R}\) be a continuous function such that \[\begin{align} \label{eq:UglyConstantsEstimate} h(t) \geq b_1 e^{c_1 \sqrt{k}t}- b_2 ke^{c_2 \sqrt{k}t} \int_0^t \left( e^{-c_3\sqrt{k}} + \frac{e^{-\frac{c_4}{s}}}{\sqrt{s}} \right) d s \end{align}\qquad{(20)}\] for all \(k\in \mathbb{N}\) and \(m\in \mathbb{N}_0\) and \(t\in [0,1]\). Then there exists a \(K=K(b_1,b_2,c_1,c_2,c_3,c_4) \in \mathbb{N}\) such that for all \(k\geq K\), it holds \[\begin{align} h(t) \geq \frac{b_1}{2} e^{c_1 \sqrt{k}t} \end{align}\] for all \(t \in [0, \tfrac{\mathcal{C}}{\sqrt[4]k} ]\) where \(\mathcal{C}= \frac{1}{2} \left( \frac{c_4}{c_2} \right)^{1/2}\).

Proof. At first, we apply the mean value theorem in integral form and simplify ?? to \[\begin{align} h(t) & \geq b_1 e^{c_1\sqrt{k}t} - \underbrace{b_2kt e^{c_2\sqrt{k}t-\frac{c_4}{t}} }_{=:I} -\underbrace{b_2k t e^{(c_2t-c_3)\sqrt k}}_{:=II} . \end{align}\] For the first term, we point out that \(c_2\sqrt{k}t-\frac{c_4}{t}\leq -\frac{\sqrt{c_1c_2}}{\sqrt[4]k}\) is true for \(0\leq t \leq \frac{1}{\sqrt[4]k}\frac{1}{2} \left( \frac{c_4}{c_2} \right)^{1/2}\). Therefore, there exists a \(K_I\in \mathbb{N}\) such that \[\begin{align} I&\leq b_2k e^{-\sqrt{c_1c_2}\sqrt[4]{k}} \leq \frac{b_1}{4} \end{align}\] for all \(k\geq K_{I}\) and \(0\leq t \leq \frac{1}{2} \left( \frac{c_4}{c_2} \right)^{1/2} \frac{1}{\sqrt[4]{k}}\). With respect to the second term, we note that \(0\leq t \leq \frac{c_3-1}{c_2}\) implies \(c_2t-c_3\leq -1\). Hence there exists \(K_{II} \in \mathbb{N}\) such that \[\begin{align} II & \leq t b_2 k e^{-\sqrt{k}} \leq t \frac{b_1 c_1 \sqrt{k} }{4} \leq \frac{b_1}{4} e^{c_1 \sqrt k t} \end{align}\] for all \(k\geq K_{II}\) and \(0\leq t\leq \frac{c_3-1}{c_2}\). Setting \(K := \max\{K_I,K_{II}, \lceil\left( \frac{1}{2} \left( \frac{c_4}{c_2} \right)^{1/2}\right)^{-1/4} \rceil\}\), the statement follows. ◻

Proof of Theorem 1. Let \(u_k\) be the solution of 5 subject to the initial datum \(u_k(0, \cdot) =\partial_y\phi_{k}(0,\cdot)\) of ?? . Then it holds \(u_k=u_k^{\rm fr}+ \delta u_k\) by linearity. It follows that \[\begin{align} \left\| u_k(t,\cdot) \right\|_{H^{m-1}_\lambda( \mathbb{R}^+)} & \geq \left\| u_k(t,\cdot) \right\|_{L^2( \mathbb{R}^+)} \\ & \geq \left\| u_k^{\mathrm{fr}}(t,\cdot) \right\|_{L^2(\mathbb{R}^+)} - \left\| \delta u_k(t,\cdot) \right\|_{L^2(\mathbb{R}^+)} \end{align}\] Next, we substitute the estimate for the initial datum ?? , the lower bound on the forced solution ?? and the estimate for \(\delta u_k\) given by ?? with respect to the family of remainders derived in Proposition [prop:remainder-exact-form] to obtain \[\begin{align} \left\| u_k(t,\cdot) \right\|_{H^{m-1}_\lambda( \mathbb{R}^+)} &\geq \frac{1}{2} \| U_s' \|_{L^2(a_0+\frac{d}{2}, \infty)} e^{t \sqrt{k} \frac{|U_s''(a_0)|^{1/2}}{2\sqrt{2}}} \\ &\phantom{=}- 4 \bigg( 1 + k\frac{e^{2\lambda^2 t}}{\sqrt{\lambda}} \| U_s'' \|_{L^2_\lambda } \bigg) \times \\ & \phantom{=}\phantom{=} \times \int_0^ t \left\| \sum_{j=1}^7 \mathcal{R}^1_{j,k}(\tau ,\cdot)+\sum_{j=1}^3 \mathcal{R}^2_{j,k}(\tau,\cdot)\, \right\|_{L^2} |e^{\int_0^\tau \sigma_k(l) dl} | e^{ (t-\tau) \sqrt{k} (1+2\|U_s''\|_{L^\infty} )} \,d\tau. \end{align}\] Furthermore, by inserting the estimates for the remainders, ?? and ?? , and restricting ourselves to \(0\leq t\leq 1\), we have \[\begin{align} \label{eq:EstimateProofMainTheorem} \begin{aligned} \left\| u_k(t,\cdot) \right\|_{H^m_\lambda( \mathbb{R}^+)} &\geq \frac{1}{2} \| U_s' \|_{L^2(a_0+\frac{d}{2}, \infty)} e^{t \sqrt{k} \frac{|U_s''(a_0)|^{1/2}}{2\sqrt{2}}} \\ &\phantom{=}- 4 \bigg( 1 + \frac{e^{2\lambda^2 }}{\sqrt{\lambda}} \| U_s'' \|_{L^2_\lambda } \bigg) \mathcal{C}_\mathcal{R}(1+d^{-1}) k \, e^{ (1+2\|U_s''\|_{L^\infty} )\sqrt{k} t} \\ &\phantom{===}\times \int_0^ t \left[ \frac{ e^{ - \frac{d^2}{16 \tau} } }{\sqrt{\tau }}+ e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}} \right] \underbrace{|e^{\int_0^\tau \sigma_k(l) dl} | e^{-\tau \sqrt{k} (1+2\|U_s''\|_{L^\infty} )}}_{\leq 1} \,d\tau. \end{aligned} \end{align}\tag{44}\] The inequality in the last line follows from the Definition of \(\sigma_k\) in ?? . With 44 at hand, all assumptions of Lemma 4 are satisfied. It follows that there exists a \(K\in \mathbb{N}\) which only depend on \(m, U_s, d,\lambda\) and \(\mathcal{C}_\mathcal{R}\) such that the solution \(u_k\) satisfies \[\begin{align} \left\| u_k(t,\cdot) \right\|_{H^{m-1}_\lambda(\mathbb{R}^+)} \geq \frac{ \left\| U_s' \right\|_{L^2(a_0+\tfrac{d}{2},\infty)} }{2} \exp \left( \frac{1}{2}\sqrt{k\frac{|U''(a_0)|}{2}} t \right) \end{align}\] for all \(k\geq K\) and all \(0\leq t\leq \frac{\mathcal{C}}{\sqrt[4]{k}}\) with \(\mathcal{C}= \frac{d}{16(1+ \left\| U''_s \right\|_{L^\infty} )}\). This proves the claim. ◻

