Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

We address a physically-meaningful extension of the Prandtl system, also known as hyperbolic Prandtl equations. We show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k^(1/3) in the frequencies of the tangential direction, akin the pioneering result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was of order k^(1/2)). We emphasise however that this growth rate does not imply ill-posedness in Gevrey-class m, with m>3 and we relate also these aspects to the original Prandtl equations in Gevrey-class m, with m>2. By relaxing certain assumptions on the shear flow and on the solutions, namely allowing for unbounded flows in the vertical direction, we however determine a suitable shear flow for the mentioned ill-posedness in Gevrey spaces.


Introduction 1.Presentation of the problem
In this work we are mainly concerned with the following hyperbolic extension of the linearised Prandtl equations (cf.[1,8,21,28,29]) (0, T ) × T. (1.1) The main unknown is the scalar component u = u(t, x, y) of the velocity field (u, v) T , while v = v(t, x, y) is formally determined by the divergence-free condition v(t, x, y) = − y 0 ∂ x u(t, x, z)dz.
The function U s = U s (y) is prescribed and defines a suitable shear flow (U s (y), 0) around which the model have been linearised.We aim to develop an ill-posedness theory of (1.1) in Sobolev spaces akin to the one in the seminal paper of Gérard-Varet and Dormy [12], for the linearization of the Prandtl equations around a shear flow.Our main result concerns a norm inflation phenomenon in Sobolev regularities: some initial data, that are of unitary size in H m along the variable x ∈ T, generate solutions of inflated size δ −1 after a time t = δ > 0 arbitrarily small (cf.Theorem 1.1).
In [8], we proved that System (1.1) is locally well-posed when the initial data have regularity Gevreyclass 3 along x ∈ T. With this work, we aim to provide further insights on the well-or ill-posedness issue when the initial data are Gevrey-class m, m > 3. We relate moreover these aspects to the original Prandtl equations in Gevrey-class m, m > 2, supporting with certain remarks (cf.Section 1.5) the main result of Gérard-Varet and Dormy in [12].

A brief overview on the analysis of the Prandtl equations
System (1.1) shares several similarities with the classical Prandtl system, whose leading equation is always on the state variable u = u(t, x, y) and replaces the first equation in (1.1) (with initial data only on u).Nowadays, the Prandtl model belongs to the mathematical and physical folklore, predicting the dynamics of boundary layers in fluid mechanics.Numerous mathematical investigations have been performed during the past decades, in order to reveal the underlying instabilities of the solutions.In this paragraph, we shall briefly recall some of the results about the related well-and ill-posedness issues.For further analytical problems like asymptotic behaviour, boundary-layer separations, boundary-layer expansions, homogenisation and more refined models, we refer for instance to [7,9,11,13,18,20,27,30] and references therein.
Regarding the well-posedness of the classical Prandtl equations, there are two branches of assumptions that one might impose on the initial data: they are monotonic in the vertical variable (hence they do not allow for non-degenerate critical points) or they belong to suitable function spaces that control all derivatives along the tangential variable (i.e.analytic or Gevrey initial data).
In case the initial data are monotonic, the system is known to be well-posed in standard functions spaces, such as Sobolev and Hölder.Roughly speaking this monotonicity precludes certain instabilities of the solutions (at least locally in time), such as the so-called boundary-layer separations.An analytic result in this direction was shown by Oleinik (see e.g.[25,26]) using the Crocco transform.More recently, the result was renovated by Masmoudi and Wong in [24], and Alexandre, Wang, Xu and Yang in [3] without employing the latter.On the other hand, requiring the initial data to be highly regular suffices as well for the well-posedness of the Prandtl equations.Caflish and Sammartino [31] proved local-in-time well-posedness for initial data that are analytic in all directions.In reality, a refinement elaborated in [23] showed that the analyticty condition is only required in the tangential variable.Motivated by findings on ill-posedness, the set of functions spaces allowing for local existence theory was enlarged from analytic to the Gevrey scale.We mention here the works of Gérard-Varet and Masmoudi [14] in the class 7/4, Li and Yang [22] for small initial data in the class 2, and Dietert and Gérard-Varet [10] achieving the well-posedness also for large data in the Gevrey class 2. The latter regularity property is strongly suggested to be the borderline case.Indeed, smooth solutions might blow-up without analyticity or monotonicity assumptions as shown in [32].But more drastically, Gérard-Varet and Dormy proved in [12] the ill-posedness of the linearized Prandtl system in any Sobolev space by constructing solutions which experience a norm inflation as e σ (1.1) inhere better properties as a result of the finite speed propagation.In [8] we provided a positive answer to this dilemma, enlarging the Gevrey scale from the class 2 to the class 3.With the current work we show however that System (1.1) is still not desirable for the well-posedness in Sobolev spaces.

