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# On hyperbolicity and Gevrey well-posedness.
Part two: Scalar or degenerate transitions.

Baptiste Morisse Université Paris Diderot, Sorbonne Paris Cité, Institut de
Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, F- 75025, France -
<EMAIL_ADDRESS>The author thanks his advisor Benjamin Texier for
all the remarks on this work, and Jeffrey Rauch for interesting discussions.

###### Abstract

For first-order quasi-linear systems of partial differential equations, we
formulate an assumption of a transition from initial hyperbolicity to
ellipticity. This assumption bears on the principal symbol of the first-order
operator. Under such an assumption, we prove a strong Hadamard instability for
the associated Cauchy problem, namely an instantaneous defect of Hölder
continuity of the flow from $G^{\sigma}$ to $L^{2}$, with
$0<\sigma<\sigma_{0}$, the limiting Gevrey index $\sigma_{0}$ depending on the
nature of the transition. We restrict here to scalar transitions, and non-
scalar transitions in which the boundary of the hyperbolic zone satisfies a
flatness condition. As in our previous work for initially elliptic Cauchy
problems [B. Morisse, On hyperbolicity and Gevrey well-posedness. Part one:
the elliptic case, arXiv:1611.07225], the instability follows from a long-time
Cauchy-Kovalevskaya construction for highly oscillating solutions. This
extends recent work of N. Lerner, T. Nguyen, and B. Texier [The onset of
instability in first-order systems, to appear in J. Eur. Math. Soc.].

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Overview of the paper
  2. 2 Main assumptions and results
    1. 2.1 Branching eigenvalues and defect of hyperbolicity
    2. 2.2 The case of a smooth transition
    3. 2.3 The case of a stiff transition
    4. 2.4 Statement of the results
  3. 3 Highly oscillating solutions and reduction to a fixed point equation
    1. 3.1 Highly oscillating solutions
    2. 3.2 Remainder terms
    3. 3.3 Upper bounds for the propagators
    4. 3.4 Free solutions
    5. 3.5 Fixed point equation
  4. 4 Contraction estimates
    1. 4.1 Functional spaces: definitions
    2. 4.2 Functional spaces: properties
    3. 4.3 Estimates of remainder terms
    4. 4.4 Contraction estimates
  5. 5 Estimates from below and Hadamard instability
    1. 5.1 Existence of solutions
    2. 5.2 Bounds from below
    3. 5.3 Conclusion: Hadamard instability in Gevrey spaces
  6. 6 Appendix: on the Airy equation
    1. 6.1 Reduction to the scalar Airy equation and resolution
    2. 6.2 Upper bounds for the propagator: proof of Lemma (3.4)
    3. 6.3 Growth of the free solution: proof of Lemma (3.6)

##  1 Introduction

We consider the following Cauchy problem, for first-order quasi-linear systems
of partial differential equations:

$\partial_{t}u=\sum_{j=1}^{d}A_{j}(t,x,u)\partial_{x_{j}}u+f(t,x,u)\;,\qquad
u(0,x)=h(x).$ (1.1)

The system is of size $N$, that is $u(t,x)$ and $f(t,x,u)$ are in
${\mathbb{R}}^{N}$ and the $A_{j}(t,x,u)\in{\mathbb{R}}^{N\times N}$. The time
$t$ is nonnegative, and $x$ is in ${\mathbb{R}}^{d}$. We assume throughout the
paper that the $A_{j}$ and $f$ are analytic in a neighborhood of some point
$(0,x_{0},u_{0})\in{\mathbb{R}}_{t}\times{\mathbb{R}}^{d}_{x}\times{\mathbb{R}}^{N}_{u}$.

Under assumptions of weak defects of hyperbolicity for the first-order
operator, we prove ill-posedness of (1.1) in Gevrey spaces. Our results extend
Métivier’s ill-posedness theorem in Sobolev spaces for initially elliptic
operators [Mét05], our own ill-posedness result in Gevrey spaces for initially
elliptic operators [Mor16a], Lerner, Nguyen and Texier’s theorem on systems
transitioning from hyperbolicity to ellipticity [LNT17], and echo Lu’s
construction of WKB profiles [Lu16] which are destabilized by terms not
present in the initial data.

Our proofs use Métivier’s method developed in [Mét05] based on majoring
series, hence the assumption of analyticity for the $A_{j}$ and $f$. Our
assumptions of weak defects of hyperbolicity mean that the operator in (1.1)
experiences a transition in time from hyperbolicity to non hyperbolicity. The
transition is possibly not uniform in space. Our assumptions bearing on the
principal symbol, and the associated normal forms, are presented in Section 2.
Our results are Theorems 2.11 and 2.12, stated in Section 2.4. The proofs
comprise Sections 3 to 8.

In the companion paper [Mor17], we consider the case of genuinely non-scalar
transitions.

###  1.1 Background

####  1.1.1 A long-time Cauchy-Kovaleveskaya result for elliptic Cauchy
problems

Our article [Mor16a] contains a long-time Cauchy-Kovalevskaya theorem, based
on the paper [Mét05] by Métivier, which proves an Hadamard instability result
for initially elliptic quasi-linear systems in Gevrey spaces. Precisely, the
result of [Mor16a] asserts that the flow associated to the Cauchy problem
(1.1) fails to be Hölder from a highly regular $\sigma$-Gevrey space to the
very lowly regular $L^{2}$ space, locally in the $x$ variable and for $\sigma$
less than a critical exponent $\sigma_{0}$ depending on initial spectrum,
under the assumption of initial ellipticity for the first-order differential
operator.

Here initial ellipticity is understood as an initial defect of hyperbolicity.
That is for some
$(x_{0},\vec{u}_{0},\xi_{0})\in{\mathbb{R}}_{x}^{d}\times{\mathbb{R}}_{u}^{N}\times{\mathbb{R}}_{\xi}^{d},$
the principal symbol at $(0,x_{0},\vec{u}_{0},\xi_{0})$:

$A_{0}:=\sum_{j=1}^{d}A_{j}(0,x_{0},\vec{u}_{0})\xi_{0,j}$

has at least one couple of non-real eigenvalues, with imaginary part $\pm
i\gamma_{0}$ associated to eigenvectors $\vec{e}_{\pm}$.

In [Mor16a] we posit in $(3.2)$ the ansatz
$u_{\varepsilon}(t,x)={\mathbf{u}}(t/\varepsilon,x,(x-x_{0})\cdot\xi_{0}/\varepsilon)$,
where ${\mathbf{u}}(s,x,\theta)$ is periodic in the $\theta$ variable. We
transform then the Cauchy problem (1.1) into the equation

$\partial_{s}{\mathbf{u}}-\underline{A}(\varepsilon
s,x)\partial_{\theta}{\mathbf{u}}=G({\mathbf{u}})$ (1.2)

for some non-linear remainder term $G({\mathbf{u}})$. The leading term
$\underline{A}(t,x)$ is here the principal symbol

$\underline{A}(t,x)=\sum_{j}A_{j}(t,x,\vec{u}_{0})\xi_{0,j}.$ (1.3)

The ellipticity condition is an open condition bearing on the principal symbol
$\underline{A}$. In particular, ellipticity at $(0,x_{0})$ implies ellipticity
around $(0,x_{0})$. The proof of [Mor16a] introduces the propagator $U$
defined by

$\partial_{s}U(s^{\prime},s,\theta)-\underline{A}(\varepsilon
s,x)\partial_{\theta}U(s^{\prime},s,\theta)=0.$ (1.4)

By ellipticity, the propagator $U$ has an exponential growth. We introduce an
appropriate Banach space of functions of $(s,x,\theta)$ which are analytical
in the $x$ variable and which Fourier coefficients in $\theta$ have an
exponential growth which reflects the growth of the propagator. A fixed point
argument shows existence and uniqueness, and exponential growth in this space,
which implies the Hadamard instability.

The main issue in [Mor16a], compared to the previous analysis of Metivier
[Mét05], is that in Gevrey spaces, the Hadamard instability is recorded at
much longer times than in Sobolev spaces.

####  1.1.2 Lerner, Morimoto and Xu’s result on transition to ellipticity for
scalar equations

In [LMX10], Lerner, Morimoto and Xu introduce the notion of transition to
ellipticity for initially hyperbolic systems. A prototypical example is the
Burgers equation with a complex forcing: $\partial_{t}u+u\partial_{x}u=i$. In
the case of real data, the principal symbol is initially hyperbolic. Due to
the complex forcing, the principal symbol is elliptic for ulterior times. For
general equations (1.1) (with $N=1$: scalar equations), under bracket
conditions generalizing the situation for Burgers with complex forcing, and
describing a transition from hyperbolicity to ellipticity, the authors in
[LMX10] prove a strong form of instability, namely that if local $C^{2}$
solutions exist, then the complement of the analytic wave-front set of the
datum is not empty. In particular, if the bracket conditions are formulated at
$(x_{0},u_{0})\in{\mathbb{R}}^{d}\times{\mathbb{R}}$, it is shown in [LMX10]
that for any analytical datum $\underline{h}$ such that
$\underline{h}(x_{0})=u_{0}$ , there exists smooth initial data $h$ close to
$\underline{h}$ which do not generate local $C^{2}$ solutions, a result
analogous to Lebeau’s theorem for Kelvin-Helmholtz [Leb02]. The proof of
[LMX10] relies strongly on a representation of solutions based on the method
of characteristics, specific to scalar equations, which was developed earlier
in [Mét85].

####  1.1.3 Lerner, Nguyen and Texier’s result on transition to ellipticity
for general systems

In [LNT17], Lerner, Nguyen and Texier extend the analysis of [LMX10] to
systems (1.1). The result of [LNT17] shows an instantaneous lack of Hölder
well-posedness of the flow, with an arbitrarily large loss of derivatives,
under appropriate assumptions of transition to ellipticity. The analysis of
[LNT17] is based on the method of approximation of pseudo-differential flows
introduced in [Tex16]. One key observation in [LNT17] is that for systems,
many types of transitions may occur. The focus in [LNT17] is on genuinely non-
scalar transitions (more about this specific point in Remark 1.1). For these,
the propagator generically grows in time like the Airy function.

####  1.1.4 Defect of hyperbolicity in Maxwell systems

There is a strong analogy between the progression from [Mét05] to our present
results and recent results [LT15], [Lu16] in geometric optics. In [LT15], Lu
and Texier study large-amplitude solutions to Maxwell-based systems in the
small wavelength limit. They show that in appropriate coordinates, resonances
in frequency correspond to points of weak hyperbolicity. Thus at the
resonances, the subprincipal symbol plays a role in the stability analysis.
Under a Levi condition, hyperbolicity is violated around the resonances, and
WKB solutions do not approximate exact solutions issued from appropriate
nearby initial data, no matter how precise the order of the WKB approximation.
This result is somehow analogous to Métivier’s initial ellipticity result.
Following [LT15], Lu studied in [Lu16] a situation in which WKB solutions are
destabilized by terms which are not present in the initial data. That is, the
Levi condition of [LT15] is satisfied initially, but higher-order harmonics of
the WKB solutions, which are generated by the nonlinearities in the course of
the propagation, are associated with higher-order resonances. For these
resonances, the Levi conditions may not be satisfied, leading to instability.
This framework is somehow similar to ours, with an instability which develops
in time, starting from an initially hyperbolic situation.

###  1.2 Overview of the paper

Our assumptions are based on the framework set out in [LNT17], the results of
which we extend in two distinct ways: we prove existence of solutions up to
the observation time at which the Hadamard instability is recorded, and we
measure the deviation in Gevrey spaces.

We assume that for a specific frequency $\xi_{0}\in{\mathbb{R}}^{d}$ the
linear part of the principal symbol at $u=\vec{u}_{0}\in{\mathbb{R}}^{N}$
defined by (1.3) has a real spectrum at time $t=0$ while non real eigenvalues
appear for $t>0$. In this sense the operator experiences a transition from
initial hyperbolicity ($t=0$, real eigenvalues) to eventual ellipticity
($t>0$, non-real eigenvalues).

A sharp difference with the initially elliptic case lies in the normal forms
of the operators. Indeed, the elliptic case is reducible to the case where
$\underline{A}$ is a triangular matrix with non real and conjugated diagonal
entries.

By contrast, transitions in time appear in many ways. There is not one single
normal form. Section 2 will be devoted to the descriptions of such transitions
in time and the associated normal forms for systems of size $N=2$. In
particular, this paper focuses on two particular normal forms, described in
the next paragraphs.

####  1.2.1 The smoothly diagonalizable case

Under Assumptions 2.2, 2.4 and 2.5 (see Proposition 2.6 below), there holds

$\underline{A}(t,x)\approx\underline{A}^{{\rm S}}(t):=\begin{pmatrix}0&t\\\
-\gamma_{0}^{2}t&0\end{pmatrix}$

with $\gamma_{0}>0$. Here $\approx$ means equality up to higher order terms in
the Taylor expansion in time and space, and up to a change of basis. The
matrix $\underline{A}^{{\rm S}}(t)$ is smoothly diagonalisable in
${\mathbb{C}}$, with smooth eigenvalues $\pm i\gamma_{0}t$. This case is
mostly scalar ; it is analogous to a degenerate Cauchy-Riemann problem.

Our analysis shows that our method in [Mor16a] is robust enough to allow for
such a weak defect of hyperbolicity. We replace ansatz $(3.2)$ therein by
$u_{\varepsilon}(t,x)={\mathbf{u}}(t/\varepsilon^{1/2},x,(x-x_{0})\cdot\xi_{0}/\varepsilon)$.
For such $\underline{A}^{{\rm S}}(t)$, the growth for the associated
propagator solving

$\partial_{s}U^{{\rm S}}(s^{\prime},s,\theta)-\underline{A}^{{\rm
S}}(s)\partial_{\theta}U^{{\rm S}}(s^{\prime},s,\theta)=0$

is like

$|U_{n}^{{\rm
S}}(s^{\prime},s)|\lesssim\exp\left(\int_{s^{\prime}}^{s}\gamma_{{\rm
S}}^{\sharp}(\tau)d\tau\right)\;,\quad\forall\,0\leq s^{\prime}\leq
s\;,\;\forall\,n\in{\mathbb{Z}}^{d}$ (1.5)

for the Fourier coefficients of $U^{{\rm S}}(s^{\prime},s,\theta)$, with
$\gamma_{{\rm S}}^{\sharp}(\tau)=\gamma_{0}\tau$.