section1 -3.5ex 1.5ex Proof of Theorem 2 This section is dedicated to the proof of Theorem 2. We show that for the specific choice of initial data \(u_{\rm in}\) as in 7 , \[\mathbf{U}_{\rm in}(x,y) := \sum_{k = 1}^\infty e^{-\sigma_0 \sqrt[4]{k}} \frac{d}{dy} \phi_{{\rm unst,k}} (y) e^{ {i}k x} \in \mathbb{C},\quad (x,y) \in \mathbb{T} \times \mathbb{R}_+,\] there cannot exist a corresponding weak solution. A contrario, suppose there existed a weak solution \(u\) in the sense of Definition 2 with initial datum either \(u_{\rm in } = \operatorname{Re}(\mathbf{U}_{\rm in})\) or \(u_{\rm in } = \operatorname{Im}(\mathbf{U}_{\rm in})\) where \(\mathbf{U}_{\rm in}=\mathbf{U}_{\rm in}(x,y)\) is defined in 7 . These solutions, in particular, satisfy \(u \in L^\infty(0,T;L^2(\mathbb{T}\times \mathbb{R}_+))\) for some \(\delta>0\). Furthermore, there exists a Fourier expansion \[\begin{align} u(t,x,y) = \sum_{k\in \mathbb{Z}} u_k(t,y) e^{{i}k x} \end{align}\] which holds for almost every \(t\in (0,\delta)\). Consider a \(t_0\in (0,\delta)\) such that the Fourier series converges. Since \(u\) is a weak solution to the linearised Prandtl equations, the coefficients solve 5 for every \(k\in \mathbb{Z}\) with initial value \(u_k(0,y) = e^{-\sigma_0 \sqrt[4]{k}}\frac{d}{dy} \phi_{\mathrm{unst},k}(y)\). The solutions \((u_k)_k\) are uniquely determined and by Theorem 1 satisfy \[\begin{align} \left\| u_k(t,\cdot) \right\|_{L^2(\mathbb{R}^+)} \geq \mathcal{C}e^{-\sigma_0 \sqrt[4] k + \frac{1}{2}\sqrt{k\frac{|U''_s(a_0)|}{2}} ~t} \end{align}\] for every \(k\geq K\), where \(K\in \mathbb{N}\) is fixed, and for every \(0\leq t \leq \tfrac{d}{16(1+ \left\| U'' \right\|_{L^\infty} )} \frac{1}{\sqrt[4] k}\) with \(\mathcal{C}= \frac{1}{4} \bigg( \int_{a_0+\frac{d}{2}}^\infty | \, U_s'(y)\,|^2 dy \,\bigg)^\frac{1}{2}\). We set \(k_{t_0} := \left\lfloor \left( \frac{d}{16(1+ \left\| U''_s \right\|_{L^\infty} )} \cdot \frac{1}{t_0}\right)^4 \right\rfloor\). By Parseval’s identity, it follows \[\begin{align} \left\| u(t_0,\cdot) \right\|_{L^2(\mathbb{T} \times \mathbb{R}^+)} & \geq \left\| u_k(t_0,\cdot) \right\|_{L^2(\mathbb{R}^+)} \\ &\geq \mathcal{C}e^{- \sigma_0 \sqrt[4] k_{t_0} + \tfrac12 \sqrt{k_{t_0}\frac{|U''_s(a_0)|}{2}} ~t_0} \\ & \geq \mathcal{C}\exp \left( \sqrt[4]{k_{t_0}} \left[ \frac{d\sqrt{|U_s''(a_0)}}{\sqrt 2 \cdot 16 (1+ \left\| U''_s \right\|_{L^\infty} )}- \sigma_0\right] \right) \end{align}\] Now, if \(t_0 \to 0^+\) and if \(\sigma_0 <\frac{d|U''(a_0)|^{1/2}}{32(1+ \left\| U''_s \right\|_{L^\infty} )} < \frac{d\sqrt{|U_s''(a_0)}}{\sqrt 2 \cdot 16 (1+ \left\| U''_s \right\|_{L^\infty} )}\), this yields a contradiction to \(u\in L^\infty(0,T;L^2(\mathbb{T}\times \mathbb{R}_+))\). Therefore, the claim is proved.

section1 -3.5ex 1.5ex A collection of auxiliary lemmata

This section is devoted to some technical lemmas used in the previous sections. The results depend on the notation and assumptions introduced throughout the paper, which we recall at the relevant steps.