Some physical aspects of the model
System (1.1) formally arises as a (linearised) model for the boundary layers of the Navier-Stokes equations, whose Cauchy stress tensor is "delayed" through a first-order Taylor expansion: We refer to [2,8] and references therein for a deeper discussion on the physical motivations and derivation of the model.We point out that this delayed relation on S was introduced in fluid-dynamics by Carrassi and Morro [4], inspired by the celebrated work of Cattaneo [5,6] on heat diffusion.Moreover, System (1.2) represents somehow a simplified version of the celebrated Oldroyd-B model (or Maxwell equations, cf. for instance [19]): where the constant γ represents the time scale for the elastic relaxation.Equations (1.2) neglects nevertheless several terms of Oldroyd-B, in particular the convective and upper convective derivatives.We might therefore interpret our results as precursors for the analytical understanding of boundary layers of viscoelastic fluids.

Main Result
For the sake of comparison, we employ a similar Ansatz as the one of Gérard-Varet and Dormy in [12].All along the present manuscript, we denote by W s,∞ α , with s ≥ 0 and α > 0, the following weighted Sobolev spaces in For the entirety of this manuscript α > 0 is fixed and might be imposed equal to 1 (we keep the notation W s,∞ α , mainly because of [12]).Since our solutions are periodic in the tangential direction x ∈ T, we also set , making use of the Fourier transform in x ∈ T: Our main result expresses the ill-posedness of System (1.1) in H m W 0,∞ α when the shear flow U s (y) is non-monotonic.It develops around a family of solutions that are of unitary size in Sobolev spaces at t = 0 and experience an inflation of the norm after any short time t = δ > 0.
Theorem 1.1.Assume that the shear flow α (0, ∞) and satisfies the structural relations U s (a) = U ′ s (a) = 0 together with U ′′ s (a) = 0, for a given a ∈ (0, ∞).Then for all m ≥ 0, µ ∈ [0, 1/3) and small time δ > 0, there exists a pair of initial data that generates a global-in-time smooth solution u of (1.1), which satisfies Remark 1.2.We allow the solutions u in Theorem 1.1 and the suitable family of initial data to take values in C. Nevertheless, since the shear flow as well as all other coefficients (1.1) are real valued, taking the real (or imaginary part respectively) provide (non-trivial) real-valued solutions which undergo the same instability mechanism.The parameter µ ∈ [0, 1/3) allows to enlarge the Sobolev space for the values of the solutions.As long as the regularity index of H m−µ differs less than 1/3 from the original space H m (a reminiscent of the Gevrey-class 3 regularity), the problem is still ill-posed.An analogous threshold was shown in [12] with µ ∈ [0, 1/2) (reminiscent of Gevrey-class 2).
As for its homologous problem in [12], the instability process described by Theorem 1.1 is mainly enabled by a meaningful family of initial data, which highly oscillate in the tangential variable x ∈ T.More precisely, at high frequencies k ≫ 1, we select initial data (u in , u t,in ) on the eigenspace of e ikx and determine a suitable non-trivial asymptotic of the (t, y)-dependent Fourier coefficients as frequencies k → ∞ (more details are provided starting from Section 2).We show that the corresponding solutions grow as e σ 3 √ kt at least for a very short time depending on the frequencies (cf.Section 1.5, Proposition 2.1 and Lemma 2.3).Moreover, as k → ∞, the profiles in y ∈ R + of the initial data relate to a suitable "spectral condition" for the following ordinary differential equation: Lemma 1.3.There exists a complex number γ ∈ C with Im(γ) < 0 and a complex solution W : R → C of the ordinary differential equation such that lim z→−∞ W (z) = 0 and lim z→+∞ W (z) = 1.