####  1.2.2 The degenerate Airy case

Under Assumptions 2.2 and 2.7 (see Proposition 2.8 below), there holds

$\underline{A}(t,x)\approx\underline{A}^{{\rm Ai}}(t,x):=\begin{pmatrix}0&1\\\
-\gamma_{0}^{2}(t-t_{\star}(x))&0\end{pmatrix}$

where $\approx$ means equality up to higher order terms in the Taylor
expansion in time and space and $t_{\star}(x)\geq 0$ in a whole neighborhood
of $x=0$111If there were $x_{0}$ such that $t_{\star}(x_{0})<0$, we would be
in the case of initial ellipticity and Métivier’s result would apply..

The time transition function $t_{\star}(x)$ defines the boundary between the
elliptic and hyperbolic zones. Indeed, for $t<t_{\star}(x)$, the eigenvalues
are $\pm\sqrt{t_{\star}(x)-t}$ while for $t>t_{\star}(x)$ the eigenvalues are
$\pm i\sqrt{t-t_{\star}(x)}$.

The transition between hyperbolicity and ellipticity is thus not uniform in
space, and depends on the space-dependent transition time $t_{\star}(x)$. The
non degenerate case $t_{\star}(x)=O(x^{2})$ is out of reach of the method
presented in this paper, and requires special attention - we devoted two
companion papers [Mor16b] and [Mor17] to the subject - more about this
specific point in Remark 1.1 below. We will focus here on the degenerate case

$t_{\star}(x)=O(x^{4}).$

Note that the cases of odd power of $x$ are in contradiction with the
assumption of non-negativity of $t_{\star}$ around $x=0$.

We emphasize also the fact that the eigenvalues of $\underline{A}^{{\rm Ai}}$
are $C^{0}$ in time but not $C^{1}$, hence the stiffness of this case.

In this framework, we replace ansatz $(3.2)$ of [Mor16a] by
$u_{\varepsilon}(t,x)={\mathbf{u}}(t/\varepsilon^{2/3},x,(x-x_{0})\cdot\xi_{0}/\varepsilon)$.
As such a transition is not semi-simple, the previous ansatz induces the
following equation for the propagator

$\partial_{s}U^{{\rm
Ai}}(s^{\prime},s,\theta)-\varepsilon^{-1/3}\underline{A}^{{\rm
Ai}}(s,0)\partial_{\theta}U^{{\rm S}}(s^{\prime},s,\theta)=0$

which as then a growth like

$|U_{n}^{{\rm
Ai}}(s^{\prime},s)|\lesssim\varepsilon^{-1/3}\exp\left(\int_{s^{\prime}}^{s}\gamma_{{\rm
Ai}}^{\sharp}(\tau)d\tau\right)\;,\quad\forall\,0\leq s^{\prime}\leq
s\;,\;\forall\,n\in{\mathbb{Z}}^{d}$ (1.6)

for the Fourier coefficients of $U^{{\rm Ai}}(s^{\prime},s,\theta)$ with
$\gamma_{{\rm Ai}}^{\sharp}(\tau)=\gamma_{0}\tau^{1/2}$ which is typical of
the Airy growth.

######  Remark 1.1.

In [LNT17] , the authors allow for generic non-scalar transitions, for which
$t_{\star}(x)=O(x^{2})$. In particular, the space-time domain
$\\{(t,x),x^{2}\leq t\\}$ is included in the domain of hyperbolicity. As we
will see precisely in the course of the proof of Proposition 5.4, in our
context this space-time domain is too large for the standard Cauchy-
Kovalevskaya theorem to apply. Thus, in the case $t_{\star}(x)=O(x^{2})$, we
need a specific Gevrey well-posedness result in that space-time domain before
observing the instability develop in the elliptic domain. This Gevrey well-
posedness result is the object of the article [Mor16b], and the completion of
the instability proof in the case $t_{\star}(x)=O(x^{2})$ is the object of the
article [Mor17].

####  1.2.3 Example: compressible Euler with Van der Waals pressure law

Transitions of the principal symbol from hyperbolicity to ellipticity, as
described in the above paragraphs, are observed in physical equations
describing phase transitions. One such system (mentioned in both [Mét05] and
[LNT17]) is the compressible Euler equations in one spatial dimension, with a
Van der Waals pressure law:

$\begin{cases}&\partial_{t}u_{1}+\partial_{x}u_{2}=0\\\
&\partial_{t}u_{2}+\partial_{x}(p(u_{1}))=0\end{cases}$ (1.7)

where $p$ follows a Van der Waals equation of state, for which there holds
$p^{\prime}(u_{1})\leq 0$, for some $u_{1}\in{\mathbb{R}}$. The system is
hyperbolic (resp. elliptic) for $p^{\prime}(u_{1})>0$ (resp. for
$p^{\prime}(u_{1})<0$). For solutions which leave the hyperbolic zone, a phase
transition occurs. This corresponds for us to the catastrophic growth recorded
in the elliptic zone. If for instance the elliptic zone is defined by
$\\{|u_{1}|\leq\delta\\}$, for some $\delta>0$, then solutions may enter the
elliptic zone only to leave it immediately, due to the exponential growth.

##  2 Main assumptions and results

###  2.1 Branching eigenvalues and defect of hyperbolicity

We look at the possible cases of a defect of hyperbolicity, that is
transitions from initial hyperbolicity to ellipticity at time $t>0$, following
the work of Lerner, Nguyen and Texier of [LNT17].

We introduce first

$A(t,x,u,\xi)=\sum_{j}A_{j}(t,x,u)\xi_{j}.$ (2.1)

We assume there are
$(x_{0},\vec{u}_{0},\xi_{0})\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{N}\times{\mathbb{R}}^{d}$
and $r_{0}>0$ such that the principal symbol defined by

$\underline{A}(t,x,\xi)=A(t,x,\vec{u}_{0},\xi)$ (2.2)

satisfies

${\rm
Sp}\left(\underline{A}(0,x,\xi)\right)\subseteq{\mathbb{R}}\;,\quad\forall
x\in B_{r_{0}}(x_{0})\,,\forall\,\xi\in{\mathbb{R}}^{d}$ (2.3)

which stands for initial and local hyperbolicity around
$(x_{0})\in{\mathbb{R}}^{d}$. Note that, as soon as there is some
$(x_{1},\xi_{1})\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ such that
$\underline{A}(0,x_{1},\xi_{1})$ has non real spectrum, we are in the case of
initial ellipticity treated in [Mor16a].

We assume also that, for small times $t>0$, there are some $(x,\xi)$ close to
$(x_{0},\xi_{0})$ such that

${\rm
Sp}\left(\underline{A}(t,x,\xi)\right)\not\subseteq{\mathbb{R}}\qquad\forall
t>0$ (2.4)

Condition (2.3) stands for initial and local hyperbolicity around $x_{0}$ ;
condition (2.4) expresses the ellipticity of $\underline{A}$ at time $t>0$. Up
to translations in $x$ and $u$, which do not affect our forthcoming
assumptions, and by homogeneity in $\xi$, we may assume

$x_{0}=0\quad,\quad\vec{u}_{0}=0\quad,\quad|\xi_{0}|=1.$ (2.5)

Since the $A_{j}$ have real coefficients, non-real eigenvalues of
$\underline{A}(t,x,\xi)$ appear in conjugate pairs. For such a pair
$\lambda_{\pm}(t,x,\xi)$, by reality of the eigenvalues at $t=0$ we have a
double eigenvalue $\lambda_{-}(0,x,\xi)=\lambda_{+}(0,x,\xi)\in{\mathbb{R}}$
of $\underline{A}(0,x,\xi)$. To avoid higher order transitions (which would
involve eigenvalues of multiplicity $3$ or greater), we assume the eigenvalues
of $\underline{A}(0,0,\xi_{0})$ to be distinct and simple, except for one
double eigenvalue:

######  Assumption 2.1.

We assume the eigenvalues of $\underline{A}(0,0,\xi_{0})$ to be distinct and
simple, except for one double eigenvalue.

We block diagonalize the principal symbol into $\underline{A}(t,x)^{(0)}$ and
$\underline{A}(t,x)^{(1)}$. The block $\underline{A}(t,x)^{(0)}$ is a $2\times
2$ matrix corresponding to the double eigenvalue, and the block
$(N-2)\times(N-2)$ block $\underline{A}^{(1)}$ has simple real eigenvalues in
a whole neighborhood of $(t,x)=(0,0)$. Thanks to Assumption 2.1 the block
diagonalization is smooth. Therefore we focus our discussion on
$\underline{A}^{(0)}$, and we may assume $N=2$, that is
$\underline{A}\equiv\underline{A}^{(0)}$.

The question is now to describe the possible matrices $\underline{A}(t,x,\xi)$
satisfying conditions (2.3) and (2.4). Following [LNT17], we reformulate the
conditions (2.3) and (2.4) in terms of the characteristic polynomial of
$\underline{A}$ defined as

$P(\lambda,t,x,\xi)={\rm det}\left(\lambda-\underline{A}(t,x,\xi)\right)$
(2.6)

which is simply in the case $N=2$

$P(\lambda,t,x,\xi)=\left(\lambda-\frac{1}{2}{\rm
Tr}\underline{A}(t,x,\xi)\right)^{2}+\Delta(t,x,\xi)$ (2.7)

where we define

$\Delta(t,x,\xi)={\rm det}\underline{A}(t,x,\xi)-\left(\frac{1}{2}{\rm
Tr}\underline{A}(t,x,\xi)\right)^{2}.$ (2.8)

Thus the real or complex nature of the spectrum depends on the sign of
$\Delta$. So condition (2.3) is equivalent in terms of $\Delta$ to

$\Delta(0,x,\xi)\leq 0\qquad\forall x\in
B_{r_{0}}(0)\;,\forall\xi\in{\mathbb{R}}^{d}.$ (2.9)

As a double eigenvalue
$\lambda_{-}(0,x,\xi)=\lambda_{+}(0,x,\xi)\in{\mathbb{R}}$ of
$\underline{A}(0,x,\xi)$ corresponds to a double root of $P(t=0,x,\xi)$, we
formulate the following Assumption:

######  Assumption 2.2 (Branching eigenvalues).

In addition to (2.9), we assume that there exists some
$\lambda_{0}\in{\mathbb{R}}$ such that

$P(\lambda_{0},0,0,\xi_{0})=0\quad,\quad\partial_{\lambda}P(\lambda_{0},0,0,\xi_{0})=0.$
(2.10)

######  Remark 2.3.

Note that condition (2.10) is equivalent to

$\lambda_{0}=\frac{1}{2}{\rm
Tr}\underline{A}(0,0,\xi_{0})\quad,\quad\Delta(0,0,\xi_{0})=0.$

For condition (2.4) to be satisfied, that is for a conjugate pair of
eigenvalues to appear as $t>0$, $\Delta(t,x,\xi)$ has to be positive for
$t>0$. The eigenvalues of $\underline{A}$, which are the zeroes of $P$, are
then expressed by the square roots of $\Delta$. Even though the regularity of
$\Delta$, being an algebraic combination of the coefficients of
$\underline{A}$, is analytic, the regularity of the square roots of $\Delta$
can be of course much weaker. How much rougher than $\Delta$ may
$\sqrt{\Delta}$ be has been studied in particular by Glaeser [Gla63]. The
question of the regularity of the eigenvalues and of the eigenvectors is here
of importance as we work in the analytic framework: we may not use non-smooth
(in time and space) changes of basis, since the methods we use, following
[Mét05], strongly rely on analyticity. In particular, we may not diagonalize
the principal symbol if the eigenvectors are not smooth.

###  2.2 The case of a smooth transition

For the square roots of $\Delta$ to be as smooth as $\Delta$, the discriminant
$\Delta$ has to be the square of a smooth function $\delta(t,x,\xi)$:

$\Delta(t,x,\xi)=\delta(t,x,\xi)^{2}.$

In this case, note that $\Delta(0,x,\xi)=\delta(0,x,\xi)^{2}\geq 0$. Since we
assume also that $\Delta(0,x,\xi)\leq 0$ by (2.9), we get

$\delta(0,x,\xi)=0\qquad\forall x\in
B_{r_{0}}(0)\;,\forall\xi\in{\mathbb{R}}^{d}.$

This is equivalent to the existence of some analytic function
$\tilde{\delta}(t,x,\xi)$ such that

$\delta(t,x,\xi)=t\tilde{\delta}(t,x,\xi).$

We sum up all this in the following

######  Assumption 2.4 (Smooth transition).

There is a function $\delta(t,x,\xi)$ analytic in the $t$ and $x$ variables
such that

$\Delta(t,x,\xi)=\left(t\delta(t,x,\xi)\right)^{2}$ (2.11)

with

$\delta(0,0,\xi_{0})=\gamma_{0}>0.$ (2.12)

Under Assumption 2.4, since $\Delta(0,x,\xi)\equiv 0$ the eigenvalues of
$\underline{A}(0,x,\xi)$ are the double eigenvalue $\frac{1}{2}{\rm
Tr}\underline{A}(0,x,\xi)$. There are two cases222As opposed to the case of a
stiff transition, described in Section 2.3, where $\underline{A}(0,0,\xi_{0})$
is not semi-simple., as $\underline{A}(0,x,\xi)$ could be semi-simple or not.
In what follows we add the assumption

######  Assumption 2.5 (Semi-simplicity).

The unique eigenvalue of $\underline{A}(0,0,\xi_{0})$ is semi-simple, for all
$(x,\xi)$ near $(x_{0},\xi_{0})$.