Lemma 5. For any time \(t \in [0, T_{\rm max}]\), any index \(n \in \{0,1,2\}\) and any frequency \(k \in \mathbb{N}\): \[\begin{align} & \max_{a_0-\frac{d}{2}\leq y \leq a_0+ \frac{d}{2}} \bigg| \partial_y^n \Big( u_s(t,y) - \alpha(t) - \beta(t) (y-a(t))^2 \Big) \bigg| \leq d^{3-n} \big\| U_s^{(3)}\big\|_{L^1} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}} ,\label{eq:ineq-in-lemma-us-appx} \\ & |a'(t)| \leq \frac{2}{|U_s''(a_0)|} \| U_s^{(3)} \|_{L^1(\mathbb{R}_+)} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}}, \qquad |\gamma'(t)| \leq \frac{2}{|U_s''(a_0)|^\frac{3}{4}} \| U_s^{(3)} \|_{W^{1,1}} \frac{e^{ - \frac{d^2}{16 t} }}{\sqrt{\pi t}} ,\label{eq:ineq-in-lemma-us-a39-gamma39-appx} \\ & \max_{a_0-\frac{d}{2}\leq y \leq a_0+ \frac{d}{2}} \Big| \Upsilon^{(n)}\big(\sqrt[4]{k}\,z (t,y) \big) \Big| \leq 4 \big( 1+ |U_s''(a_0)|^{\frac{2-n}{4}} d^{2-n} \big) k^\frac{2-n}{4}.\label{eq:ineq-in-lemma-us-Upsilon-appx} \end{align}\] {#eq: sublabel=eq:eq:ineq-in-lemma-us-appx,eq:eq:ineq-in-lemma-us-a39-gamma39-appx,eq:eq:ineq-in-lemma-us-Upsilon-appx}

Proof. We begin by addressing ?? . Since \(u_s \in \mathcal{C}^\infty(\big[a_0-\frac{d}{2}, \, a_0+ \frac{d}{2}\big] \times \mathbb{R}_+)\), invoking the Taylor theorem, there exists \(\xi_n(t,y)\in \big[a_0-\frac{d}{2}, \, a_0+ \frac{d}{2}\big]\) between \(y\) and \(a(t)\) such that \[\begin{align} \partial_y^n \Big[ u_s(t,y) - u_s(t,a(t)) - \frac{\partial_y^2 u_s(t,a(t))}{2} &(y-a(t))^2 \Big] = \frac{\partial_y^{3} u_s(t,\xi_n(t,y))}{(3-n)!} (y-a(t))^{3-n} \\ &= \frac{(y-a(t))^{3-n}}{(3-n)!} \frac{1}{\sqrt{4\pi t}} \int_0^\infty \bigg( e^{-\frac{|\xi_n(t,y) - \omega|^2}{4t}} - e^{-\frac{|\xi_n(t,y) + \omega|^2}{4t}} \bigg) U_s^{(3)}(\omega) d\omega, \end{align}\] using Green’s formula of the heat kernel on the half-line with Dirichlet conditions. From the quadratic assumption, \(U_s^{(3)}\equiv 0\) in \(\big[a_0-d, \, a_0+ d\big]\), thus the last integral is on \(\mathbb{R}_+\setminus \big[a_0-d, \, a_0+ d\big]\). Inequality ?? follows thus from \[|y-a(t)| \leq d,\qquad |\xi_n(t,y) - \omega|\geq \frac{d}{2}, \qquad |\xi_n(t,y) + \omega| = \xi_n(t,y)+\omega \geq \xi_n(t,y)\geq a_0 -\frac{d}{2} \geq \frac{d}{2},\] for any \(\omega \in \mathbb{R}_+ \setminus [a_0-d, \, a_0+ d]\) and any \(t \in [0, T_{\rm max}]\).

We next address ?? and recall that \(\partial_y^2 u_s(t,a(t))+ \partial_y^3 u_s(t,a(t)) = 0\) is for any \(t\in [0, T_{\rm max}]\). Hence, \[\begin{align} |a'(t)| = \left|\frac{\partial_y^3 u_s(t,a(t))}{\partial_y^2 u_s(t,a(t))} \right| &= \frac{1}{|\partial_y^2 u_s(t,a(t))|} \frac{1}{\sqrt{4\pi t}} \left| \int_0^\infty \bigg( e^{-\frac{|a(t) - \omega|^2}{4t}} - e^{-\frac{|a(t) + \omega|^2}{4t}} \bigg) U_s^{(3)}(\omega) d\omega \right|. \end{align}\] The first inequality of ?? follows from \(|\partial_y^2 u_s(t,a(t))| > \frac{|U_s''(a_0)|}{2}\) and, once more, \(|a(t) - \omega| \geq \frac{d}{2}\) and \(|a(t) + \omega| \geq a(t) \geq a_0 - \frac{d}{2}\geq \frac{d}{2}\), for any \(t \in [0, T_{\max}]\) and \(\omega \in \mathbb{R}_+ \setminus [a_0-\frac{d}{2}, \, a_0+ \frac{d}{2}]\). For what concerns \(\gamma'(t)\) in ?? , we use the fact that \(u_s\) satisfies the heat equation and therefore \(\partial_t \partial_y^2 u_s = \partial_y^4 u_s\). We obtain \[\gamma'(t) = \frac{d}{dt} \Big[ \Big( -\frac{\partial_y^2 u_s(t,a(t))}{2}\Big)^\frac{1}{4} \Big] = \Big( -\frac{\partial_y^2 u_s(t,a(t))}{2}\Big)^{-\frac{3}{4}} \Big( \partial_y^4 u_s(t, a(t)) + \partial_y^3u_s (t, a(t)) \Big).\] The second inequality in ?? follows by applying Green’s formula to the resulting expression and using estimates analogous to those before.