The ODE (1.4) is similar to its homologous (1.7) in [12] (cf.also here (3.18)).They do present however some technical differences, that are mainly due to the fact that the leading operator of Prandtl is a heat equation, whereas the leading term in (1.1) is a wave equation (at high frequencies k ≫ 1).

About the ill-posedness of the Prandtl equation
The pioneering work of Dietert and Gérard-Varet in [10] established the local-in-time well-posedness of the Prandtl equations, when the initial data have Gevrey-class-2 regularity along the tangential direction x ∈ T. Before their result, the exponent of Gevrey-class 2 was attained only in the special setting prescribed by [22], in light also of the previous investigation of Gérard-Varet and Dormy [12] on the illposedness of Prandtl in Sobolev spaces.Indeed, for any sufficiently-large frequency k ≫ 1, Gérard-Varet and Dormy succeeded in establishing solutions of the linearised Prandtl equations around a non-monotonic shear flow (U s (y), 0) experiencing a dispersion relation proportional to √ k.Their result was later on strengthened by further investigations on related problems, we report for instance Gérard-Varet and Nguyen in [15] and Ghoul, Ibrahim, Lin and Titi in [16], emphasising that there exist solutions experiencing growth of order e √ kt in the frequencies: • The authors . . .construct O(k −∞ ) approximate solutions that grow like e √ kt for high frequencies k in x, • its linearisation around a special background flow has unstable solutions of similar form, but with σ k ∼ λ √ k, for k ≫ 1 arbitrarily large and some positive λ ∈ R + .
Because of this growth rate, one may wonder why Gérard-Varet and Dormy addressed the ill-posedness in Sobolev spaces rather than in any Gevrey-class m, with m > 2. With this section, we want to highlight some quantitative aspects related to the proof in [12] that play a fundamental role in the present manuscript as well.Furthermore, in our case we build solutions of the hyperbolic extension (1.1) with a dispersive relation of order 3 √ k, thus these remarks reveal also our choice of Sobolev rather than Gevrey-class m, with m > 3. Roughly speaking, the most compelling reason resides in the maximal lifespan for which the mentioned growths occur: it is proportional to t ∼ ln(k)/ √ k for the Prandtl equation and to t ∼ ln(k)/ 3 √ k for the hyperbolic extension.To be more specific, we state the following corollary of Theorem 1 in [12], when applied to the autonomous system (1.5).
Corollary 1.4 (due to Theorem 1 in [12]).Let U s ∈ W 4,∞ α (R + ) and assume that U ′ s (a) = 0 and U ′′ s (a) = 0 for some a > 0. Denote by T the semigroup of (1.5) for analytic solutions (cf.Proposition 1 in [12]).There exists σ > 0, such that The result of Gérard-Varet and Dormy implies in particular that T does not extend to an operator in the Sobolev space H m W 0,∞ α .Following the proof in [12], one can obtain in reality a more refined version of Corollary 1.4, which provides indeed an explicit lower bound of the semigroup T k (the projection of T on the eigenspace generated by e ikx ), associated to with v k = u k = 0 in y = 0, and an explicit upper bound for the time in which the instability occurs.