This assumption is Hypothesis 1.5 in [LNT17]. We can now prove the following
normal form result

######  Proposition 2.6 (Normal form for the smooth transition).

Under Assumptions 2.2, 2.4 and 2.5, there is an analytical change of basis
$Q_{0}(t,x,\xi)\in{\mathbb{R}}^{2\times 2}$ such that

$Q_{0}^{-1}(t,x,\xi)\left(\underline{A}(t,x,\xi)-\frac{1}{2}{\rm
Tr}\underline{A}(t,x,\xi)\,{\rm Id}\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&t\\\
-t\delta^{2}&0\end{pmatrix}.$ (2.13)

###### Proof.

We denote

$\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&-a_{11}\end{pmatrix}=\underline{A}(t,x,\xi)-\frac{1}{2}{\rm
Tr}\underline{A}(t,x,\xi){\rm Id}.$

By definition (2.8) there holds $\Delta=-a_{11}^{2}-a_{12}a_{21}$ and then, by
(2.11) in Assumption 2.4

$-a_{11}^{2}-a_{12}a_{21}=t^{2}\delta(t,x,\xi)^{2}.$ (2.14)

By Assumption 2.5 the matrix $(a_{ij})_{i,j}$ satisfies

$a_{11}(0,x,\xi)=a_{12}(0,x,\xi)=a_{21}(0,x,\xi)=0$

so that there are smooth functions $\widetilde{a_{ij}}(t,x,\xi)$ such that
$a_{ij}(t,x,\xi)=t\widetilde{a_{ij}}(t,x,\xi)$. Hence by (2.14) we get

$\delta(t,x,\xi)^{2}=-\widetilde{a_{11}}^{2}-\widetilde{a_{12}}\widetilde{a_{21}}.$

As $\delta(0,0,\xi_{0})^{2}>0$ by Assumption 2.4 (2), the term
$\widetilde{a_{12}}\widetilde{a_{21}}(0,0,\xi_{0})$ cannot be $0$. Assuming
that $\widetilde{a_{21}}(0,0,\xi_{0})\neq 0$, the matrix

$Q_{0}(t,x)=\begin{pmatrix}a_{11}&1\\\ a_{12}&0\end{pmatrix}$

is such that (2.13) holds.

∎

###  2.3 The case of a stiff transition

If $\Delta$ is not the square of a function, its square roots are typically
not as smooth as $\Delta$. In fact, for any $k\in{\mathbb{N}}$ it is possible
to find $\Delta$ such that it is analytic, but its square roots are $C^{k}$
and not $C^{k+1}$. The first non degenerate case of this kind is when

$\partial_{t}\Delta(0,0,\xi_{0})>0$ (2.15)

which implies that $\Delta(t,0,\xi_{0})^{1/2}\sim t^{1/2}$ which is $C^{0}$
but not $C^{1}$ at $t=0$. With $\Delta(0,0,\xi_{0})=0$ by Assumption 2.2,
condition (2.15) and the implicit function theorem give the existence of an
analytic function $t_{\star}(x,\xi)$ such that

$\Delta(t,x,\xi)=0\Longleftrightarrow t=t_{\star}(x,\xi)\quad\text{locally
around }(t,x,\xi)=(0,0,\xi_{0}).$ (2.16)

Introducing

$e(t,x,\xi)=\int_{0}^{1}\partial_{t}\Delta((1-\tau)t_{\star}(x,\xi)+\tau
t,x,\xi)d\tau$

there holds

$\Delta(t,x,\xi)=\left(t-t_{\star}(x,\xi)\right)e(t,x,\xi).$

As $\Delta$ is analytic, $e$ is also analytic, and satisfies

$e(0,0,\xi_{0})=\partial_{t}\Delta(0,0,\xi_{0})>0$

so that $e$ is positive around $(0,0,\xi_{0})$. Then the sign of
$\Delta(t,x,\xi)$, hence the real or complex nature of the spectrum of
$\underline{A}(t,x,\xi)$, is given by the sign of $t-t_{\star}(x,\xi)$, a
situation comparable to the one described in Section 1.2.3 of [LNT17]:

  * •

For $(t,x)$ under the transition curve $\\{(t_{\star}(x,\xi),x)\\}$ the
eigenvalues of $\underline{A}(t,x,\xi)$ are real.

  * •

For $(t,x)$ above the transition curve, the eigenvalues of
$\underline{A}(t,x,\xi)$ have a non-zero imaginary part like $\pm
i(t-t_{\star}(x,\xi))^{1/2}$.

The question is then to describe $t_{\star}$. First, because
$\Delta(0,0,\xi_{0})=0$,

$t_{\star}(0,\xi_{0})=0.$ (2.17)

As $\Delta(0,x,\xi)\leq 0$ for $x\in B_{r_{0}}(0)$ and
$\xi\in{\mathbb{R}}^{d}$, we have

$t_{\star}(x,\xi)\geq 0\qquad\forall x\in
B_{r_{0}}(0)\;,\forall\xi\in{\mathbb{R}}^{d}$

which implies

$\partial_{x}t_{\star}(0,\xi_{0})=0\quad\text{and}\quad\partial_{x}^{3}t_{\star}(0,\xi_{0})=0$
(2.18)

so that the first terms of the Taylor expansion of $t_{\star}$ around
$(0,\xi_{0})$ are

$t_{\star}(x,\xi)=\frac{1}{2}\partial_{x}^{2}t_{\star}(0,\xi)\,x^{2}+\frac{1}{4!}\partial_{x}^{4}t_{\star}(0,\xi)\,x^{4}+\text{h.o.t.}$

Derivatives of $t_{\star}$ can be expressed by derivatives of $\Delta$, and
then by derivatives of $\underline{A}$. Then conditions on derivatives of
$t_{\star}$ are equivalent to conditions on derivatives of $\Delta$. First by
definition of $t_{\star}$ (2.16)

$\partial_{x}\left(\Delta(t_{\star}(x,\xi),x,\xi)\right)=0$

which implies that

$\partial_{x}t_{\star}(0,\xi_{0})\partial_{t}\Delta(0,0,\xi_{0})+\partial_{x}\Delta(0,0,\xi_{0})=0$

and then by (2.18)

$\partial_{x}\Delta(0,0,\xi_{0})=0.$

By Faà di Bruno formula on iterate derivatives applied to (2.16), there holds
for any $k\in{\mathbb{N}}$

$\partial_{x}^{k}\left(\Delta(t_{\star}(x,\xi),x,\xi)\right)=\sum_{0\leq
k_{1},k_{2}\leq
k}c_{k_{1},k_{2}}\prod_{p_{1}+\cdots+p_{k_{1}}+k_{2}=k}\partial_{x}^{p_{j}}t_{\star}(x,\xi)\left(\partial_{t}^{k_{1}}\partial_{x}^{k_{2}}\Delta\right)(t_{\star}(x,\xi),x,\xi)=0$

for some constants $c_{k_{1},k_{2}}$. For $k=2$ and $x=0$, we get

$\partial_{x}^{2}t_{\star}(0,\xi_{0})\partial_{t}\Delta(0,0,\xi_{0})+\partial_{x}^{2}\Delta(0,0,\xi_{0})=0.$

The non degenerate case $\partial_{x}^{2}t_{\star}(0,\xi_{0})>0$ cannot be
dealt with our method, as we will see in the proof of Proposition 5.4. We have
to assume

$\partial_{x}^{2}t_{\star}(0,\xi_{0})=0$

which is equivalent to

$\partial_{x}^{2}\Delta(0,0,\xi_{0})=0.$

Next we get

$\partial_{x}^{3}t_{\star}(0,\xi_{0})\partial_{t}\Delta(0,0,\xi_{0})+\partial_{x}^{3}\Delta(0,0,\xi_{0})=0$

which implies that

$\partial_{x}^{3}\Delta(0,0,\xi_{0})=0$

as $t_{\star}(x)\geq 0$. And finally,

$\partial_{x}^{4}t_{\star}(0,\xi_{0})\partial_{t}\Delta(0,0,\xi_{0})+\partial_{x}^{4}\Delta(0,0,\xi_{0})=0.$

In view of the above discussion, we formulate the following assumption on the
derivatives of $\Delta$

######  Assumption 2.7 (Degenerate stiff transition).

For $k=0$ and $k=2$

$\partial_{x}^{k}\Delta(0,0,\xi_{0})=0$ (2.19)

and

$\partial_{x}^{4}\Delta(0,0,\xi_{0})<0.$ (2.20)

Note that (2.9) implies $\partial_{x}^{k}\Delta(0,0,\xi_{0})=0$ for $=1$ and
$k=3$, as explained above.

######  Proposition 2.8 (Normal form for the stiff transition).

Under Assumptions 2.2 and 2.7, there are an analytical change of basis
$Q_{0}(t,x,\xi)$ and real analytical functions $t_{\star}(x,\xi)$ and
$e(t,x,\xi)$ such that

$Q_{0}^{-1}\left(\underline{A}(t,x,\xi)-\frac{1}{2}{\rm
Tr}\underline{A}(t,x,\xi)\,{\rm Id}\right)Q_{0}=\begin{pmatrix}0&1\\\
-(t-t_{\star})e&0\end{pmatrix}.$ (2.21)

###### Proof.

By definition of $\Delta$ and denoting

$\underline{A}-\frac{1}{2}{\rm Tr}\underline{A}\,{\rm
Id}=\begin{pmatrix}a_{11}&a_{12}\\\ a_{21}&-a_{11}\end{pmatrix}$

we get $\Delta=-a_{11}^{2}-a_{12}a_{21}$. As $\Delta\sim t$ both $a_{12}$ and
$a_{21}$ cannot both be zero at $(0,0,\xi_{0})$. Assuming that
$a_{21}(0,0,\xi_{0})\neq 0$, the matrix

$Q_{0}(t,x,\xi)=\begin{pmatrix}a_{11}&1\\\ a_{21}&0\end{pmatrix}$

is an analytical change of basis such that (2.21) holds.

∎

######  Remark 2.9.

Note that, on the contrary of the normal form of the smooth transition given
in Proposition 2.6, the normal form of the stiff transition is not semi-
simple. This is of importance, as non-semisimplicity introduces an additional
factor $\varepsilon^{-1/3}$ in the upper bound (3.26) of the Airy propagator,
to be compared with the upper bound (3.18) in the smooth case.

We add the following assumption:

######  Assumption 2.10 (Genuinely nonlinear zeroth-order perturbation).

We assume that $f(t,x,u)$ is quadratic in $u$ locally around $u=\vec{u}_{0}$,
that is

$\partial_{u}f(t,x,u)\big{|}_{u=\vec{u}_{0}}\equiv 0$

in a neighborhood of $(t,x)=(0,0)$.

###  2.4 Statement of the results

We recall first the definition of conical domain of ${\mathbb{R}}^{d+1}$, as
in Definition $2.2$ in [Mor16a]. For $x_{0}\in{\mathbb{R}}^{d}$ we denote

$\Omega_{R,\rho}(x_{0})=\bigcup_{t\geq
0}\\{t\\}\times\Omega_{R,\rho,t}(x_{0})=\left\\{(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{d}\;\Big{|}\;0\leq
t<\rho^{-1},\;R|x-x_{0}|_{1}+\rho t<1\right\\}.$ (2.22)

######  Theorem 2.11 (Gevrey ill-posedness of the smooth case).

Under Assumptions 2.1, 2.2, 2.4, 2.5 and 2.10, the Cauchy problem (1.1) is not
Hölder well-posed in Gevrey spaces $G^{\sigma}$ for all
$\sigma\in(0,\sigma_{0})$ with

$\sigma_{0}=1/3.$

That is for all $c>0$, $K$ compact of ${\mathbb{R}}^{d}$ and $\alpha\in(0,1]$,
there are sequences $R_{\varepsilon}^{-1}\to 0$ and
$\rho_{\varepsilon}^{-1}\to 0$, a family of initial conditions
$h_{\varepsilon}\in G^{\sigma}$ and corresponding solutions $u_{\varepsilon}$
of the Cauchy problem on domains
$\Omega_{R_{\varepsilon},\rho_{\varepsilon}}(0)$ such that

$\lim_{\varepsilon\to
0}||u_{\varepsilon}||_{L^{2}(\Omega_{R_{\varepsilon},\rho_{\varepsilon}}(0))}/||h_{\varepsilon}||_{\sigma,c,K}^{\alpha}=+\infty.$
(2.23)

The time of existence of the solutions $u_{\varepsilon}$ is at least of size
$\varepsilon^{1/2-\sigma/2}$.

######  Theorem 2.12 (Gevrey ill-posedness of the Airy case).

Under Assumptions 2.1, 2.2, 2.7 and 2.10, the result of Theorem 2.11 holds for
any Gevrey index $\sigma\in(0,\sigma_{0})$, with

$\sigma_{0}=2/13.$

Recall that a function $f$ defined on an open set $B$ of ${\mathbb{R}}^{d}$ is
said to belong to the Gevrey space $G^{\sigma}$ if for all compact $K\subset
B$, there are constants $C_{K}>0$ and $c_{K}>0$ that satisfy

$|\partial^{\alpha}f|_{L^{\infty}(K)}\leq
C_{K}c_{K}^{|\alpha|}|\alpha|!^{1/\sigma}\qquad\forall\alpha\in{\mathbb{N}}^{d}.$
(2.24)

We then define a family of norms on $G^{\sigma}$, for all compact $K\subset B$
and $c>0$

$||f||_{\sigma,c,K}=\sup_{\alpha}|\partial^{\alpha}f|_{L^{\infty}(K)}c^{-|\alpha|}|\alpha|!^{-1/\sigma}.$
(2.25)

######  Remark 2.13.

The limiting Gevrey index $\sigma_{0}$ is in both cases due in part to
technical limitations. In the proof, in each case remainder terms which are
proved to be small in the spaces described later. The limiting index
$\sigma_{0}$ is directly influenced by this smallness of the remainders. In
the smooth case, a null remainder would imply $\sigma_{0}=1/2$, which is the
expected limiting Gevrey index in this case. In the Airy case, a smaller
remainder would imply a greater index $\sigma_{0}$, but it is not clear if the
limit $1/2$ could be attained.

Also, as pointed out in Remark 2.9, one main difference between both cases is
the extra weight for the Airy propagator in the ansatz of highly oscillating
solutions, as shown in Lemma 3.5. This implies a stronger constraint on the
smallness of the remainder terms appearing in the Airy case, as explained in
the proof of Proposition 5.4.

The proofs are given in Sections 3 to 5, with an appendix devoted to the Airy
equation in Section 6. We introduce a functional framework that is flexible
enough to simultaneously cover the smooth, semi-simple case (Theorem 2.11) and
the stiff, non-semi-simple case (Theorem 2.12). We develop in Section 3 the
ansatz of highly oscillating solutions which reduces the Cauchy problem (1.1)
to a fixed point equation. In Sections 4 we recall properties of the spaces
developed in [Mor16a], and use them to prove contraction estimates and
existence of solutions. Finally, in Section 5.3 we prove that the constructed
solutions satisfy a lower bound that leads to the Hadamard instability for
Gevrey regularity $\sigma\in(0,\sigma_{0})$.