We finally address ?? . Recalling that \(z(t,y)= \gamma(t) (y-a(t))e^{-\frac{i\pi}{8}}\), we first remark that by substitution: \[\begin{align} \frac{1}{2} \bigg| 1 + \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\, z(t,y)}{\sqrt{2}} \bigg) \bigg| = \frac{1}{\sqrt{2\pi}} \bigg| \int_{-\infty}^{\sqrt[4]{k}\,\gamma(t) (y-a(t))} \exp\left( -e^{-\frac{i \pi}{4}}\frac{\omega^2}{2}\right)d\omega \, e^{-\frac{ i \pi}{8}} \bigg| \leq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty e^{ - \frac{\omega^2}{2\sqrt{2}}}d\omega = \sqrt[4]{2}. \end{align}\] Hence, using the definition of \(\Upsilon\) in 13 , we find that for \(n = 0\): \[\begin{align} |\Upsilon(&\sqrt[4]{k}\, z(t,y) ) | = \bigg|\; \frac{1}{\sqrt{2\pi}}\,\zeta\, e^{-\frac{\zeta^2}{2}} + (1+\zeta^2) \frac{\, \mathrm{erf}\left( \zeta/\sqrt{2} \right) + 1}{2} \;\bigg|_{\zeta =\sqrt[4]{k}\, z(t,y) = \sqrt[4]{k}\,\gamma(t) (y-a(t)) e^{-\frac{i\pi}{8}}} \\ &\leq \frac{1}{\sqrt{2\pi}} \Big( \sqrt[4]{k}\,\gamma(t) |\,y-a(t)| \, \Big) e^{- \frac{1}{2\sqrt{2}} \big(\sqrt[4]{k}\,\gamma(t) |y-a(t)|\,\big)^2} + \Big( 1+ \sqrt{k} \,\gamma(t)^2 (y-a(t))^2 \Big) \frac{1}{2} \bigg| 1 + \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\, z(t,y)}{\sqrt{2}} \bigg) \bigg| \\ &\leq \frac{\sqrt[4]{2}}{\sqrt{2\pi e}} + \sqrt[4]{2}+ \sqrt[4]{2} \sqrt{k} \,\gamma(t)(y-a(t))^2 \leq 1 + |U_s''(a_0)|^\frac{1}{2} d^2 \sqrt{k} \leq 4 \big( 1 + |U_s''(a_0)|^\frac{2-0}{4} d^{2-0} \big) k^\frac{2-0}{4} , \end{align}\] where we also used \(\max\limits_{\mathrm{x}\in \mathbb{R}} \big|\mathrm{x} \, e^{-\mathrm{x}^2/(2\sqrt{2})}\big|=\sqrt[4]{2}/\sqrt{e}\) and \(|\gamma(t)|\leq |U_s''(a_0)|^{1/4}\). For \(n = 1\), we proceed similarly: \[\begin{align} |\Upsilon'(\sqrt[4]{k}\, z(t,y) ) | &= \bigg| \sqrt{\frac{2}{\pi}}e^{-\frac{\zeta^2}{\sqrt{2}}} + \zeta \Big( 1 + \, \mathrm{erf}\Big( \frac{\zeta}{2}\Big) \Big) \bigg|_{\zeta = \sqrt[4]{k}\,\gamma(t) (y-a(t)) e^{-\frac{i\pi}{8}}} \leq \sqrt{\frac{2}{\pi}} + \sqrt[4]{k}\,\gamma(t) |y-a(t)|\,2 \sqrt[4]{2} \\ &\leq \sqrt{\frac{2}{\pi}} +\sqrt[4]{k} |U_s''(a_0)|^{\frac{1}{4}}d \,2 \sqrt[4]{2} \leq 1 + 4 |U_s''(a_0)|^{\frac{1}{4}}d\sqrt[4]{k} \leq 4 \big( 1 + |U_s''(a_0)|^{\frac{2-1}{4}}d^{2-1} \big)k^\frac{2-1}{4}. \end{align}\] The final case \(n = 2\) is straightforward, since we simply obtain: \[\begin{align} |\Upsilon''(\sqrt[4]{k}\, z(t,y) ) | = \bigg| 1 + \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\, z(t,y)}{\sqrt{2}} \bigg) \bigg| \leq 2\sqrt[4]{2} \leq 4 \big( 1 + |U_s''(a_0)|^{\frac{2-2}{4}}d^{2-2} \big)k^\frac{2-2}{4}. \end{align}\] This concludes the proof of 5. ◻

Lemma 6. For any time \(t\in [0, T_{\rm max}]\) and any \(y \in \mathbb{R}_+ \setminus [a_0 - \tfrac{d}{4}, a_0+\tfrac{d}{4}]\), the following inequality holds true: \[\label{eq:erf-H-estimate-appx} \bigg| \, \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \, \bigg| \leq \frac{13}{|U_s''(a_0)|^\frac{1}{4}} \frac{1}{\sqrt[4]{k}\,d} e^{ -\sqrt{k} \,\big(\frac{d}{16}\big)^2 \big(\frac{|U_s''(a_0)|}{2} \big)^{1/2}}.\qquad{(21)}\]

Proof. We first address the case of \(y > a_0+\tfrac{d}{4}\), which implies \(y-a(t) >\frac{d}{8}>0\), since \(a(t) \in [a_0-\tfrac{d}{8}, a_0 + \tfrac{d}{8}]\) for any \(t\in [0, T_{\rm max}]\). In particular \(H(y-a(t)) = 1\), for \(y > a_0+\tfrac{d}{4}\).