Theorem 1.5 implies in particular Corollary 1.4.Indeed, at any small time t = δ > 0, we may consider a sufficiently large frequency Remarkably, Theorem 1.5 provides an explicit inflation of the norms at any frequency k ∈ N, as well as a related maximum lifespan.This time is proportional to t ∼ ln(k)/ √ k and was somehow already observed in the proof of [12] (cf.pag.602 in [12], where the authors obtained a contradiction considering a time On a more technical level, the proof in [12] develops around the following ansatz for a velocity field u(t, x, y) = e ik(−Us(a)+τ k −1/2 )t+ikx ûk (y), where τ ∈ C and σ 0 := Im(τ ) < 0. This function u experiences therefore a-priori a growth as e σ 0 √ kt , at any time t > 0. However, we shall emphasise that this flow is a solution u fr = u fr k (t, y)e ikx of a "forced" version of the Prandtl equations, which depends on a non-trivial remainder r k = R k (t, y)e ikx (cf.(4.2) in [12], with ε = 1/k and u ε = u fr ).Hence, in order to transfer the instability of u fr to a homogeneous solution u of Prandtl, the authors invoked the Duhamel's identity (cf. the identity below (4.2) in [12], where Ũε = u fr k and U ε = u k ).Roughly speaking, u k behaves similarly as u fr k , as long as the integral at the r.h.s of Duhamel is sufficiently small.Theorem 1.5 translates this smallness relation in terms of the semigroup T k and the maximal lifespan t = We obtain a similar result also to the hyperbolic extension (1.1) (cf.Section 2 and Lemma 2.3).
Remark 1.6.Finally, we shall remark that the norm inflation of Theorem for any δ > 0. Inequality (1.7) does not automatically imply (1.8), since it would require that (1.7) holds true at a time t k is hence too restrictive for the ill-posedness of the linearised Prandtl equations in any Gevrey-class m, with m > 2.
On the other hand, if we allow U s (y) to be unbounded in y ∈ [0, ∞), we may consider a specific shear flow of the form U s (y) = (y − a) 2 /2, which satisfies indeed the assumptions of Theorem 1.5 at y = a (but does not decay exponentially as y → ∞).In this case, the forced solution u fr k (derived in the proof of [12]) corresponds in reality to an exact solution of the Prandtl equation: u fr k = u k (the forcing term R k vanishes at any time for this specific choice of U s , cf.Remark A.2).In this case, the growth e σ 0 √ kt holds true at any time t > 0, which therefore implies ill-posedness in any Gevrey-class m, with m > 2, along the tangential direction x ∈ T.However, also in this case there are some drawbacks: the built solution u k is unbounded and diverge to ∞ as y → ∞.This increase in y ∈ [0, ∞) is a reminiscent of the analogous behaviour of U s (y) = (y −a) 2 /2, making the Prandtl solution somehow unphiysical.Nevertheless, it might suggest that the time barrier t ∼ ln(k)/ √ k is (momentarily) only mathematical and that the dispersion rate might hold true at any time t > 0 also when considering non-monotonic shear flows U s (y) that decade as y → ∞.The main goal of this paper is to address the ill-posedness of the hyperbolic extension (1.1) in Sobolev spaces.We however provide a short proof of Theorem 1.5 , as further support of the pioneeric work of Gérard-Varet and Dormy in [12] (cf.Appendix A).

Outline of the proof
In this section, we illustrate the general principles that we set as basis of our proof, and we postpone the technical details to the remaining paragraphs.Our proof develops along three major axes: (i) We project the main equation (1.1) into frequency eigenspaces in x ∈ T (cf.(2.1)) and we perturb the resulting equations with some meaningful forcing terms (cf.(2.2) and Proposition 2.1).This ensures a specific exponential growth (typical of Gevrey-class 3 regularities) on certain inhomogeneous solutions.In particular, this exponential growth holds true, globally in time.
(ii) Locally in time, we transpose the growth described in (i) to the original homogeneous equation.Contrarily to (i), this holds uniquely for a very short time, which depends in particular upon the frequency of each eigenspace (cf.Lemma 2.3).
(iii) Finally, making use of the homogeneous solutions built in (ii), we derive an inflation of the Sobolev norms of the solutions of (1.1) as described by Theorem 1.1.
We postpone the major aspects of part (i) to Section 3 and we devote this section to some underlying remarks and the details of parts (ii) and (iii).