##  3 Highly oscillating solutions and reduction to a fixed point equation

###  3.1 Highly oscillating solutions

As in Section 3.1 of [Mor16a], we first reduced (1.1) to the new Cauchy
problem

$\partial_{t}u=\sum_{j}A_{j}(t,x,u)\partial_{x_{j}}{u}+F(t,x,u)u\qquad
u(0,x)=h(x)$ (3.1)

with $F$ analytic in a neighborhood of
$(0,0,0)\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}^{d}\times{\mathbb{R}}^{N}_{u}$,
and $h$ small analytic functions satisfying $h_{|x=0}=0$, as perturbations of
the trivial datum $h\equiv 0$.

Next, we adapt the ansatz of highly oscillating solutions of [Mét05] and
[Mor16a]. As the transitions may not occur for a fixed frequency $\xi_{0}$, we
have to deal with a family of functions indexed by both $\varepsilon$ and
$\xi$. In this view we posit

$u(t,x\,;\varepsilon,\xi)=\varepsilon{\mathbf{u}}(\varepsilon^{-1}\,t,x,x\cdot\xi/\varepsilon\,;\varepsilon,\xi)$
(3.2)

where

  * •

The small parameter $\varepsilon>0$ corresponds to high frequencies.

  * •

The parameter $\xi\in\mathbb{S}^{1}$ is close to the distinguished frequency
$\xi_{0}$.

  * •

The function ${\mathbf{u}}(s,x,\theta;\varepsilon,\xi)$ is $2\pi$-periodic in
$\theta$.

We introduce for any analytical function $H(t,x,u)$ the compact notation

${\mathbf{H}}(s,x,{\mathbf{u}}\,;\xi)=H\left(\varepsilon^{1/(1+\eta)}s,x,\varepsilon^{2/(1+\eta)}{\mathbf{u}}\,;\xi\right).$
(3.3)

For $u(t,x\,;\varepsilon,\xi)$ to be solution of (3.1) it is then sufficient
that ${\mathbf{u}}(s,x,\theta\,;\varepsilon,\xi)$ solves the following
equation

$\partial_{s}{\mathbf{u}}=\varepsilon^{-\eta/(1+\eta)}{\mathbf{A}}(\xi)\,\partial_{\theta}{\mathbf{u}}+\varepsilon^{1/(1+\eta)}\left(\sum_{j}{\mathbf{A}}_{j}\partial_{x_{j}}{\mathbf{u}}+{\mathbf{F}}\,{\mathbf{u}}\right)$
(3.4)

where we use the notation (3.3) for $\mathbf{A}$ and $\mathbf{F}$, and $A$ is
defined by (2.1).

###  3.2 Remainder terms

We focus here on the term
$\varepsilon^{-\eta/(1+\eta)}{\mathbf{A}}(\xi)\,\partial_{\theta}{\mathbf{u}}$
of the previous equation. To prove the expected growth of solutions of the
initial problem, we decompose the symbol $A(t,x,u,\xi)$ in several pieces to
highlight the leading term denoted by $\underline{A}^{{\rm S}}(t)$ for the
smooth case, $\underline{A}^{{\rm Ai}}(t)$ for the Airy case, which will lead
to the exponential growth.

First, by analyticity of the $A_{j}$ and Taylor expansion formula, there is a
family of analytical matrices $(A_{u_{j}})_{j=1,\ldots,N}$ such that locally
around
$(0,0,0)\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}^{d}\times{\mathbb{R}}_{u}^{N}$
there holds

$A(t,x,u,\xi)=\underline{A}(t,x,\xi)+\sum_{j}A_{u_{j}}u_{j}.$ (3.5)

In both smooth and Airy cases, we perform an analytical Taylor expansion on
$\underline{A}(t,x,\xi)$ in order to highlight the principal term that lead to
the exponential growth. This is made precise in the two following lemmas.

######  Lemma 3.1 (Expansion formula: smooth case).

We introduce the leading term $\underline{A}^{{\rm S}}(t)$, defined up to a
change of basis and a trace terme as

$Q_{0}^{-1}(t,x,\xi)\left(\underline{A}^{{\rm S}}(t)-\frac{1}{2}{\rm
Tr}\,\underline{A}(t,x,\xi)\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&t\\\
-\gamma_{0}^{2}t&0\end{pmatrix}$ (3.6)

and the analytical error term

$Q_{0}^{-1}\,{\rm R}^{{\rm S}}\,Q_{0}=\begin{pmatrix}0&0\\\
-t(\delta^{2}-\delta(0,0,\xi_{0})^{2})&0\end{pmatrix}.$ (3.7)

Then there holds

$\underline{A}(t,x,\xi)=\underline{A}^{{\rm S}}(t)+{\rm R}^{{\rm S}}(t,x,\xi)$
(3.8)

and there are analytical matrices ${\rm R}^{{\rm S}}_{t}(t,x)$, ${\rm R}^{{\rm
S}}_{x}(t,x)$ and ${\rm R}^{{\rm S}}_{\xi}(t,x)$ such that

${\rm R}^{{\rm S}}(t,x,\xi)=t^{2}{\rm R}^{{\rm S}}_{t}(t,x,\xi)+tx\cdot{\rm
R}^{{\rm S}}_{x}(t,x,\xi)+t(\xi-\xi_{0})\cdot{\rm R}^{{\rm S}}_{\xi}(t,x,\xi)$
(3.9)

locally around
$(0,0,\xi_{0})\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}^{d}\times{\mathbb{R}}_{\xi}^{d}$.

###### Proof.

First the equality (2.13) of Lemma 2.6 implies that

$Q_{0}^{-1}(t,x,\xi)\left(\underline{A}^{{\rm S}}-\frac{1}{2}{\rm
Tr}\,\underline{A}(t,x,\xi)\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&t\\\
-t\delta^{2}&0\end{pmatrix}$

hence (3.8). Second, by analyticity of $\delta$ and Taylor expansion formula,
there are analytical functions ${\rm r}^{{\rm S}}_{t}$, ${\rm r}^{{\rm
S}}_{x_{j}}$ and ${\rm r}^{{\rm S}}_{\xi}$ such that

$\delta^{2}(t,x)-\delta(0,0)^{2}=t\,{\rm r}^{{\rm S}}_{t}(t,x)+x\cdot{\rm
r}^{{\rm S}}_{x}(t,x)+(\xi-\xi_{0})\cdot{\rm r}^{{\rm S}}_{\xi}.$

Introducing finally the matrices

$Q_{0}^{-1}\,{\rm R}^{{\rm S}}_{t}\,Q_{0}=\begin{pmatrix}0&0\\\ -{\rm r}^{{\rm
S}}_{t}&0\end{pmatrix}\quad Q_{0}^{-1}\,{\rm R}^{{\rm
S}}_{x}\,Q_{0}=\begin{pmatrix}0&0\\\ -{\rm r}^{{\rm
S}}_{x}&0\end{pmatrix}\quad Q_{0}^{-1}\,{\rm R}^{{\rm
S}}_{\xi}\,Q_{0}=\begin{pmatrix}0&0\\\ -{\rm r}^{{\rm
S}}_{\xi}&0\end{pmatrix}$

which leads to (3.9) and ends the proof. ∎

######  Lemma 3.2 (Expansion formula: Airy case).

We introduce the leading term $\underline{A}^{{\rm S}}(t)$, defined up to a
change of basis and a trace terme as

$Q_{0}^{-1}(t,x,\xi)\left(\underline{A}^{{\rm Ai}}(t)-\frac{1}{2}{\rm
Tr}\,\underline{A}(t,x,\xi)\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&1\\\
-\gamma_{0}^{2}t&0\end{pmatrix}$ (3.10)

and the analytical error term

$Q_{0}^{-1}(t,x,\xi)\left({\rm R}^{{\rm Ai}}-\frac{1}{2}{\rm
Tr}\,\underline{A}(t,x,\xi)\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&0\\\
-t(e-e(0,0))+t_{\star}e&0\end{pmatrix}.$ (3.11)

Then there holds

$\underline{A}(t,x)=\underline{A}^{{\rm Ai}}(t)+{\rm R}^{{\rm Ai}}(t,x).$
(3.12)

and there are analytical matrices ${\rm R}^{{\rm Ai}}_{t}$, ${\rm R}^{{\rm
Ai}}_{x}$, ${\rm R}^{{\rm Ai}}_{\xi}$ and ${\rm R}^{{\rm Ai}}_{e}$ such that

${\rm R}^{{\rm Ai}}(t,x)=t^{2}{\rm R}^{{\rm Ai}}_{t}(t,x)+tx\cdot{\rm R}^{{\rm
Ai}}_{x}(t,x)+t(\xi-\xi_{0})\cdot{\rm R}^{{\rm Ai}}_{\xi}(t,x)+t_{\star}{\rm
R}^{{\rm Ai}}_{e}(t,x)$ (3.13)

locally around $(0,0)$.

###### Proof.

First the equality (2.21) of Lemma 2.8 implies that

$Q_{0}^{-1}(t,x,\xi)\left(\underline{A}^{{\rm Ai}}-\frac{1}{2}{\rm
Tr}\,\underline{A}(t,x,\xi)\right)Q_{0}(t,x,\xi)=\begin{pmatrix}0&1\\\
-(t-t_{\star}(x))e(t,x)&0\end{pmatrix}$

hence (3.12). Second, by analyticity of $e$ and Taylor expansion formula,
there are analytical functions $e_{t}$, $e_{x_{j}}$ and $e_{\xi}$ such that

$e(t,x,\xi)-e(0,0,\xi_{0})=te_{t}+x\cdot e_{x}+(\xi-\xi_{0})\cdot e_{\xi}$

locally around $(0,0)$. Introducing the matrices

$Q_{0}^{-1}\,{\rm R}^{{\rm Ai}}_{t}\,Q_{0}=\begin{pmatrix}0&0\\\
-e_{t}&0\end{pmatrix}\quad,\quad Q_{0}^{-1}\,{\rm R}^{{\rm
Ai}}_{x}\,Q_{0}=\begin{pmatrix}0&0\\\ -e_{x}&0\end{pmatrix}$

and

$Q_{0}^{-1}\,{\rm R}^{{\rm Ai}}_{\xi}\,Q_{0}=\begin{pmatrix}0&0\\\
{}-e_{\xi}&0\end{pmatrix}\quad,\quad Q_{0}^{-1}\,{\rm R}^{{\rm
Ai}}_{e}\,Q_{0}=\begin{pmatrix}0&0\\\ e&0\end{pmatrix}$

we end the proof. ∎

In both Airy and smooth cases, we have then an expansion formula of the form

$A(t,x,u,\xi)=\underline{A}^{\eta}(t)+{\rm R}^{\eta}(t,x)+A_{u}\cdot u$

where $\eta$ corresponds to the parameter introduced in the ansatz (3.2),
adapted to each specific case. This parameter will be be precised in Lemma 3.4
in the smooth case, and in Lemma 3.5 for the Airy case. The remainder term
${\rm R}^{\eta}$ is ${\rm R}^{{\rm S}}$ defined by (3.7) in the smooth case,
and is ${\rm R}^{{\rm Ai}}$ defined by (3.11) in the Airy case.

We rewrite now equation (3.4) as

$\partial_{s}{\mathbf{u}}-\varepsilon^{-\eta/(1+\eta)}\underline{A}^{\eta}(\varepsilon^{1/(1+\eta)}s)\partial_{\theta}{\mathbf{u}}={\mathbf{G}}^{\eta}(s,x,{\mathbf{u}}).$
(3.14)

where we define the source term

$\displaystyle{\mathbf{G}}^{\eta}(s,x,{\mathbf{u}})$ $\displaystyle=$
$\displaystyle\varepsilon^{-\eta/(1+\eta)}\left({\mathbf{R}}^{\eta}+\varepsilon^{2/(1+\eta)}{\mathbf{A}}_{u}\cdot{\mathbf{u}}\right)\partial_{\theta}{\mathbf{u}}$
$\displaystyle+\varepsilon^{1/(1+\eta)}\left({\mathbf{A}}(s,x,{\mathbf{u}})\cdot\partial_{x}{\mathbf{u}}+{\mathbf{F}}(s,x,{\mathbf{u}}){\mathbf{u}}\right)$

using the notation (3.3).

######  Remark 3.3.

Note that in [Mor16a] there are no remainder terms ${\mathbf{R}}^{\eta}$, as
we consider the full varying-coefficient operator $\underline{A}(\varepsilon
s,x)\partial_{\theta}$ in equation (3.14).

###  3.3 Upper bounds for the propagators

To solve the Cauchy problem of the equation (3.14) with initial datum
$h_{\varepsilon}$ specified in Section 3.4, we first study the case
${\mathbf{G}}^{\eta}\equiv 0$, that is

$\partial_{s}{\mathbf{u}}-\varepsilon^{-\eta/(1+\eta)}\underline{A}^{\eta}(\varepsilon^{1/(1+\eta)}s)\partial_{\theta}{\mathbf{u}}=0.$
(3.16)

Note that this equation is linear, non autonomous and non scalar. For a
general $\underline{A}^{\eta}(t)$ we define the matrix propagator
$U^{\eta}(s^{\prime},s,\theta)$ as the solution of

$\partial_{s}U^{\eta}(s^{\prime},s,\theta)-\varepsilon^{-\eta/(1+\eta)}\underline{A}^{\eta}(\varepsilon^{1/(1+\eta)}s)\partial_{\theta}U^{\eta}(s^{\prime},s,\theta)=0\;,\quad
U^{\eta}(s^{\prime},s^{\prime},\theta)\equiv{\rm Id}.$

and $U^{\eta}(s^{\prime},s,\theta)$ is periodic in $\theta$, following the
ansatz (3.2). The choice of the time scaling $s=\varepsilon^{-1/(1+\eta)}t$,
that is the choice of $\eta$, is such that solutions of (3.16) have a typical
exponential growth independent of $\varepsilon$. Both following Lemmas
explicit the growth of the propagators in both cases.