We write the \(\, \mathrm{erf}\)-function as a path-integral in the complex plane with respect to its derivative \(\, \mathrm{erf}'(\zeta) = \frac{2}{\sqrt{\pi}}e^{\zeta^2}\). Since \[\frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}} = \frac{\sqrt[4]{k}}{\sqrt{2}} \, (y-a(t)) \, \gamma(t) e^{-\frac{i \pi }{8}}\in \mathbb{C},\quad \text{with} \quad \gamma(t)= \bigg( \frac{|\partial_y^2 u_s(t,a(t))|}{2}\bigg)^\frac{1}{4} \geq \bigg( \frac{|U_s''(a_0)|}{4}\bigg)^\frac{1}{4} = \frac{|U_s''(a_0)|^\frac{1}{4}}{\sqrt{2}} >0,\] we choose the complex line path \(\Gamma\) given by \(\Gamma(\omega) = \omega e^{-\frac{i \pi}{8}}\) with \(\omega \in [0, \frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t)(y-a(t))]\subset \mathbb{R}_+\). Hence, also using \(\, \mathrm{erf}(0) = 0\), the following identity holds true: \[\label{eq:lemma-erf-H-erf-part} \begin{align} \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) &= \frac{2}{\sqrt{\pi}} \int_\Gamma e^{-\zeta^2} d\zeta = \frac{2}{\sqrt{\pi}} \int_0^{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))} e^{-\Gamma(\omega)^2} \Gamma'(\omega) d\omega \\ &= \frac{2}{\sqrt{\pi}} \int_0^{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))} \exp\big( -e^{-\frac{i\pi}{4}}\omega^2\big) d \omega\, e^{-\frac{i\pi}{8}}. \end{align}\tag{45}\] Next, we apply Cauchy’s theorem to shift the half-line Gaussian integral to an integral along the complex half-line at an angle of \(e^{-\frac{i\pi}{8}}\) (where the integrand still retains exponential decay): \[\label{eq:lemma-erf-H-1-is-int-et2} 1 = \frac{2}{\sqrt{\pi}} \int_0^\infty e^{-l^2} dl = \frac{2}{\sqrt{\pi}} \int_0^\infty \exp\big( -e^{-\frac{i\pi}{4}}\omega^2\big) d \omega\, e^{-\frac{i\pi}{8}}.\tag{46}\] Here, the contribution from the circular arc at infinity (required to close the contour) is zero due to the exponential decay of \(e^{-\zeta^2}\) in the sector \(\arg(\zeta)< \frac{\pi}{4}\). Combining 45 with 46 implies \[\label{eq:lemma-erf-H-to-cite-at-the-end} \begin{align} \bigg| \,\, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \, \bigg| &= \frac{2}{\sqrt{\pi}} \bigg| \int_{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))}^\infty \exp\big( -e^{-\frac{i\pi}{4}}\omega^2\big) d \omega\, e^{-\frac{i\pi}{8}} \bigg| \\ &\leq \frac{2}{\sqrt{\pi}} \int_{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))}^\infty e^{-\frac{\omega^2}{\sqrt{2}}} d\omega = \sqrt{\frac{2}{\pi}} \int_{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))}^\infty \frac{1}{\omega} \frac{d}{d\omega} \bigg[ - e^{-\frac{\omega^2}{\sqrt{2}}} \bigg] d\omega, \end{align}\tag{47}\] where all terms in the last integrand are positive. Since \(\omega \geq \frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))\), the following upper bound holds true: \[\frac{1}{\omega} \leq \frac{\sqrt{2}}{\sqrt[4]{k}\gamma(t) (y-a(t))} \leq \frac{16}{|U_s''(a_0)|^\frac{1}{4}\sqrt[4]{k} \,d}.\] Remarking that \(16\sqrt{2}/\sqrt{\pi} \leq 13\), we eventually obtain \[\begin{align} \bigg| \,\, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \, \bigg| &\leq \frac{13}{|U_s''(a_0)|^\frac{1}{4}\sqrt[4]{k} \,d} \int_{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))}^\infty \frac{d}{d\omega} \bigg[ - e^{-\frac{\omega^2}{\sqrt{2}}} \bigg] d\omega \\ &\leq \frac{13}{|U_s''(a_0)|^\frac{1}{4}\sqrt[4]{k} \,d} \exp \bigg( -\sqrt{k} \,\gamma(t)^2 \frac{(y-a(t))^2}{2\sqrt{2}} \bigg). \end{align}\] Hence, when \(y \geq a_0+\tfrac{d}{2}\), the estimate ?? follows from the fact that \(\gamma(t)^2>|U_s''(a_0)|^{1/2}/\sqrt{2}\) and \(|y-a(t)|\geq d/8\).

For the case \(y < a_0 - \frac{d}{4}\), the argument is analogous. Here, \(y-a(t) < 0\) and thus \(H(y-a(t)) = -1\). The integral representation becomes \[\bigg| \, \mathrm{erf}\bigg( \frac{\sqrt[4]{k}\,z(t,y)}{\sqrt{2}}\bigg) - H(y-a(t)) \bigg| = \frac{2}{\sqrt{\pi}} \bigg| \int^{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (y-a(t))}_{-\infty} e^{ -e^{-\frac{i\pi}{4}}\omega^2 }d \omega \bigg| = \frac{2}{\sqrt{\pi}} \bigg| \int_{\frac{\sqrt[4]{k}}{\sqrt{2}} \gamma(t) (a(t)-y)}^{\infty} e^{ -e^{-\frac{i\pi}{4}}\omega^2 }d \omega \bigg|.\] Since this expression has the same form as in 47 , the same procedure applies, thereby completing the proof. ◻

Lemma 7. For any \(n \in \mathbb{N}\) with \(n \geq 3\) and any \(t\in [0, T_{\rm max}]\), the following inequality holds true: \[\max_{a_0-\frac{d}{2}\leq y \leq a_0 + \frac{d}{2}} \big|\, \Upsilon^{(n)}(\sqrt[4]{k}\, z(t,y) )\, \big| \leq (n-3)! \, 2\big(\, 1 + |\,U_s''(a_0)\,|^\frac{n-3}{4} d^{n-3}\,\big) k^\frac{n-3}{4}.\]

Proof. By definition, the third derivative of \(\Upsilon\) is given by \(\Upsilon^{(3)}(\zeta) = \sqrt{ \frac{2}{\pi}} \, e^{-\frac{\zeta^2}{2}}\). Therefore, for \(n\geq 3\) and \(\zeta \in \mathbb{C}\), the \(n\)-th derivative of \(\Upsilon\) can be expressed as: \[\label{eq:appx-Upsilon9440n41} \Upsilon^{(n)}(\zeta ) = \sqrt{ \frac{2}{\pi}} \frac{d^{n-3}}{d \zeta^{n-3}} \Big[ e^{-\frac{\zeta^2}{2}} \Big] = \sqrt{ \frac{2}{\pi}} \,\, \text{He}_{n-3} \big( \zeta \big) \, e^{-\frac{\zeta^2}{2}},\tag{48}\] where \(\text{He}_{n-3}\) is the probabilistic Hermite polynomial of degree \(n-3\in \mathbb{N}\), satisfying \[\text{He}_{n-3}(\zeta) = (n-3)! \sum_{m=0}^{\lfloor \frac{n-3}{2} \rfloor} \frac{(-1)^m}{m!(n-3-2m)!}\frac{\zeta^{n-3-2m}}{2^m}.\] We use the following elementary estimate for \(\zeta \in \mathbb{C}\): \[| \,\text{He}_{n-3}(\zeta) \,| \leq (n-3)! \big( 1 + |\zeta|^{n-3} \big) \sum_{m = 0}^\infty \frac{1}{2^m} = (n-3)!\, 2 \big( 1 + |\zeta|^{n-3} \big)\] We set \(\zeta =\sqrt[4]{k}\,z(t,y) = \sqrt[4]{k} \gamma(t) (y-a(t)) e^{-\frac{i\pi}{8}}\) with \(\gamma(t) = |\partial_y^2u_s(t,a(t))/2|^{1/4}\) and remark that the exponential term satisfies \[\Big| \exp \Big( -\sqrt{k} \,\frac{\gamma(t)^2 (y-a(t))^2 e^{-\frac{i\pi}{4}}}{2} \Big) \Big| = \exp \Big( -\sqrt{k} \,\frac{\gamma(t)^2 (y-a(t))^2 }{2\sqrt{2}} \Big) \leq 1.\] Additionally, from the assumptions in 16 , we have \(\gamma(t) \leq |U_s''(a_0)|^{1/4}\) and \(a(t) \in [a_0-\tfrac{d}{8},\, a_0+\tfrac{d}{8}]\). Thus, since \(|y-a(t)|\leq d\) for any \(y \in [a_0-\tfrac{d}{2},\, a_0+\tfrac{d}{2}]\), it follows that \[\begin{align} \max_{a_0-\frac{d}{2}\leq y \leq a_0 + \frac{d}{2}} \big| \,\text{He}_{n-3}\big(\sqrt[4]{k}\, z(t,y) \big)\, \big| &\leq (n-3)! \,2 \max_{a_0-\frac{d}{2}\leq y \leq a_0 + \frac{d}{2}} \Big( 1 + \gamma(t)^{n-3} |y-a(t)|^{n-3} k^\frac{n-3}{4} \,\Big)\\ &\leq (n-3)! \, 2\big(\, 1 + |\,U_s''(a_0)\,|^\frac{n-3}{4} d^{n-3}\,\big) k^\frac{n-3}{4}. \end{align}\] Combining this with the expression for \(\Upsilon^{(n)}\) in 48 and noting that \(\sqrt{2/\pi}\leq 1\), completes the proof of the lemma. ◻