(i) The inhomogeneous equation and the global-in-time growth of Gevrey-3 type We first formally project the main equation (1.1) to the eigenspace of each positive frequency k ∈ N in the x-variable, by means of the Ansatz We hence look for a suitable family of (t, y)-dependent profiles u k , which solve the homogeneous system Since this equation is (roughly) a linear damped wave equation at each fixed frequency k ∈ N, we infer that any initial data , for any lifespan T > 0. Furthermore, we also infer that u k can be written in terms of a semigroup T k for any T > 0 and any t ∈ (0, T ).Certainly, u k satisfies additional regularities, however our central goal is to estimate u k in L ∞ (0, T ; W 0,∞ α ) and to determine a suitable growth of u k (t) W 0,∞ α as time increases.As depicted in part (i), we momentarily allow for perturbation of equation (2.1) by means of a general forcing term f k ∈ L ∞ (0, T ; W 0,∞ α ).To avoid confusion in the notation, we set this inhomogeneous version in the state variable u fr k = u fr k (t, y), which reflects a "forced" version of the hyperbolic Prandtl equation: We hence aim to determine a meaningful non-trivial forcing term f k , such that (2.2) admits a solution u fr k of (2.2), whose norm u fr k (t) W 0,∞ α experiences an exponential growth in time as e σ 0 tk 1/3 , for a suitable constant σ 0 > 0. This growth holds true at any time t ∈ (0, T ), as described by the following proposition.
Proposition 2.1.Assume that the shear flow α (0, ∞) and satisfies the relations U s (a) = U ′ s (a) = 0 together with U ′′ s (a) = 0, for a given a ∈ (0, ∞).For any positive frequency k ∈ N, there exist two non-trivial profiles U k , R k : R + → C in W 2,∞ α (R + ) and W 0,∞ α (R + ), respectively, and there exists a complex number τ ∈ C only depending on U s with negative imaginary part Im(τ ) < 0, such that the initial data and the forcing term 2), which can be written explicitly in the following form The sequences (U k ) k∈N is in W 2,∞ α (R + ) and it is uniformly bounded from above and below in W 0,∞ α (R + ), namely there exist two constants C, C > 0 such that (2.5) Furthermore, at high frequencies, the sequence (k there exists a constant C R > 0, which depends uniquely on the shear flow U s , such that for any k ∈ N. (2.6) Remark 2.2.Denoting by σ 0 := −Im(τ ) > 0, Proposition 2.1 implies that u fr k (t) W 0,∞ α ∼ e σ 0 k 1/3 t , at any t ∈ (0, T ), thanks to the explicit form (2.4) of the forced solution u fr k .Indeed, invoking also the uniform estimates from below in (2.5), we gather Unfortunately, this method of determining an explicit solution as in (2.4) seems to work uniquely for the inhomogeneous system (2.2) and it is somehow inefficient with its homogeneous counterpart (1.1) (thus with the original system).This is mainly due to the sequence of remainders (R k ) k∈N , which define non-trivial forces (f k ) k∈N in (2.4).From (2.4) and (2.5), f k (t) W 0,∞ α does not vanish (a-priori) as k → ∞.We will determine indeed an explicit form of R k (cf.where H stands for the Heaviside step function.The contribution of f k to the global-in-time instability seems to be not negligible.However, we show in (ii) that we can still obtain a similar result with the homogeneous equation, at least for a very short time that depends on the frequencies.Since the proof of Proposition 2.1 is rather technical, we postpone it to Section 3. We anticipate that it follows a similar approach as the one used in [12]: we plug the ansatz (2.3) to the main equations (2.2), we analyse the asymptotic limit as frequencies k → ∞ and we reduce the problem to a "spectral condition" on a related ODE.We shall hence devote the remaining parts of this section to address the details of (ii) and (iii) and thus to the proof of Theorem 1.1.
(ii) An instability of the homogeneous equation at a short time t ∼ ln(k)/

√ k
Based on the result given by Proposition 2.1, we aim to obtain a similar instability for the homogeneous system (2.1), which will lead in (iii) to the ill-posedness of (1.1) in Sobolev spaces.We first invoke the following Duhamel's formula, which relates a general forced solution u fr k of (2.2) with the semigroup T k of the homogeneous system (2.1): Here we shall interpret u k as the unique solution of the homogeneous problem (2.1), with same initial data of u fr k .Roughly speaking, in order to transfer the instability of u fr k to u k , we need to ensure that the integral on the r.h.s. of (2.8) remains sufficiently small.With the following lemma, we translate this condition directly as a property of the semigroup T k (t) at a time t, which is (at maximum) proportional to ln(k)/ 3  √ k (thus a time that vanishes as the frequency k → ∞).