######  Lemma 3.4 (Growth of the propagator: the smooth case).

Under Assumptions 2.2, 2.4 and 2.5, we put $\eta=1$. The matrix propagator
$U^{{\rm S}}(s^{\prime},s,\theta)$ defined by

$\partial_{s}U^{{\rm S}}(s^{\prime},s,\theta)-\underline{A}^{{\rm
S}}(s)\partial_{\theta}U^{{\rm S}}(s^{\prime},s,\theta)=0\quad,\quad U^{{\rm
S}}(s^{\prime},s^{\prime},\theta)={\rm Id}$ (3.17)

satisfies the following growth of its Fourier modes in the $\theta$ variable:

$|U^{{\rm
S}}_{n}(s^{\prime},s)|\lesssim\exp\left(\int_{s^{\prime}}^{s}\gamma^{\sharp}_{{\rm
S}}(\tau)d\tau\,|n|\right)\quad,\quad\forall\,0\leq s^{\prime}\leq s\text{ and
}\forall\,n\in{\mathbb{Z}}$ (3.18)

with

$\gamma^{\sharp}_{{\rm S}}(\tau)=\gamma_{0}\tau.$ (3.19)

###### Proof.

First, as $\underline{A}^{{\rm S}}$ is given by (3.6) up to a change of basis
$Q_{0}(t,x,\xi)$ and a trace term $\frac{1}{2}{\rm Tr}\underline{A}$, we
introduce

$V_{n}(s^{\prime},s,x\,;\varepsilon,\xi)=\exp\left(in\int_{s^{\prime}}^{s}\frac{1}{2}{\rm
Tr}\underline{A}(\varepsilon^{1/(1+\eta)}\tau,x,\xi)d\tau\right)Q_{0}^{-1}(\varepsilon^{1/(1+\eta)}s,x,\xi)U_{n}(s^{\prime},s,x\,;\varepsilon,\xi)$
(3.20)

which solves

$\partial_{s}V_{n}(s^{\prime},s)-in\begin{pmatrix}0&\varepsilon^{(1-\eta)/(1+\eta)}s\\\
-\gamma_{0}^{2}\varepsilon^{(1-\eta)/(1+\eta)}s&0\end{pmatrix}V_{n}(s^{\prime},s)=-\varepsilon^{1/(1+\eta)}Q_{0}^{-1}\partial_{t}Q_{0}\,V_{n}(s^{\prime},s)$
(3.21)

with initial condition
$V_{n}(s^{\prime},s^{\prime},x\,;\varepsilon,\xi)=Q_{0}^{-1}(\varepsilon^{1/(1+\eta)}s^{\prime},x,\xi)$.
We focus then on the autonomous differential system

$\partial_{s}\widetilde{V}_{n}(s^{\prime},s)-in\begin{pmatrix}0&\varepsilon^{(1-\eta)/(1+\eta)}s\\\
-\gamma_{0}^{2}\varepsilon^{(1-\eta)/(1+\eta)}s&0\end{pmatrix}\widetilde{V}_{n}(s^{\prime},s)=0$
(3.22)

which becomes, with the choice $\eta=1$, the $\varepsilon$-free matrix
equation

$\partial_{s}\widetilde{V}_{n}(s^{\prime},s)-in\,s\,\begin{pmatrix}0&1\\\
-\gamma_{0}^{2}&0\end{pmatrix}\widetilde{V}_{n}(s^{\prime},s)=0.$

The complex constant change of basis

$\underline{Q}=\begin{pmatrix}1&1\\\ -i\gamma_{0}&i\gamma_{0}\end{pmatrix}$

leads us to the exact solution

$\widetilde{V}_{n}(s^{\prime},s)=\underline{Q}\begin{pmatrix}\exp(n\gamma_{0}(s^{2}-s^{\prime
2})/2)&0\\\ 0&\exp(-n\gamma_{0}(s^{2}-s^{\prime
2})/2)\end{pmatrix}\underline{Q}^{-1}$

which satisfies the upper bound

$|\widetilde{V}_{n}(s^{\prime},s,\,;\varepsilon,\xi)|\lesssim\exp(|n|\gamma_{0}(s^{2}-s^{\prime
2})/2)$ (3.23)

uniformly in $x$ and $\xi$. Getting back to (3.21), we use Duhamel formula to
write

$V_{n}(s^{\prime},s)=\widetilde{V}_{n}(s^{\prime},s)Q_{0}^{-1}(\varepsilon^{1/2}s^{\prime})-\int_{s^{\prime}}^{s}\varepsilon^{1/2}\widetilde{V}_{n}(\tau,s)\,\left(Q_{0}^{-1}\partial_{t}Q_{0}\right)(\varepsilon^{1/2}\tau)\,V_{n}(s^{\prime},\tau)d\tau.$
(3.24)

Factorizing by the exponential growth $e^{|n|\gamma_{0}(s^{2}-s^{\prime
2})/2}$, we get

$\displaystyle e^{-|n|\gamma_{0}(s^{2}-s^{\prime
2})/}|V_{n}(s^{\prime},s)|\,\lesssim\,e^{-|n|\gamma_{0}(s^{2}-s^{\prime
2})/}|\widetilde{V}(s^{\prime},s)|$
$\displaystyle\quad+\int_{s^{\prime}}^{s}\varepsilon^{1/2}e^{-|n|\gamma_{0}(s^{2}-\tau^{2})/}|\widetilde{V}_{n}(\tau,s)|\,\left|Q_{0}^{-1}\partial_{t}Q_{0}\right|(\varepsilon^{1/2}\tau)\,e^{-|n|\gamma_{0}(\tau^{2}-s^{\prime
2})/}|V_{n}(s^{\prime},\tau)|d\tau.$

We denote for conveniency $y(s^{\prime},s)=e^{-|n|\gamma_{0}(s^{2}-s^{\prime
2})/}|V_{n}(s^{\prime},s)|$. Thanks to the upper bound (3.23) there holds

$y(s^{\prime},s)\lesssim
1+\int_{s^{\prime}}^{s}\varepsilon^{1/2}\,\left|Q_{0}^{-1}\partial_{t}Q_{0}\right|(\varepsilon^{1/2}\tau)\,y(s^{\prime},\tau)d\tau$

and we make use of the Gronwall inequality to get

$y(s^{\prime},s)\lesssim\exp\left(\varepsilon^{1/2}c(s-s^{\prime})\right)$

with $c=\max\left|Q_{0}^{-1}\partial_{t}Q_{0}\right|>0$. As
$\varepsilon^{1/2}s$ is small in our setting, we get the announced upper bound
(3.18).

∎

######  Lemma 3.5 (Growth of the propagator: the Airy case).

Under Assumptions 2.2 and 2.7 we put $\eta=1/2$. The matrix propagator
$U^{{\rm Ai}}(s^{\prime},s,\theta)$ defined by

$\partial_{s}U^{{\rm
Ai}}(s^{\prime},s,\theta)-\varepsilon^{-1/3}\underline{A}^{{\rm
Ai}}(\varepsilon^{2/3}s)\partial_{\theta}U^{{\rm
Ai}}(s^{\prime},s,\theta)=0\quad,\quad U^{{\rm
Ai}}(s^{\prime},s^{\prime},\theta)\equiv{\rm Id}$ (3.25)

satisfies the following growth of its Fourier modes in the $\theta$ variable:

$|U^{{\rm
Ai}}_{n}(s^{\prime},s)|\lesssim\varepsilon^{-1/3}\exp\left(\int_{s^{\prime}}^{s}\gamma^{\sharp}_{{\rm
Ai}}(\tau)d\tau\,|n|)\right)\quad,\quad\forall\,0\leq s^{\prime}\leq
s<\varepsilon^{-2/3}\text{ and }\forall\,n\in{\mathbb{Z}}$ (3.26)

$\gamma^{\sharp}_{{\rm Ai}}(\tau)=\gamma_{0}\tau^{1/2}.$ (3.27)

###### Proof.

We proceed as in the previous proof of Lemma 3.4, by defining

$V_{n}(s^{\prime},s,x\,;\varepsilon,\xi)=\exp\left(in\int_{s^{\prime}}^{s}\frac{1}{2}{\rm
Tr}\underline{A}(\varepsilon^{1/(1+\eta)}\tau,x,\xi)d\tau\right)Q_{0}^{-1}(\varepsilon^{1/(1+\eta)}s,x,\xi)U_{n}^{{\rm
Ai}}(s^{\prime},s,x\,;\varepsilon,\xi)$

and looking at the matrix equation

$\partial_{s}\widetilde{V}_{n}(s^{\prime},s)-in\begin{pmatrix}0&\varepsilon^{-\eta/(1+\eta)}\\\
-\gamma_{0}^{2}\varepsilon^{(1-\eta)/(1+\eta)}s&0\end{pmatrix}\widetilde{V}_{n}(s^{\prime},s)=0.$
(3.28)

As the eigenvalues of

$\begin{pmatrix}0&\varepsilon^{-\eta/(1+\eta)}\\\
-\gamma_{0}^{2}\varepsilon^{(1-\eta)/(1+\eta)}s&0\end{pmatrix}$

are $\pm i\gamma_{0}\varepsilon^{(1-2\eta)/(1+\eta)}\sqrt{s}$, the choice
$\eta=1/2$ is natural in this case.

As in the proof of Lemma 3.4, to prove the upper bound (3.26) it suffices to
prove the same bound for $\widetilde{V}_{n}(s^{\prime},s)$. This is postponed
to Section 6.2 of the Appendix. It uses classical bounds on the Airy function.

∎

######  Remark 3.6.

The limitation $s<\varepsilon^{-2/3}$ for the previous estimate is not a
constraint for the following of the paper. Indeed, as $t=\varepsilon^{2/3}s$
is bound to be small, $s<\varepsilon^{-2/3}$ is valid in our scaling.

###  3.4 Free solutions

As in Section 3.5 in [Mor16a], we seek for high-oscillating, small and well-
polarized initial data of the form

$h^{\eta}_{\varepsilon}(x)=\varepsilon^{2/(1+\eta)}e^{-M(\varepsilon)}{\rm
Re}\left(e^{ix\cdot\xi_{0}/\varepsilon}\vec{e}^{\,\eta}_{+}+e^{-ix\cdot\xi_{0}/\varepsilon}\vec{e}^{\,\eta}_{-}\right)$
(3.29)

which correspond in the ansatz (3.2) of high-oscillating solutions to

${\mathbf{h}}^{\eta}_{\varepsilon}(\theta)=e^{-M(\varepsilon)}{\rm
Re}\left(e^{i\theta}\vec{e}^{\,\eta}_{+}+e^{-i\theta}\vec{e}^{\,\eta}_{-}\right).$
(3.30)

Here $\vec{e}^{\,\eta}_{+}$ and $\vec{e}^{\,\eta}_{-}$ are vectors chosen in
each case such that $U^{\eta}(0,s){\mathbf{h}}^{\eta}_{\varepsilon}$ satisfies
the maximal growth of $U^{\eta}$, in either smooth or Airy case. The parameter
$M(\varepsilon)$ is chosen such that the Gevrey norm of
$h^{\eta}_{\varepsilon}$ is small. Following Lemma 3.3 of [Mor16a], we posit

$M(\varepsilon)=\varepsilon^{-\delta},\quad\delta\in(\sigma,1).$ (3.31)

Remark 3.4 in [Mor16a] explains in particular the link between the long time
of existence of solutions and the Gevrey weight $e^{-M(\varepsilon)}$, hence
the constraint $\sigma<\delta$.

We introduce also

${\mathbf{f}}^{\eta}(s,\theta)=U^{\eta}(0,s){\mathbf{h}}^{\eta}_{\varepsilon}(\theta)$
(3.32)

which we call the free solution of equation (3.14) as it solves the equation
for ${\mathbf{G}}^{\eta}\equiv 0$. In both smooth and Airy cases, we prove
that for well-chosen $\vec{e}^{\,\eta}_{+}$ and $\vec{e}^{\,\eta}_{-}$, Lemma
3.2 of [Mor16a] still holds with

$\gamma^{\flat}_{\eta}(\tau)=\gamma_{0}\tau^{\eta}$ (3.33)

for $\eta=1$ and $\eta=1/2$.

######  Lemma 3.7 (Growth of the free solution: the smooth case).

We define

$\vec{e}_{+}^{\,{\rm S}}=Q_{0}(0,x,\xi)\begin{pmatrix}1\\\
-i\gamma_{0}\end{pmatrix}\quad\text{and}\quad\vec{e}_{-}^{\,{\rm
S}}=Q_{0}(0,x,\xi)\begin{pmatrix}1\\\ i\gamma_{0}\end{pmatrix}.$ (3.34)

Then the free solution ${\mathbf{f}}^{{\rm S}}$ of the smooth case satisfies

$|{\mathbf{f}}^{{\rm S}}(s)|\approx
e^{-M(\varepsilon)}e^{\int_{0}^{s}\gamma_{{\rm
S}}^{\flat}(\tau)d\tau}\quad,\quad\forall s\geq 0$ (3.35)

where $\approx$ means equality up to constant and with

$\gamma_{{\rm S}}^{\flat}(\tau)=\gamma_{0}\tau.$ (3.36)

###### Proof.

We use the steps of the proof of Lemma 3.4. First, it is explicit that there
holds

$\widetilde{V}_{+1}(0,s)\begin{pmatrix}1\\\
-i\gamma_{0}\end{pmatrix}=e^{\gamma_{0}s^{2}/2}\begin{pmatrix}1\\\
-i\gamma_{0}\end{pmatrix}$ (3.37)

and also

$\widetilde{V}_{-1}(0,s)\begin{pmatrix}1\\\
i\gamma_{0}\end{pmatrix}=e^{\gamma_{0}s^{2}/2}\begin{pmatrix}1\\\
i\gamma_{0}\end{pmatrix}.$ (3.38)

Then, thanks to the Duhamel formula (3.24) there holds

$V_{+1}(0,s)\vec{e}_{+}^{\,{\rm S}}=\widetilde{V}_{+1}(0,s)\begin{pmatrix}1\\\
-i\gamma_{0}\end{pmatrix}-\int_{0}^{s}\varepsilon^{1/2}\widetilde{V}_{+1}(\tau,s)\,\left(Q_{0}^{-1}\partial_{t}Q_{0}\right)(\varepsilon^{1/2}\tau)\,V_{n}(0,\tau)\vec{e}_{+}^{\,{\rm
S}}d\tau.$

Using the upper bounds of $\widetilde{V}$ and $V$ proved in Lemma 3.4, we get
the following estimate for the integral term

$\displaystyle\left|\int_{0}^{s}\varepsilon^{1/2}\widetilde{V}_{+1}(\tau,s)\,\left(Q_{0}^{-1}\partial_{t}Q_{0}\right)(\varepsilon^{1/2}\tau)\,V_{n}(0,\tau)\vec{e}_{+}^{\,{\rm
S}}d\tau\right|$
$\displaystyle\lesssim\varepsilon^{1/2}\int_{0}^{s}\exp(\gamma_{0}(s^{2}-\tau^{2})/2)\exp(\gamma_{0}\tau^{2}/2)d\tau$
$\displaystyle\lesssim\varepsilon^{1/2}s\exp(\gamma_{0}s^{2}/2).$

By (3.37) and as $\varepsilon^{1/2}s$ is small, we get

$\left|V_{+1}(0,s)\vec{e}_{+}^{\,{\rm S}}\right|\approx e^{\gamma_{0}s^{2}/2}$

and the same holds for $V_{-1}(0,s)$. We end the proof by using formula
(3.20).