section1 -3.5ex 1.5ex Basic estimates for the heat equation In this section we provide some elementary estimates in weighted Sobolev spaces of the heat equation in the half-line. For general initial data \(U_s \in L^2(\mathbb{R}_+)\), the unique solution \(u_s \in \mathcal{C}(\mathbb{R}_+, L^2(\mathbb{R}_+))\cap L^2(\mathbb{R}_+, H^1_0(\mathbb{R}_+))\) of System 1 is given by the Dirichlet Green function: \[\label{appx-eq:green-formula} u_s(t,y) = \int_0^{\infty} G_D(t,y, z) \,U_s (z) \,d z, \qquad G_D(t,y,z) = \frac{1}{\sqrt{4\pi t}}\Big( e^{-\frac{|y-z|^2}{4t}} - e^{-\frac{|y+z|^2}{4t}} \Big),\tag{49}\] for any \(t,\,y >0\). The following lemma concerns the propagation of the \(H^m_\lambda(\mathbb{R}_+)\)-norm.

Lemma 8. Let \(m \in \mathbb{N}_0\), \(\lambda > 0\) and \(U_s \in H^m_\lambda(\mathbb{R}_+)\). If \(m\geq 1\), assume that \(U_s(0) = \dots = U_s^{2\lfloor \frac{m}{2} \rfloor }(0) = 0\). Then \(u_s\) defined by 49 belongs to \(\mathcal{C}([0, +\infty[, H^m_\lambda (\mathbb{R}_+))\) and satisfies \[\| \,\partial_y^j u_s(t , \cdot ) \,\|_{L^2_\lambda(\mathbb{R}_+)} \leq e^{2 \lambda^2 t} \big\| \,U_{s}^{(j)}\, \|_{L^2_\lambda (\mathbb{R}_+)},\] for any \(t\geq 0\) and \(j\in \{0, \dots, m\}\).

Proof. We denote by \(\tilde{U}_s\) and \(\tilde{u}_s\) the odd extensions of \(U_s\) and \(u_s\) to the whole line \(y \in \mathbb{R}\). Due to the compatibility conditions of \(U_s\) at \(y=0\), we have \(\tilde{U}_s \in H^m(\mathbb{R})\). Since \(\tilde{u}_s\) solves the heat equation on the whole line with initial datum \(\tilde{U}_s\), it follows that \(\tilde{u}_s \in \mathcal{C}(\mathbb{R}_+, H^m(\mathbb{R}))\). Moreover, \(\tilde{u}_s\) and its derivatives can be expressed using the heat kernel: \[\partial_y^j \tilde{u}_s(t,y) = \frac{1}{\sqrt{4\pi t}} \int_{\mathbb{R}} e^{-\frac{|y-z|^2}{4t}} \tilde{U}_s^{(j)}(z) dz,\] for all \(t>0\), \(y \in \mathbb{R}\) and \(j \in \{0, \dots,m \}\). We apply the triangular inequality to the following expression: \[\begin{align} \big|\,e^{\lambda |y|} \partial_y^j \tilde{u}_s(t,y)\,\big| &= \bigg|\, \frac{1}{\sqrt{4\pi t}} \int_{\mathbb{R}} e^{-\frac{|y-z|^2}{4t} +\lambda |y- z +z |} \tilde{U}_s^{(j)} (z) d z \,\bigg| \leq \int_{\mathbb{R}} K_\lambda (t, y-z) e^{\lambda |z| } |\, \tilde{U}_s^{(j)}(z) |dz , \end{align}\] where \(K_\lambda\) denotes the kernel \(K_\mu(t,z ):= \frac{1}{\sqrt{4\pi t}} e^{-\frac{z^2}{4t}+ \lambda |z|}\), for any \(t>0\) and \(z \in \mathbb{R}\). The \(L^1\)-norm of \(K_\lambda\) satisfies \[\begin{align} \int_{\mathbb{R}} | \, K_\lambda (t, z) \, |dz &= \frac{1}{\sqrt{\pi t}} \int_{0}^\infty e^{-\frac{z^2}{4t}+ \lambda z}dz = \frac{e^{\lambda^2 t}}{\sqrt{\pi t}} \int_{0}^\infty e^{-\left( \frac{z}{2 \sqrt{t}} - \lambda \sqrt{t} \right)^2}dz = e^{\lambda^2 t} \frac{2}{\sqrt{\pi}} \int_{-\lambda \sqrt{t}}^\infty e^{-w^2}dw\\ & = e^{\lambda^2 t} \bigg( \frac{2}{\sqrt{\pi}} \int_{0}^\infty e^{-w^2}dw + \frac{2}{\sqrt{\pi}} \int_{0}^{\lambda \sqrt{t}} e^{-w^2}dw \bigg) = e^{\lambda^2 t} \Big( 1 + \, \mathrm{erf}\big( \lambda \sqrt{t} \big) \Big)\leq e^{2 \lambda^2 t} . \end{align}\] Thanks to Young’s inequality, \(\partial_y^j \tilde{u}_s(t, \cdot) \in L^2_\lambda (\mathbb{R})\) for any \(t>0\) and satisfies \[\begin{align} \big\| \, \partial_y^j \tilde{u}_s(t, \cdot)\, \big\|_{L^2_\lambda(\mathbb{R})} \leq \| K_\lambda (t, \cdot ) \|_{L^1(\mathbb{R})} \| \,\tilde{U}_s^{(j)} \|_{L^2_\lambda (\mathbb{R})} \leq 2 \, e^{2 \lambda^2 t} \big\| \,U_s^{(j)} \|_{L^2_\lambda (\mathbb{R}_+)}. \end{align}\] It remains to prove that \(\tilde{u}_s\) in \(\mathcal{C}([0, \infty[, H^m_\lambda(\mathbb{R}))\). Thanks to Scheffé’s lemma, \(K_\lambda \in \mathcal{C}( \, ]0, \infty[, L^1(\mathbb{R}))\) which implies that \(\partial_y^j \tilde{u}_s\) in \(\mathcal{C}(]0, \infty[, L^2_\lambda(\mathbb{R}))\) for any \(j \in \{0, \dots, m\}\). As \(t\to 0+\), \(K_\lambda\) behaves like the heat kernel and converges distributionally to the Dirac-delta: \[\lim_{t\to 0+ } K_\lambda(t, \cdot ) = \delta_0,\quad \text{in} \quad \mathcal{D}'(\mathbb{R}).\] The continuity at \(t = 0\) is hence achieved by the density of \(\mathcal{D}(\mathbb{R})\) in \(H^m_\lambda(\mathbb{R})\) and the boundness of \(\| K_\lambda(t, \cdot) \|_{L^1(\mathbb{R})}\) around \(t= 0\). Details are left to the reader. ◻