Lemma 2.3.Assume that the shear flow U s = U s (y) is in W 3,∞ α (0, ∞) and satisfies the relations U s (a) = U ′ s (a) = 0 together with U ′′ s (a) = 0, for a given a ∈ (0, ∞).Let τ ∈ C be as in Proposition 2.1 and denote by σ 0 := −Im(τ ) > 0. For any k ∈ N and any σ ∈ (0, σ 0 ) the following inequality holds true where the constants C and C R are as in Proposition 2.1.
Remark 2.4.Before establishing the proof of Lemma 2.3, some remarks are here in order.Inequality (2.9) is written in terms of the semigroup T k .It implies however that there exist two profiles ≤ 1 (which may differ with respect to the ones of Proposition 2.1), such that the generated homogeneous solution u k of (2.1) satisfies (2.10) Unfortunately this inequality presents a major disadvantage: it is unclear at what time the inflation e σtk 1/3 holds true.This is deeply in contrast with the instability (2.7) of u fr k , which is indeed satisfied globally in time.Certainly, (2.10) is not achieved at t = 0 because of the initial data, however the inflation may occur at a time t very close to the origin (for which e −σtk 1/3 ∼ 1).Furthermore, also in case that the inflation occurs at the largest time t = , we may obtain at best that , which somehow implies only an inflation of Sobolev type.In other words, because of the time limitation of estimate (2.9), we deal in this work only with ill-posedness in Sobolev spaces and our approach seems to be inconclusive in Gevrey-class m, with m > 3.
Proof of Lemma 2.3.Assume by contradiction that there exists a frequency k ∈ N, so that sup where we have used the abbreviation . We consider the global-in-time solution u fr k (t, y) = e iτ k 1/3 t U k (y) provided by Proposition 2.1, which by uniqueness (at a fixed frequency) also satisfies the Duhamel's relation thanks to (2.8).Hence, applying the W 0,∞ α -norm to this identity and making use of the triangular inequality, we gather that where we have estimated u fr k (t) W 0,∞ α with the inequality in (2.7).Hence applying (2.11) and invoking the uniform bound (2.6) on the forcing term R k , we obtain Multiplying both l. and r.h.s. by e −σ 0 k 1/3 t /c and calculating explicitely the integral on the r.h.s, we get for any time t ∈ 0, , which in particular implies that By bringing the last term on the r.h.s to the l.h.s., we finally obtain that which is indeed a contradiction.This concludes the proof of Lemma 2.3.

(ii) Proof of Theorem 1.1 and the inflation of the Sobolev norms
Thanks to Lemma 2.3, we are now in the condition to conclude the proof of Theorem 1.1.Let σ be a fixed value in (0, σ 0 ) = (0, −Im(τ )) and let δ > 0 be an arbitrary short time.We first recall (2.9) for any k ∈ N, where Recalling that µ ∈ [0, 1/3), we consider a general frequency k ∈ N that satisfies By multiplying (2.12) with k −µ , we remark that there exists a time t δ ∈ (0, δ), such that Next we write the above estimate in terms of a specific solution u k = u k (t, y) of the homogeneous system (2.1): we consider two profiles u in,k and u t,in,k in W 1,∞ α and W 0,∞ α , which satisfy We hence set a solution u = u(t, x, y) with of the original system (1.1) by means of In particular, the initial data of u satisfy On the other hand, the following inflation of the Sobolev norm holds true at t = t δ : This concludes the proof of Theorem 1.1.
3 Proof of Proposition 2.1 Without loss of generality, we may assume that U ′′ s (a) < 0. Contrarily, we might set Ũs (y) := −U s (y), ũ(t, x, y) := −u(t, −x, y), ṽ(t, x, y) := v(t, −x, y) and f (t, x, y) := −f (t, −x, y).Thus (ũ, ṽ) is solution of For simplicity, we denote by ε = ε(k) := 1/k > 0 the inverse of a positive frequency k ∈ N, so that ε → 0 when k → ∞.Furthermore, throughout our proof, we will repeatedly use the following abuse of notation: we interchange any index k of a general function with its corresponding index ε, for instance and so on.