∎

In the Airy case, we make a careful analysis of the prefactor term coming from
the crossing of eigenvalues.

######  Lemma 3.8 (Growth of the free solution: Airy case).

We define

$\vec{e}_{+}^{\,{\rm Ai}}=Q_{0}(0,x,\xi)\begin{pmatrix}{\rm Ai_{1}}(0)\\\
-i\varepsilon^{1/3}j{\rm
Ai_{1}}^{\prime}(0)\end{pmatrix}\quad\text{and}\quad\vec{e}_{-}^{\,{\rm
Ai}}=Q_{0}(0,x,\xi)\begin{pmatrix}{\rm Ai_{1}}(0)\\\ i\varepsilon^{1/3}j{\rm
Ai_{1}}^{\prime}(0)\end{pmatrix}\quad$ (3.39)

where ${\rm Ai_{1}}$ is the Airy function defined in Lemma 6.1 of the
Appendix, and $j=e^{2i\pi/3}$. Then the free solution ${\mathbf{f}}^{{\rm
Ai}}$ of the Airy case satisfies

$|{\mathbf{f}}^{{\rm Ai}}(s)|\approx
s^{-1/4}e^{-M(\varepsilon)}e^{\int_{0}^{s}\gamma_{{\rm
Ai}}^{\flat}(\tau)d\tau}\quad,\quad\forall 0\leq s<\varepsilon^{-2/3}$ (3.40)

with

$\gamma_{{\rm Ai}}^{\flat}(\tau)=\gamma_{0}\tau^{1/2}.$ (3.41)

We postpone the proof of this Lemma to Section 6.3 of the Appendix. For the
limitation $s<\varepsilon^{-2/3}$, see Remark 3.6.

######  Remark 3.9.

Note that, on the contrary of [Mor16a], in each case there holds
$\gamma_{\eta}^{\sharp}=\gamma_{\eta}^{\flat}$. This is due to the fact that
we do not consider here the full varying-coefficient operator
$\underline{A}(\varepsilon^{1/(1+\eta)}s,x)\partial_{\theta}$ but the reduced
operator $\underline{A}^{\eta}(s)\partial_{\theta}$, which is a first-order
approximation of operator
$\underline{A}(\varepsilon^{1/(1+\eta)}s,x)\partial_{\theta}$.

###  3.5 Fixed point equation

Using the propagator $U^{\eta}(s^{\prime},s,\theta)$, the free solution (3.32)
and the Duhamel formula, we can express now (3.14) as the fixed point equation

${\mathbf{u}}(s,x,\theta)={\mathbf{f}}^{\eta}(s,\theta)+\int_{0}^{s}U^{\eta}(s^{\prime},s,\theta){\mathbf{G}}^{\eta}(s^{\prime},{\mathbf{u}}(s^{\prime}x,\theta))ds^{\prime}$
(3.42)

where ${\mathbf{G}}^{\eta}({\mathbf{u}})$ is defined by (3.2). We denote the
integral term

$T^{\eta}(s,{\mathbf{u}})=\int_{0}^{s}U^{\eta}(s^{\prime},s){\mathbf{G}}^{\eta}(s^{\prime},{\mathbf{u}}(s^{\prime}))ds^{\prime}$
(3.43)

which we split into three parts thanks to definition (3.2) like

$\displaystyle\quad T^{\eta}(s,{\mathbf{u}})$ (3.44)
$\displaystyle=\int_{0}^{s}U^{\eta}(s^{\prime},s)\left[\left(\varepsilon^{-\eta/(1+\eta)}{\mathbf{R}}^{\eta}+\varepsilon^{(2-\eta)/(1+\eta)}{\mathbf{A}}_{u}\cdot{\mathbf{u}}\right)\partial_{\theta}{\mathbf{u}}+\varepsilon^{1/(1+\eta)}\left({\mathbf{A}}\cdot\partial_{x}{\mathbf{u}}+{\mathbf{F}}\right)\right]ds^{\prime}$
$\displaystyle=T^{[\eta,\theta]}(s,{\mathbf{u}})+T^{[\eta,x]}(s,{\mathbf{u}})+T^{[\eta,{\mathbf{u}}]}(s,{\mathbf{u}})$
(3.45)

where we define

$\displaystyle T^{[\eta,\theta]}(s,x,{\mathbf{u}})$ $\displaystyle=$
$\displaystyle\int_{0}^{s}U^{\eta}(s^{\prime},s)\,\left[\varepsilon^{-\eta/(1+\eta)}{\mathbf{R}}^{\eta}+\varepsilon^{(2-\eta)/(1+\eta)}{\mathbf{A}}_{u}\cdot{\mathbf{u}}\right]\,\partial_{\theta}{\mathbf{u}}(s^{\prime})ds^{\prime}$
(3.46) $\displaystyle T^{[\eta,x]}(s,x,{\mathbf{u}})$ $\displaystyle=$
$\displaystyle\int_{0}^{s}U^{\eta}(s^{\prime},s)\,\left[\varepsilon^{1/(1+\eta)}{\mathbf{A}}(s^{\prime},x,{\mathbf{u}}(s^{\prime}))\right]\cdot\partial_{x}{\mathbf{u}}(s^{\prime})ds^{\prime}$
(3.47) $\displaystyle T^{[\eta,{\mathbf{u}}]}(s,x,{\mathbf{u}})$
$\displaystyle=$
$\displaystyle\int_{0}^{s}U^{\eta}(s^{\prime},s)\,\left[\varepsilon^{1/(1+\eta)}{\mathbf{F}}(s^{\prime},x,{\mathbf{u}}(s^{\prime}))\right]\,{\mathbf{u}}(s^{\prime})ds^{\prime}$
(3.48)

We have now reduced the initial question of finding a family of initial data
$h_{\varepsilon}$ generating a family of appropriately growing analytic
solutions $u_{\varepsilon}$ to the fixed point equation (3.42) for operator
$T^{\eta}$. In order to prove Theorems 2.11 and 2.12 we refer to the proof of
Gevrey instability in the case of initial ellipticity in [Mor16a]. A sketch of
the proof can be found in Section 3.6 therein.

##  4 Contraction estimates

We make use of spaces ${\mathbb{E}}$ constructed in Section 4 in [Mor16a] and
their properties developed in Section 5 therein. The method is robust enough
to be used in our context of transitions from hyperbolicity to ellipticity.

###  4.1 Functional spaces: definitions

We refer to Section $4.1$ of [Mor16a] for definition and properties of the
majoring series relation denoted by $\prec$. We recall the definition of
$\Phi$ as the reference series in one variable

$\Phi(z)=\sum_{k\geq 0}\frac{c_{0}}{k^{2}+1}z^{k}$

with $c_{0}>0$ such that $\Phi^{2}\prec\Phi$. We recall the notation

$\Phi_{k}(t)=R^{|k|}\sum_{p\in{\mathbb{N}}}\frac{c_{0}}{(|k|+p)^{2}+1}\binom{|k|+p}{k,p}\rho^{p}t^{p}\qquad\forall
k\in{\mathbb{N}}^{d}$ (4.1)

of the coefficients of $\Phi(RX_{1}+\cdots+RX_{d}+\rho t)$, with $R$ and
$\rho$ both positive parameters. From now on, we will denote for convenience
and with an abuse of notation $\Phi(RX+\rho t)=\Phi(RX_{1}+\cdots+RX_{d}+\rho
t)$.

Recall that, for any formal series
$\phi(t,x)=\sum_{k\in{\mathbb{N}}^{d}}\phi_{k}(t)x^{k}$ in the $x$ variable,
with $t$ a parameter, notation $\phi(t,x)\prec_{t}\Phi(RX+\rho t)$ means

$|\phi_{k}(t)|\leq\Phi_{k}(t)\quad,\quad\forall\,k\in{\mathbb{N}}^{d}\;,\;\forall
t<\rho^{-1}.$

We consider trigonometric series in one variable $\theta$ with coefficients in
the space of formal series in $d$ variables $x$ in the sense of Section 4.1
with $\xi\in B_{\varepsilon}(\xi_{0})$ a parameter, and we denote $F_{2d+1}$
the space of all such trigonometric series:

$F_{2d+1}=\left\\{{\mathbf{v}}(x,\theta\,;\xi)=\sum_{n\in\mathbb{Z}}{\mathbf{v}}_{n}(x)e^{in\theta}\;\Big{|}\;{\mathbf{v}}_{n}(x\,;\xi)=\sum_{k\in{\mathbb{N}}^{d}}{\mathbf{v}}_{n,k}(\xi)x^{k}\right\\}.$

We now define fixed time spaces ${\mathbb{E}}_{s}$ as in [Mor16a], which we
slightly modified as to take into account three differences:

  * •

The solutions ${\mathbf{u}}(s,x,\theta\,;\xi)$ depend on
$\xi\in\mathbb{S}^{1}$.

  * •

The renormalization in time is $t=\varepsilon^{1/(1+\eta)}s$ instead of
$t=\varepsilon s$ in [Mor16a].

  * •

The growth of the propagator as described in Lemmas 3.4 and 3.5 still depends
on a nonnegative function $\gamma_{\eta}^{\sharp}(\tau)$, and spaces
${\mathbb{E}}$ are precisely developed on such functions. The difference lies
in the final growth of time, as it will be more precise in (4.5).

######  Definition 4.1 (Fixed time spaces ${\mathbb{E}}_{s}$).

Given $\eta\geq 0$, $M^{\prime}>0$,
$s\in[0,\left(\varepsilon^{1/(1+\eta)}\rho\right)^{-1}[$, $\beta>0$, $R>0$ and
$\rho>0$, we denote
${\mathbb{E}}_{s}={\mathbb{E}}_{s}(\eta,R,\rho,M^{\prime},\beta)$ the space of
trigonometric series ${\mathbf{v}}\in F_{2d+1}$ such that for some constant
$C>0$ there holds

${\mathbf{v}}_{n}(x\,;\xi)\prec C\frac{\displaystyle c_{1}}{\displaystyle
n^{2}+1}\exp\left(\left(M^{\prime}-\int_{0}^{s}\gamma_{\eta}(\tau)d\tau\right)\left<n\right>\right)\Phi\left(RX+\varepsilon^{1/(1+\eta)}\rho
s\right)\quad\forall n\in{\mathbb{Z}}$ (4.2)

uniformly in $\xi\in B_{\varepsilon}(\xi_{0})$ and where we denote

$\gamma_{\eta}(\tau)=\gamma_{\eta}^{\sharp}(\tau)+\beta.$ (4.3)

We define a norm on ${\mathbb{E}}_{s}$ by

$\|{\mathbf{v}}\|_{s}=\inf\left\\{C>0\,|\,\quad\eqref{defi.EEs}\text{ is
satisfied }\right\\}$ (4.4)

As in the discussion following the Definition 4.7 of [Mor16a], we introduce
the growth time $\underline{s}_{1}(\eta)$ defined implicitely as

$M^{\prime}=\int_{0}^{\underline{s}_{1}(\eta)}\gamma_{\eta}(\tau)d\tau.$ (4.5)

and the final time as

$\underline{s}(\eta)=\min\left\\{\underline{s}_{1}(\eta),\left(\varepsilon^{1/(1+\eta)}\rho\right)^{-1}\right\\}.$
(4.6)

To simplify the notations, in all the following we will omit the parameters
$R$, $\rho$, $M^{\prime}$ and $\beta$ in
${\mathbb{E}}_{s}(\eta,R,\rho,M^{\prime},\beta)$.

We consider now trigonometric series

${\mathbf{u}}(s,x,\theta\,;\xi)=\sum_{n\in{\mathbb{Z}}}{\mathbf{u}}_{n}(s,x\,;\xi)e^{in\theta}$

with coefficients ${\mathbf{u}}_{n}(s,x\,;\xi)$ being formal series in $x$
whose coefficients depend smoothly on $s\in[0,\underline{s}(\eta)[$. We denote
$F_{2d+2}$ the space of all such trigonometric series:

$F_{2d+2}=\left\\{{\mathbf{u}}(s,x,\theta\,;\xi)=\sum_{n\in{\mathbb{Z}}}{\mathbf{u}}_{n}(s,x\,;\xi)e^{in\theta}\;\Big{|}\;{\mathbf{u}}_{n}(s,x\,;\xi)=\sum_{k\in{\mathbb{N}}^{d}}{\mathbf{u}}_{n,k}(s\,;\xi)x^{k}\;\text{with
}{\mathbf{u}}_{n,k}(s\,;\xi)\,C^{\infty}\text{ in }s\right\\}$

######  Definition 4.2 (Spaces ${\mathbb{E}}$).

We introduce

${\mathbb{E}}=\left\\{{\mathbf{u}}\in F_{2d+2}\,|\,\forall\,0\leq
s<\underline{s}(\eta)\,,\quad{\mathbf{u}}(s)\in{\mathbb{E}}_{s}\right\\}$
(4.7)

and the corresponding norm

$|||{\mathbf{u}}|||=\sup_{0\leq s<\underline{s}(\eta)}\|{\mathbf{u}}(s)\|_{s}$
(4.8)

Note that ${\mathbf{u}}\in{\mathbb{E}}$ is equivalent to

${\mathbf{u}}_{n}(s,x\,;\xi)\prec_{s}|||{\mathbf{u}}|||\frac{\displaystyle
c_{1}}{\displaystyle
n^{2}+1}\exp\left(\left(M^{\prime}-\int_{0}^{s}\gamma_{\eta}(\tau)d\tau\right)\left<n\right>\right)\Phi\left(RX+\varepsilon^{1/(1+\eta)}\rho
s\right)$ (4.9)

for all $n\in{\mathbb{Z}}$ and $s\in[0,\underline{s}[$ and uniformly in
$\xi\in B_{\varepsilon}(\xi_{0})$.