section1 -3.5ex 1.5ex A comparison with the regular and shear-layer velocities

In this final section, we compare our matched asymptotic profile \(\phi_{\text{inn}, k}+ \phi_{\text{out}, k}\) introduced in 3 with the “regular” and “shear-layer” profiles \(v_\varepsilon^{reg}+ v_\varepsilon^{sl}\) proposed by Gérard-Varet and Dormy in [20].

We first introduce some notation from Section 4 of [20], supported by explicit formulas derived in [24]. Let \(\varepsilon= 1/k >0\) and and let \(\tau \in \mathbb{C}\) be as in (1.7) of [20]: \({\rm Im}(\tau)<0\) and there exists a function \(W \in \mathcal{C}^\infty(\mathbb{R}, \mathbb{C})\) satisfying \[(\tau-z^2)^2 \frac{d}{dz} W(z) + i \frac{d^3}{dz^2} \big( (\tau-z^2) W(z) ) = 0,\qquad \lim_{z \to -\infty} W(z) = 0,\qquad \lim_{z \to +\infty} W(z) = 1.\] In Corollary 1.2 of [24], we showed that there is a unique solution \((\tau, W)\) given by \(\tau = -e^{\frac{i \pi}{4}}\) and \[W(z) = \frac{\Upsilon\big( \zeta \big) }{1 +\zeta^2 } = \frac{1}{\sqrt{2\pi}}\frac{\zeta}{1 +\zeta^2}\, e^{-\frac{\zeta^2}{2}} + \frac{1}{2} \left(1 + \, \mathrm{erf}\left( \frac{\zeta}{\sqrt{2}} \right) \right) ,\qquad \text{with }\zeta = e^{-\frac{i \pi}{8}} z \in \mathbb{C}.\] The function \(\sigma_k(t)\) of 3 is in [20] denoted by \(i\, \varepsilon^{-1} \omega(\varepsilon, t)\), as \(\omega(\varepsilon, t)\) is defined as \[\omega(\varepsilon, t) := -u_s(t,a(t)) + \frac{\varepsilon^{1/2}}{\sqrt{2}} |\partial_y^2 u_s(t,a(t))|^\frac{1}{2}\tau = -u_s(t,a(t)) + e^{i\frac{5 \pi}{4}} \sqrt{\frac{|\partial_y^2 u_s(t,a(t))|}{2k}} = \frac{\sigma_k(t)}{i k}.\] The regular and shear layer velocities are defined in Section 4 of [20] as follows: \[\label{eq:shear-and-regular-layers} \begin{alignedat}{4} &v_\varepsilon^{reg}(t,y) &&= H(y-a(t)) \bigg( u_s(t,y) -u_s(t,a(t)) + \frac{\varepsilon^{1/2}}{\sqrt{2}} |\partial_y^2 u_s(t,a(t))|^\frac{1}{2}\tau \bigg),\\ &v_\varepsilon^{sl}(t,y) &&= \frac{\varepsilon^{1/2}}{\sqrt{2}} \varphi(y-a(t)) |\partial_y^2 u_s(t,a(t))|^{1/2} V \left( |\partial_y^2 u_s(t,a(t))|^{1/4} \frac{y-a(t)}{(2\varepsilon)^{1/4}} \right), \end{alignedat}\tag{50}\] where \(\varphi\) is a smooth truncation function near \(0\) and \(V(z) =(\tau-\zeta^2)(W(z)-H(z))\) for any \(z \in \mathbb{R}\).

We compare these profiles with our matched asymptotic construction \(\phi_{{\rm inn},k} + \phi_{{\rm out},k}\) via the following lemma.

Lemma 9. Replace \(\varphi(y-a(t))\) in 50 with \(\chi(y)\) of 1. Define \(v_\varepsilon^{new} : [0, T_{\rm max}] \times \mathbb{R}_+ \to \mathbb{C}\) \[v_\varepsilon^{new}(t,y) := \chi(y) H(y-a(t)) \bigg( u_s(t,a(t)) + \frac{\partial_y^2 u_s(t,y(a))}{2}(y-a(t))^2 - u_s(t,y) \bigg).\] Then, for any \((t,y) \in [0, T_{\rm max}] \times \mathbb{R}_+\), the following identity holds true: \[\label{eq:relation-our-profiles-GV-profiles} \phi_{{\rm inn}, k}(t,y)+\phi_{{\rm out}, k}(t,y) = \Big( v_\varepsilon^{reg}(t,y)+ v_\varepsilon^{sl}(t,y) + v_\varepsilon^{new}(t,y) \Big) e^{i \varepsilon^{-1}\int_0^t \omega(\varepsilon, s) ds}.\qquad{(22)}\]