We aim to construct a solution (u ε (t, x, y), v ε (t, x, y)) of (2.2) with a forcing term f ε (t, x, y) of the form for a suitable ω(ε) ∈ C that we will soon determine and suitable profiles We momentarily take for given that ω is O(ε 3 ) (for the impatient reader, check (3.4) and (3.29)).In reality, the profile U ε is redundant, since the divergence-free condition implies that We hence plug the expressions (3.1) into the main equation (1.1).We obtain that U ε and V ε satisfies Dividing by e i ω(ε)t+x ε and multiplying by −ε 4/3 we get iω(ε) Thus recasting the above relation uniquely in terms of V ε (with V ′ ε = −iU ε ), we finally derive the following ordinary differential equation: We shall remark that for any Formally, by sending ε → 0, denoting the asymptotic V ε → v a , and recalling that the limit lim ε→0 ε −2/3 ω(ε) = 0, we obtain the equation Of course this limit is only formal, we shall soon analyse the difference between a solution V ε of (3.2) and a solution v a of (3.3).Since U s (a) = U ′ s (a) = 0, for a given a ∈ (0, ∞), (3.3) admits a non-trivial weak solution given by v a (y) = H(y − a)U s (y) .
We remark that v a belongs to behaves as a Dirac delta distribution in y = a, since U ′′ s (a) = 0. We next set a complex number τ ∈ C \ {0} and we introduce the following Ansatz: We hence aim to determine V ε of (3.2), making use of the following perturbation of v a , depending on τ : Hence, the unknowns are momentarily the profile V : R → C and the complex number τ ∈ C. We remark that the expansion (3.5) is similar to its homologous (2.5) in [12] for the Prandtl equations.There are however major differences in the exponents of the terms in ε.These eventually lead to different dispersion rates: of order 3 √ k = ε −1/3 for System (1.1) and of order √ k = ε −1/2 for the Prandtl equation.
Remark 3.1.Since V ε belongs to W 3,∞ , the profile V must cancel the singularities in y = a of the functions v a (y) and ε Replacing (3.4) and (3.5) into (3.2),we gather the following equation for the profile V : We first remark that several terms of (3.6) cancel out thanks to the definition of v a in (3.3) and the conditions on the shear flow U s (a) = U ′ s (a) = 0. Indeed, (3.6) can be recasted as Next, we divide the above relation with ε 4/3 > 0 and we remark that v ′′′ a (y) = U ′′ s (a)δ a (y)+H(y−a)U ′′′ s (y).Hence, we are left with the following identity, which shall be intended in terms of distributions D ′ (0, ∞): We next apply (3.7) to an appropriate test function ϕ ∈ D(0, ∞).We first consider a general test function ψ ∈ D(R) having supp ψ ⊆ (−a/ 3 √ ε, ∞) is valid.Hence we set ϕ(y) = ψ((y − a)/ 3 √ ε), for any y > 0. By applying (3.7) to this specific test function ϕ, we get the identity We perform a change of variables with z := (y−a) In particular, we deduce that V is distributional solution in D ′ (−a/ 3 √ ε, ∞) of the following equation: Remark that although V does not depend on ε > 0, the remainder R ε is still unknown and will be chosen to abosrb all the ε-dependences.We now send ε towards 0 in order to determine an equation for the profile V in D ′ (R).We recall that we assume R ε ∈ O(ε 3 ) in W 0,∞ α , that both U s (a) = U ′ s (a) = 0 and that U s ∈ W 4,∞ (0, ∞), hence for any z > −a/ 3 √ ε.Since the shear flow U s is in W 4,∞ α (0, ∞), the convergence is uniform in any compact set of (−a/ 3  √ ε, ∞).Hence, as ε > 0 vanishes, we are left with the following identity for the profile V : In particular, we aim to determine a profile V ∈ W 3,∞ (R + \ {0}) (decaying to 0 as z → ±∞), which is solution of the following linear ordinary differential equation away from the origin fulfilling also the following jump relations at the origin: We next remark that the function z ∈ R \ {0} → τ + U ′′ s (a)z 2 /2 is non-decaying solution of (3.10).Furthermore, z ∈ R → H(z)(τ + U ′′ s (a)z 2 /2) is a distributional solution of (3.9) and it satisfies the jump conditions of (3.11).We can thus get rid of the singularity at the origin by superposition introducing the function Ṽ The new profile Ṽ shall be determined in W 3,∞ loc (R), satisfying the ODE Next, recalling that we seek for a τ ∈ C with Im(τ ) < 0, we introduce the function We thus get that W ∈ W 3,∞ (R) shall satisfy with boundary conditions lim Finally, we perform the following change of variables which leads to Recalling that we assume U ′′ s (a) < 0, we finally obtain with boundary conditions lim (3.17)