For ${\mathbf{u}}$ valued in $\mathbb{C}^{N}$, ${\mathbf{u}}\in{\mathbb{E}}$
means simply that each component of ${\mathbf{u}}$ is in ${\mathbb{E}}$, and
$|||{\mathbf{u}}|||$ is then the maximum of the norms of the components. We
denote the ball of ${\mathbb{E}}$ of radius $a$, centered in
${\mathbf{u}}\in{\mathbb{E}}$ by

$B_{{\mathbb{E}}}({\mathbf{u}},a)=\left\\{{\mathbf{v}}\in{\mathbb{E}}\;|\;|||{\mathbf{v}}-{\mathbf{u}}|||<a\right\\}$
(4.10)

###  4.2 Functional spaces: properties

####  4.2.1 Basic properties

We remind here basic properties of spaces ${\mathbb{E}}$. The proofs are the
same as in [Mor16a], as those properties depend only on the nonnegativity of
$\gamma_{\eta}^{\sharp}$.

######  Proposition 4.3 (Properties of ${\mathbb{E}}_{s}(\eta)$ and
${\mathbb{E}}(\eta)$.).

For any $\eta\in[0,1]$ and $s\in[0,\underline{s}(\eta)[$, there holds

  1. 1.

The space ${\mathbb{E}}_{s}(\eta)$ is an algebra, and for any ${\mathbf{v}}$
and ${\mathbf{w}}$ in ${\mathbb{E}}_{s}(\eta)$ there holds

$||{\mathbf{v}}{\mathbf{w}}||_{s}\leq||{\mathbf{v}}||_{s}\,||{\mathbf{w}}||_{s}.$
(4.11)

  2. 2.

The space ${\mathbb{E}}_{s}(\eta)$ is a Banach space.

As an immediate corollary, there holds

  1. 1.

The space ${\mathbb{E}}(\eta)$ is an algebra, and for any ${\mathbf{v}}$ and
${\mathbf{w}}$ in ${\mathbb{E}}(\eta)$ there holds

$|||{\mathbf{v}}{\mathbf{w}}|||\leq|||{\mathbf{v}}|||\,|||{\mathbf{w}}|||.$
(4.12)

  2. 2.

The space ${\mathbb{E}}(\eta)$ is a Banach space.

The action of analytic function $H(t,x,u)$ on the space ${\mathbb{E}}$,
described in the Lemma 4.13 in [Mor16a], still holds as it relies on
properties of the majoring series relation and on the definition of $\Phi$
described in Section 4.1 of [Mor16a].

######  Lemma 4.4.

Let $H(t,x,u)$ be an analytical function on a neighborhood of
$(0,0,0)\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}^{d}\times{\mathbb{R}}_{u}^{N}$.
There are constants $C_{H}>0$, $R_{H}>0$ and $\rho_{H}>0$ which depend only on
$H$ and $c_{0}$, such that for all $R\geq R_{H}$ and $\rho\geq\rho_{H}$ and
for $\varepsilon$ small enough,

$\forall\,{\mathbf{u}}\in
B_{{\mathbb{E}}(\eta,R,\rho)}(0,1):\quad|||{\mathbf{H}}({\mathbf{u}})|||\lesssim
1$ (4.13)

where ${\mathbf{H}}$ is defined by (3.3) and $|||\cdot|||$ is defined by
(4.8).

In the operators $T^{[\theta]}$, $T^{[x]}$ and $T^{[{\mathbf{u}}]}$ defined by
(3.46), (3.47) and (3.48), there appears $A\cdot\xi_{0}$, ${\rm R}^{\eta}$,
$A_{u}$ and $F$, all of which are analytic functions in the variables
$(t,x,u)\in{\mathbb{R}}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{N}$. In the
expansion formulas of both ${\rm R}^{{\rm S}}$ and ${\rm R}^{{\rm Ai}}$ there
appear also analytical functions ${\rm R}^{{\rm S}}_{t}$, ${\rm R}^{{\rm
S}}_{t}$, ${\rm R}^{{\rm Ai}}_{t}$, ${\rm R}^{{\rm Ai}}_{x}$ and ${\rm
R}^{{\rm Ai}}_{e}$ as in Lemmas 3.1 and 3.2. The previous Lemma applies:

######  Corollary 4.5.

There are constants $R_{0}$ and $\rho_{0}$ such that for all $\eta\geq 0$,
$R\geq R_{0}$, $\rho\geq\rho_{0}$ and $\varepsilon$ small enough:

$\forall\,{\mathbf{u}}\in
B_{{\mathbb{E}}(\eta,R,\rho)}(0,1):\qquad|||{\mathbf{H}}(\cdot,\cdot,{\mathbf{u}})|||\lesssim
1$ (4.14)

with $H$ equals to $A\cdot\xi_{0}$, or ${\rm R}^{\eta}$, $A_{u}$, $F$,
$r_{x}^{{\rm S}}$, $r_{t}^{{\rm S}}$, $e$, $e_{t}$ or $e_{x}$.

####  4.2.2 Action of propagators on ${\mathbb{E}}$

To describe the action of both propagators $U^{{\rm S}}$ and $U^{{\rm Ai}}$,
we define here more general smooth matrix operator
$U^{\eta}(s^{\prime},s,\theta)$ for $\eta\geq 0$ that act diagonally on
${\mathbf{u}}\in{\mathbb{E}}$ as

$\left(U^{\eta}(s^{\prime},s,\theta){\mathbf{u}}(s^{\prime})\right)_{n}=U_{n}^{\eta}(s^{\prime},s){\mathbf{u}}_{n}(s^{\prime})\quad,\quad\forall
n\in{\mathbb{Z}},\;0\leq s^{\prime}\leq s$

and satisfied the upper bound for their Fourier modes

$|U^{\eta}_{n}(s^{\prime},s)|\lesssim
C(U^{\eta})\exp\left(\int_{s^{\prime}}^{s}\gamma_{\eta}^{\sharp}(\tau)d\tau\right)\quad,\quad\forall
n\in{\mathbb{Z}},\;0\leq s^{\prime}\leq s.$ (4.15)

for some $C(U^{\eta})>0$ depending eventually on $\varepsilon$. In the Airy
case, thanks to Lemma 3.5 we have $\eta=1/2$ and

$C(U^{{\rm Ai}})=\varepsilon^{-1/3}$ (4.16)

whereas for the smooth case, thanks to Lemma 3.4 we have $\eta=1$ and

$C(U^{{\rm S}})=1.$ (4.17)

For such matrix operator $U^{\eta}$, the same result as Lemma 4.15 in [Mor16a]
still holds:

######  Lemma 4.6.

Given $\eta\geq 0$, $\beta>0$ and ${\mathbf{u}}$ in ${\mathbb{E}}(\eta,\beta)$
then

$U^{\eta}_{n}(s^{\prime},s){\mathbf{u}}_{n}(s^{\prime},x)\prec_{s^{\prime},s}C(U^{\eta})\,C_{n}^{\eta}(s^{\prime},s)\,||{\mathbf{u}}(s^{\prime})||_{s^{\prime}}\frac{\displaystyle
c_{1}}{\displaystyle
n^{2}+1}e^{\left(M^{\prime}-\int_{0}^{s}\gamma_{\eta}(\tau)d\tau\right)\left<n\right>}\Phi\left(RX+\varepsilon^{1/(1+\eta)}\rho
s\right)$ (4.18)

where $C_{n}^{\eta}(s^{\prime},s)$ is defined by

$C_{n}^{\eta}(s^{\prime},s)=\exp\left(\beta(s-s^{\prime})\left<n\right>\right)\leq
1$ (4.19)

In particular we have

$\|U^{\eta}(s^{\prime},s){\mathbf{u}}(s^{\prime})\|_{s}\leq
C(U^{\eta})\,\|{\mathbf{u}}(s^{\prime})\|_{s^{\prime}}\qquad\forall\,0\leq
s^{\prime}\leq s<\underline{s}$ (4.20)

The proof is exactly the same as in [Mor16a], as it relies only on the
definition (4.3) of $\gamma_{\eta}$. The positive constant $\beta$ acts as a
perturbation of $\gamma_{\eta}^{\sharp}$ and introduces an error term like
$e^{\beta(s-s^{\prime})|n|}$ is the growth of the $n$-th Fourier mode of the
propagator. This explains why the prefactor term $C_{n}^{\eta}(s^{\prime},s)$
is exactly the same as in Lemma 4.15 of [Mor16a].

######  Remark 4.7.

The estimate (4.20) is not precise enough to show that $T$ is a contraction in
${\mathbb{E}}$. The more precise estimate (4.18) is very important for the
estimate (4.22) below.

####  4.2.3 Norm of the free solutions

In both smooth ($\eta=1$) and Airy ($\eta=1/2$) cases, we compute the norm in
${\mathbb{E}}$ of the free solution ${\mathbf{f}}^{\eta}$ defined in Lemmas
3.7 and 3.8. The proof of this result is the same as Lemma 4.17 of [Mor16a],
using the precise estimates described in Lemmas 3.7 and 3.8.

######  Lemma 4.8 (Norm of the free solution).

For $\eta=1$ or $\eta=1/2$ and $\beta>0$, the free solution
${\mathbf{f}}^{\eta}$ satisfies

$|||{\mathbf{f}}^{\eta}|||\lesssim e^{M^{\prime}-M(\varepsilon)}$ (4.21)

######  Remark 4.9.

Note that, on the contrary of estimate (4.33) in Lemma 4.17 of [Mor16a], the
previous estimate is not $|||{\mathbf{f}}^{{\rm
Ai}}|||\lesssim\varepsilon^{-1/3}e^{M^{\prime}-M(\varepsilon)}$ in the Airy
case, thanks to the more precise estimate (3.40).

###  4.3 Estimates of remainder terms

As pointed out in Remark 3.3, our analysis differs from [Mor16a] with the
presence of extra remainder terms ${\mathbf{R}}^{\eta}$. We compute carefully
their norms.

######  Lemma 4.10 (Smooth case).

In the framework of Lemma 3.1, the norm of the remainder term
${\mathbf{R}}^{{\rm S}}$ satisfies

$|||\varepsilon^{-1/2}{\mathbf{R}}^{{\rm
S}}|||\lesssim\varepsilon^{1/2}\underline{s}^{2}+\underline{s}R^{-1}.$

###### Proof.

By expansion formula (3.9) we have

${\rm R}^{{\rm S}}=t^{2}{\rm R}^{{\rm S}}_{t}(t,x,\xi)+tx\cdot{\rm R}^{{\rm
S}}_{x}(t,x,\xi)+t(\xi-\xi_{0})\cdot{\mathbf{R}}^{{\rm S}}_{\xi}(s,x,\xi)$

and then, as $t=\varepsilon^{1/2}s$ in the smooth case and by notation (3.3),

$\varepsilon^{-1/2}{\mathbf{R}}^{{\rm
S}}(s,x,\xi)=\varepsilon^{1/2}s^{2}{\mathbf{R}}^{{\rm
S}}_{t}(s,x,\xi)+sx\cdot{\mathbf{R}}^{{\rm
S}}_{x}(s,x,\xi)+s(\xi-\xi_{0})\cdot{\mathbf{R}}^{{\rm S}}_{\xi}(s,x,\xi).$

As the norm $|||\cdot|||$ is defined by a supremum in time, first there holds

$|||\varepsilon^{1/2}s^{2}{\mathbf{R}}^{{\rm
S}}_{t}|||\lesssim\varepsilon^{1/2}\underline{s}^{2}|||{\mathbf{R}}^{{\rm
S}}_{t}|||$

and

$|||\varepsilon^{1/2}s(\xi-\xi_{0})\cdot{\mathbf{R}}^{{\rm
S}}_{\xi}|||\lesssim\varepsilon^{1/2}\underline{s}\varepsilon|||{\mathbf{R}}^{{\rm
S}}_{\xi}|||.$

To get a precise estimate of the term $x\cdot{\mathbf{R}}^{{\rm S}}_{x}(s,x)$,
we first note that the coefficients $\Phi_{k}(t)$ defined by (4.1) satisfy

$1\leq R^{-1}\Phi_{k}(t)\quad,\quad\forall k\in{\mathbb{N}}^{d}\text{ with
}|k|=1$

so that for all $j=1,\ldots,d$

$X_{j}\prec R^{-1}\Phi(RX+\varepsilon^{1/2}\rho s)\quad,\quad\forall 0\leq
s<\underline{s}.$

By inequality (4.12) in Proposition 4.3 we get

$|||x\cdot{\mathbf{R}}^{{\rm S}}_{x}(s,x)|||\lesssim
R^{-1}|||{\mathbf{R}}^{{\rm S}}|||.$

As $C(U^{{\rm S}})=1$ by (4.17), this ends the proof. ∎

######  Lemma 4.11 (Airy case).

In the framework of Lemma 3.2, the norm of the remainder term
${\mathbf{R}}^{{\rm Ai}}$ satisfies

$|||\varepsilon^{-1/3}{\mathbf{R}}^{{\rm
Ai}}|||\lesssim\varepsilon\underline{s}^{2}+\varepsilon^{1/3}\underline{s}R^{-1}+\varepsilon^{-1/3}t_{\star}(R^{-1}).$

###### Proof.

The proof is the same as the previous one, with the differences that
$\eta=1/2$ and $C(U^{{\rm Ai}})=\varepsilon^{-1/3}$. ∎

###  4.4 Contraction estimates

####  4.4.1 Regularization results

A crucial observation is that derivation operators $\partial_{\theta}$ and
$\partial_{x_{j}}$ are not bounded operators in spaces ${\mathbb{E}}$, as
explained in Section 5.1 in [Mor16a]. The main results in our previous paper
are the description of the regularization effect of integration in time of
derivation operators. These results are precised in Sections 5.2 through 5.4
in [Mor16a].

Those results still hold in our setting. We omit the proof of the following
Lemmas, but give some indications on how to adapt the proofs of [Mor16a] of
the similar results.

######  Proposition 4.12 (Regularization of $\partial_{\theta}$).

For operator $T^{[\eta,\theta]}$ defined by (3.46), for any ${\mathbf{u}}\in
B_{{\mathbb{E}}(\eta,R,\rho)}(0,1)$ and for $\beta>0$, there holds

$|||T^{[\eta,\theta]}({\mathbf{u}})|||\lesssim
C(U^{\eta})\beta^{-1}\left(\varepsilon^{-\eta/(1+\eta)}|||{\mathbf{R}}^{\eta}|||+\varepsilon^{(2-\eta)/(1+\eta)}|||{\mathbf{u}}|||\right)\,|||{\mathbf{u}}|||$
(4.22)

The proof in the same as Proposition 5.4 in [Mor16a], as it is based on Lemma
4.6 and the expression of prefactor $C_{n}^{\eta}$ defined in (4.19) which is
the same as prefactor (4.31) in Lemma 4.15 in [Mor16a].