Proof. We first rewrite the outer layer \(\phi_{{\rm out}, k}\) of 3 as \[\label{eq:final-sec-phi-out-v-reg} \begin{align} \phi_{{\rm out}, k}(t,y) &= \big(1 - \chi(y)\big) H\big(y-a(t)\big) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) e^{\int_0^t \sigma_k(\tau) d\tau } \\ &=v_{\varepsilon}^{reg}(t,y) e^{i \varepsilon^{-1}\int_0^t \omega(\varepsilon, s) ds} - \chi(y) H\big(y-a(t)\big) \bigg( u_s(t,y) + \frac{\sigma_k(t)}{{i}k} \bigg) e^{\int_0^t \sigma_k(\tau) d\tau }. \end{align}\tag{51}\] Next, we recast \(V(z)\) in terms of \(\Upsilon(\zeta)\) with respect to the relation \(\zeta = e^{-\frac{i \pi}{8}} z\in \mathbb{C}\). Recalling that \(\tau = -e^{\frac{i \pi}{4}} = e^{i \frac{5 \pi}{4}}\): \[\begin{align} V(z) &= \big( \tau -z^2 \big) \Big( W(z) - H(z) \Big) = e^{i\frac{5 \pi}{4}} \big( 1+e^{ - i \frac{\pi}{4}} z^2 \big) \big( W(z) - H(z) \big) \\ & = e^{i\frac{5\pi}{4}} \big( 1 + \zeta^2 \big) \bigg( \frac{\Upsilon \big( \zeta \big) }{1+ \zeta^2} - H(z) \bigg) = e^{i\frac{5\pi}{4}} \bigg( \Upsilon \big( \zeta \big) - (1+\zeta^2) H(z) \bigg). \end{align}\] Hence, using the short notation \(\beta(t) = \partial_y^2 u_s(t,a(t))/2<0\), we deduce from 50 (with \(\chi\) instead of \(\varphi\)) that \[\begin{align} v^{\rm sl}_\varepsilon(t,y) & e^{i \varepsilon^{-1}\int_0^t \omega(\varepsilon, s) ds} = \frac{\varepsilon^{1/2}}{\sqrt{2}} \chi(y) |\beta(t)|^{1/2} V \left( |\beta(t)|^{1/4} \frac{y-a(t)}{\varepsilon^{1/4}} \right) e^{i \varepsilon^{-1}\int_0^t \omega(\varepsilon, s) ds} \\ &= \sqrt{\frac{|\beta(t)|}{2k}} \chi(y) e^{i\frac{5\pi}{4}} \left( \Upsilon \left( \sqrt[4]{|\beta(t)|k} (y-a(t)) e^{-\frac{i\pi}{8}} \right) - \Big( 1 + e^{-i \frac{\pi}{4}}\sqrt{k |\beta(t)|} (y-a(t))^2 \Big) \right) e^{\int_0^t \sigma_k(\tau) d\tau }. \end{align}\] From the definition of \(\phi_{\rm inn, k}\) in 3 we finally derive: \[\begin{align} \phi_{\text{inn}, k} (t,y) = \, &v^{\rm sl}_\varepsilon(t,y) e^{i \varepsilon^{-1}\int_0^t \omega(\varepsilon, s) ds} + \\ &+\chi(y) H(y-a(t)) \Bigg( \frac{\sigma_k(t)}{ik} + u_s(t,a(t)) + \frac{\partial_y^2 u_s(t,a(t))}{2} (y-a(t))^2 \Bigg)e^{\int_0^t \sigma_k(s) ds }. \end{align}\] By summation, this last identity with 51 , we establish ?? . This concludes the proof of the lemma. ◻

Remark 4. We conclude with a comment on the role of the additional term \(v_\varepsilon^{\mathrm{new}}\) in ?? . Since \(\phi_{{\rm inn}, k}\) and \(\phi_{{\rm out}, k}\) are smooth in \(y=a(t)\) for \(t>0\), if one considers just the approximate profile \(v^{sl}_\varepsilon+ v^{reg}_\varepsilon\) of [20] then \[\partial_y^4 \bigl(v^{sl}_\varepsilon+ v^{reg}_\varepsilon\bigr)\big|_{y = a(t)} = \partial_y^4 v^{new}_\varepsilon\big|_{y = a(t)} + \partial_y^4 \bigl(\phi_{{\rm inn}, k} + \phi_{{\rm out}, k} \bigr)\big|_{y = a(t)} \sim \partial_y^3 u_s(t,a(t)) \, \delta_{a(t)} .\] As a consequence, whenever \(\partial_y^3 u_s(t,a(t)) \neq 0\) for some \(t>0\), \(v^{sl}_\varepsilon+ v^{reg}_\varepsilon\) is at most in \(W^{3,\infty}\) near \(y = a(t)\). In other words, if \(\partial_y^3 u_s(t,a(t)) \neq 0\), one cannot rely solely on the profile of [20] to establish ill-posedness in Sobolev spaces with \(y\)-regularity higher than \(W^{3,\infty}\) for \(v\).

Nevertheless, we remark that, in the autonomous case \(u_s(t,y) = U_s(y)\) with \(U_s\) quadratic around \(y = a_0\), this singularity never occurs (\(v_\varepsilon^{new} \equiv 0\)). By contrast, if \(u_s\) satisfies the heat equation, the singularity does occur.

On the other hand, this issue is resolved by our work, introducing the correction \(v^{\mathrm{new}}_\varepsilon\). The mechanism becomes much more transparent when working with our profiles \(\phi_{{\rm inn}, k}\) and \(\phi_{{\rm out}, k}\) in 3, which are smooth by construction. This is one of the reasons motivating our use of matched asymptotics.

Acknowledgements: We would like to thank David Gérard-Varet for his invitation and warm hospitality at the UFR de Mathématiques and IMJ-PRG, Université Paris Cité. We are also grateful to David Gérard-Varet for several fruitful discussions on the subject of this work. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Author contributions:
Francesco De Anna: conceptualization, methodology, formal analysis, writing – original draft.
Joshua Kortum: conceptualization, formal analysis, writing - review & editing.

Declaration of competing interests:
No known competing interests.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process:
During the preparation of this work the authors used M365 Copilot (based on the GPT-5 reasoning model) to correct orthographic, grammatical, or typographical inaccuracies. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

Data availability:
No data was used for the research described in the article.

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