The spectral condition
The goal of this section is to show that the ordinary differential equation (3.16), together with its boundary conditions (3.17), admits a smooth solution W = W (z) for a suitable γ ∈ C, whose imaginary part is negative.We proceed with a similar procedure as the one used by Gérard-Varet and Dormy in [12].We shall however remark that (3.16) inherently differs from its counterpart of [12] (cf.(1.7) in [12]), since the authors derived an ODE of the form Indeed, note that by dividing (3.16) by γ, the leading third derivative in (3.16) is multiplied by 1/γ, while in (3.18) the third derivative relates uniquely to i. Nevertheless, our aim is to find a γ ∈ C with Im(γ) < 0, thus (3.16) and (3.18) share eventually similarities in terms of the behaviour of their solutions.This is exploited in details in what follows.
To begin with, we consider the auxiliary eigenvalue problem (cf.(3.2) in [12]): We aim to build the solution W in terms of the eigenfunction f .In order to do so, let us first recall certain of its underlying properties.The domain D(A) of the operator A is defined by making use of the weighted spaces Furthermore, the function f admits an extension on a simply connected domain of C and decays exponentially along suitable sectors.
We are now in the condition of defining the complex number γ ∈ C of (3.16) in terms of the positive eigenvalue α of Lemma 3.2.Recalling that the eigenvalue α is positive, we set .21)This also implies that the complex number τ = (|U ′′ s (a)|/2) 1/3 γ in (3.15) has negative imaginary part, i.e.
We next introduce the change of variable w.r.t.(3.19), and the function X : Ω → C by means of Remark 3.3.We shall remark that X decays exponentially as z ∈ R converges towards ±∞.Indeed, in virtue of (3.20), we gather that The corresponding derivatives of X are trivially computed: We can now recast the relation (3.19) of the eigenfunction f in terms of X and its derivatives.First We next multiply both left and right-hand side with α Recalling that γ = α 1 3 e −i 2π 3 from (3.21), we finally obtain the following relation on X, which can also be written as In this last identity, we have used that X is in R integrable, thanks to (3.22) and the exponential decay (3.20).We hence aim to define the function W : R → C as (3.28) The profile U k is in W 2,∞ α (R + ) for all k ∈ N. Furthermore it is uniformly bounded from below and above in W 0,∞ α (R + ), since These values converge all towards 0 as k → ∞, thus U k converges towards iU ′ s (y)H(y − a) in W 0,∞ α .We now take the difference between equation (3.8) for a positive ε > 0 and (3.9):We next set C σ in Theorem 1.5 as Assume by contradiction that there exists a frequency k ∈ N, so that sup  Hence, applying the W 0,∞ α -norm to this identity and applying the triangular inequality, we gather that where we have estimated u fr k (t) W 0,∞ α with (A.4).Hence applying (A.9) and invoking the uniform bound on R k , we obtain Multiplying both l. and r.h.s. by e −σ 0 k 1/2 t /c and calculating explicitly the integral on the r.h.s, we obtain

9 ) 0 T 1 2
We consider the global-in-time solution u fr k (t, y) = e iτ k 1/2 t U k (y) provided by Proposition A.1, which by uniqueness also satisfies the Duhamel's relationu fr k (t, y) = T k (t) U k (y) + t k (t − s)(−ke iτ k s R k (s))(y)ds,