As remainder terms ${\mathbf{R}}^{\eta}$ have different norms in spaces
${\mathbb{E}}$, given by Lemmas 4.10 and 4.11 , we give more precisely the
following two results:

######  Corollary 4.13 (Smooth case).

In the smooth case, thanks to Lemma 4.10, there holds

$|||T^{[{\rm
S},\theta]}({\mathbf{u}})|||\lesssim\beta^{-1}\left(\varepsilon^{1/2}\underline{s}^{2}+\underline{s}R^{-1}+\varepsilon^{1/2}|||{\mathbf{u}}|||\right)|||{\mathbf{u}}|||.$
(4.23)

######  Corollary 4.14 (Airy case).

In the smooth case, thanks to Lemma 4.11, there holds

$|||T^{[{\rm
Ai},\theta]}({\mathbf{u}})|||\lesssim\varepsilon^{-1/3}\beta^{-1}\left(\varepsilon\underline{s}^{2}+\varepsilon^{1/3}\underline{s}R^{-1}+\varepsilon^{-1/3}t_{\star}(R^{-1})+\varepsilon\right)|||{\mathbf{u}}|||.$
(4.24)

About the regularization of derivation operators $\partial_{x_{j}}$, the proof
relies again on the simple computation given in Section 1.2.2 of [Mor16a]. The
difference in the time renormalization $\Phi(Rx+\varepsilon^{1/(1+\eta)}\rho
s)$, instead of $\Phi(RX+\varepsilon\rho s)$ in [Mor16a], is a minor one.

######  Proposition 4.15 (Regularization of $\partial_{x_{j}}$).

For operator $T^{[\eta,x]}$ defined by (3.47) and any ${\mathbf{u}}\in
B_{{\mathbb{E}}(\eta,R,\rho)}(0,1)$, there holds

$|||T^{[\eta,x]}({\mathbf{u}})|||\lesssim
C(U^{\eta})\,R\rho^{-1}\,|||{\mathbf{u}}|||.$ (4.25)

As ${\mathbb{E}}$ is an algebra the operator $T^{[\eta,{\mathbf{u}}]}$ is
directly bounded, with no need of a regularization by time result, on the
contrary of operators $T^{[\eta,\theta]}$ and $T^{[\eta,x]}$. The following
proposition gives us precisely

######  Proposition 4.16 (Nonlinear term).

For the operator $T^{[\eta,{\mathbf{u}}]}$ defined by (3.48), for any
${\mathbf{u}}\in B_{{\mathbb{E}}}(0,1)$ and $\beta>0$ there holds

$|||T^{[\eta,{\mathbf{u}}]}({\mathbf{u}})|||\lesssim
C(U^{\eta})\,\beta^{-1}\varepsilon^{1/(1+\eta)}|||{\mathbf{u}}|||^{2}.$ (4.26)

####  4.4.2 Contraction estimates

Thanks to the results of the previous Section, we prove estimates for operator
$T^{\eta}$ defined in (3.43), as in Section 5.5 of [Mor16a]. We omit once
again the proof of this result, as it is the same as Proposition 5.9 in
[Mor16a].

######  Proposition 4.17 (Contraction estimates in ${\mathbb{E}}$).

There are $R_{0}$, $\rho_{0}>0$ such that for all $\beta>0$, $R\geq R_{0}$,
$\rho>\rho_{0}$ and $\varepsilon\in(0,1)$, we get the following estimates for
all ${\mathbf{u}}$ and ${\mathbf{v}}$ in $B_{{\mathbb{E}}}(0,1)$:

$\displaystyle|||T^{\eta}({\mathbf{u}})|||\lesssim
C(U^{\eta})\,\left(\beta^{-1}\left(\varepsilon^{-\eta/(1+\eta)}|||{\mathbf{R}}^{\eta}|||+\varepsilon^{1/(1+\eta)}|||{\mathbf{u}}|||\right)+R\rho^{-1}\right)|||{\mathbf{u}}|||$
(4.27)
$\displaystyle|||T^{\eta}({\mathbf{u}})-T^{\eta}({\mathbf{v}})|||\lesssim
C(U^{\eta})\,\left(\beta^{-1}\left(\varepsilon^{-\eta/(1+\eta)}|||{\mathbf{R}}^{\eta}|||+\varepsilon^{1/(1+\eta)}|||{\mathbf{u}}|||\right)+R\rho^{-1}\right)|||{\mathbf{u}}-{\mathbf{v}}|||$
(4.28)

For convenience we introduce

$K^{\eta}(\varepsilon)=C(U^{\eta})\,\left(\beta^{-1}\left(\varepsilon^{-\eta/(1+\eta)}|||{\mathbf{R}}^{\eta}|||+\varepsilon^{1/(1+\eta)}|||{\mathbf{f}}^{\eta}|||\right)+R\rho^{-1}\right)$
(4.29)

##  5 Estimates from below and Hadamard instability

###  5.1 Existence of solutions

Thanks to the Proposition 4.17, we can now solve the fixed point equation
(3.42) in the ball
$B_{{\mathbb{E}}(\eta)}\left(0,|||{\mathbf{f}}^{\eta}|||\right)$:

######  Corollary 5.1 (Contraction and fixed point in ${\mathbb{E}}$).

Let $\eta>0$ be fixed. Let $R(\varepsilon)>R_{0}$,
$\rho(\varepsilon)>\rho_{0}$, $\beta(\varepsilon)>0$ and $\underline{s}(\eta)$
be such that

$\lim_{\varepsilon\to
0}C(U^{\eta})\,\left(\beta^{-1}\left(\varepsilon^{-\eta/(1+\eta)}|||{\mathbf{R}}^{\eta}|||+\varepsilon^{1/(1+\eta)}|||{\mathbf{f}}^{\eta}|||\right)+R\rho^{-1}\right)=0$
(5.1)

Let assume that the propagator $U^{\eta}$ satisfy the growth (4.15). Then for
$\varepsilon$ small enough, the fixed point equation (3.42), with
${\mathbf{f}}^{\eta}$ defined by (3.32), has a unique solution ${\mathbf{u}}$
in
$B_{{\mathbb{E}}(\eta,R,\rho,\gamma)}\left(0,2|||{\mathbf{f}}^{\eta}|||\right)$.
This solution satisfies

$|||{\mathbf{u}}-{\mathbf{f}}^{\eta}|||\lesssim
K^{\eta}(\varepsilon)|||{\mathbf{f}}^{\eta}|||.$ (5.2)

The proof of the Corollary is straigthforward using the estimates of
Proposition 4.17, under the condition of smallness for $K^{\eta}(\varepsilon)$
given by (5.1). This Corollary is in some sense an abstract result, as it
deals with abstract propagator $U^{\eta}$ with specific growth (4.15),
described at the beginning of Section 4.2.2. We emphazise that $U^{\eta}$,
except for both smooth and Airy cases, have not be proved to exist. Corollary
5.1 gives a result on fixed point equations (3.42), independently of the
initial Cauchy equation.

###  5.2 Bounds from below

From now on we focus on both smooth and Airy cases, for which we have proved
the existence and the growth of the propagators (see Lemmas 3.4 and 3.5) and
the actual growth of the special free solution (see Lemmas 3.7 and 3.8).

We follow here Section 6.2 of [Mor16a]. We aim to prove that, in the smooth
case, the solutions have the same growth as ${\mathbf{f}}^{{\rm S}}$ given in
Lemma 3.7, that is

$|{\mathbf{u}}(s,x,\theta\,;\varepsilon,\xi)|\gtrsim
e^{-M(\varepsilon)}\exp\left(\int_{0}^{s}\gamma_{{\rm
S}}^{\flat}(\tau)d\tau\right)\quad,\quad\forall\,(s,x,\theta)\in\Omega_{R,\varepsilon^{1/(1+\eta)}\rho}(0)\times\mathbb{T}$
(5.3)

with $\Omega_{R,\varepsilon^{1/(1+\eta)}\rho}(0)$ defined by (2.22). In the
Airy case, thanks to Lemma 3.8, we aim to prove

$|{\mathbf{u}}(s,x,\theta\,;\varepsilon,\xi)|\gtrsim
s^{-1/4}e^{-M(\varepsilon)}\exp\left(\int_{0}^{s}\gamma_{{\rm
Ai}}^{\flat}(\tau)d\tau\right)\quad,\quad\forall\,(s,x,\theta)\in\Omega_{R,\varepsilon^{1/(1+\eta)}\rho}(0)\times\mathbb{T}.$
(5.4)

As in [Mor16a], by some computations we prove the pointwise inequality

$|({\mathbf{u}}-{\mathbf{f}}^{\eta})(s,x,\theta\,;\varepsilon,\xi))|\lesssim
K(\varepsilon)\,C(U^{{\rm\eta}})e^{-M(\varepsilon)}\exp\left(\int_{0}^{s}\gamma_{\eta}(\tau)d\tau\right)$
(5.5)

which is inequality (6.5) of [Mor16a], holding for all
$(s,x,\theta)\in\Omega_{R,\varepsilon^{1/(1+\eta)}\rho}\times\mathbb{T}$ and
$\xi\in B_{\varepsilon}(\xi_{0})$. Next, by definition (4.3) of
$\gamma_{\eta}$, inequality (5.5) becomes

$|({\mathbf{u}}-{\mathbf{f}}^{\eta})(s,x,\theta\,;\varepsilon,\xi))|\lesssim
K(\varepsilon)\,C(U^{{\rm\eta}})\exp\left(\underline{s}\beta+\int_{0}^{s}\left(\gamma_{\eta}^{\sharp}(\tau)-\gamma_{\eta}^{\flat}(\tau)\right)d\tau\right)e^{-M(\varepsilon)}\exp\left(\int_{0}^{s}\gamma_{\eta}^{\flat}(\tau)d\tau\right).$

As $\gamma_{\eta}^{\flat}=\gamma_{\eta}^{\sharp}$ (see Remark 3.9), we finally
get

$|({\mathbf{u}}-{\mathbf{f}}^{\eta})(s,x,\theta\,;\varepsilon,\xi))|\lesssim
K(\varepsilon)\,C(U^{{\rm\eta}})\,e^{\underline{s}\beta}\,e^{-M(\varepsilon)}\exp\left(\int_{0}^{s}\gamma_{\eta}^{\flat}(\tau)d\tau\right).$
(5.6)

Then in order to get (5.3) or (5.4) thanks to (5.6), limit (5.1) is not
sufficient as the term $e^{\underline{s}\beta}$ could be large as
$\varepsilon$ tends to $0$, as explained in Section 6.2 in [Mor16a]. In both
cases, we have then a stronger constraint on parameters $R$, $\rho$, $\beta$
and $M^{\prime}$:

  * •

In the smooth case, thanks to Corollary 4.13 and (5.3), we need

$\lim_{\varepsilon\to
0}\,\left(\beta^{-1}\,\left(\varepsilon^{1/2}\underline{s}^{2}+\underline{s}R^{-1}+\varepsilon^{1/2}e^{M^{\prime}-M}\right)+R\rho^{-1}\right)e^{\underline{s}\beta}=0.$
(5.7)

  * •

In the Airy case, thanks to Corollary 4.14 and (5.4), we need

$\lim_{\varepsilon\to
0}\,\varepsilon^{-1/3}\underline{s}^{1/4}\,\left(\beta^{-1}\,\left(\varepsilon\underline{s}^{2}+\varepsilon^{1/3}\underline{s}R^{-1}+\varepsilon^{-1/3}t_{\star}(R^{-1})+\varepsilon^{2/3}e^{M^{\prime}-M}\right)+R\rho^{-1}\right)e^{\underline{s}\beta}=0.$
(5.8)

The second constraint on the parameters comes from the competition between the
characteristic growth time $\underline{s}_{1}(\eta)$ defined in (4.5) and the
regularity time $(\varepsilon^{1/(1+\eta)}\rho)^{-1}$. To see the growth of
the solution, we need it to exist on a sufficiently large time compared to the
growth time, that is we need $\underline{s}$ to be $\underline{s}_{1}(\eta)$.

As $M^{\prime}$ is large in the limit $\varepsilon\to 0$, the implicit
definition (4.5) of $\underline{s}_{1}(\eta)$ and definition (3.19) of
$\gamma_{{\rm S}}^{\sharp}$ and definition (3.27) of $\gamma_{{\rm
Ai}}^{\sharp}$ lead to the equivalent

$\underline{s}_{1}(\eta)\approx M^{\prime 1/(1+\eta)}$ (5.9)

for $\eta=1$ and $\eta=1/2$. Hence the following constraints:

  * •

In the smooth case

$\lim_{\varepsilon\to 0}M^{\prime 1/2}\varepsilon^{1/2}\rho=0.$ (5.10)

  * •

In the Airy case

$\lim_{\varepsilon\to 0}M^{\prime 2/3}\varepsilon^{2/3}\rho=0.$ (5.11)

We focus now on both cases separately from now on, even if the way we find
suitable $R$, $\rho$, $\beta$ and $M^{\prime}$ which would satisfy both
constraints is very similar for the two cases. We sum up all of this in the
two following Propositions.

######  Proposition 5.2 (Estimate from below: smooth case).

With the limitation of the Gevrey index

$\sigma<\delta<1/3,$ (5.12)

both constraints (5.7) and (5.10) are satisfied. Then the fixed point equation
(3.42) has a unique solution ${\mathbf{u}}$ in ${\mathbb{E}}$ and

$|{\mathbf{u}}(s,x,\theta)|\gtrsim
e^{-M(\varepsilon)}e^{\gamma_{0}\frac{1}{2}s^{2}}\quad,\quad\forall\,(s,x,\theta)\in\Omega_{R,\varepsilon^{1/2}\rho}(0)\times\mathbb{T}.$
(5.13)

There holds also

$\underline{s}\approx\varepsilon^{-\delta/2}.$ (5.14)

###### Proof.

As in [Mor16a], we use notation $\ll$ defined in (1.32) to rewrite all the
constraints in a more useful way. Constraints constraints (5.7) and (5.10) are
equivalent to

$\displaystyle\beta^{-1}\varepsilon^{1/2}\varepsilon^{-\delta}\,e^{\beta\varepsilon^{-\delta/2}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.15)
$\displaystyle\beta^{-1}\varepsilon^{-\delta/2}R^{-1}\,e^{\beta\varepsilon^{-\delta/2}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.16)
$\displaystyle\beta^{-1}\varepsilon^{1/2}e^{M^{\prime}-M}\,e^{\beta\varepsilon^{-\delta/2}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.17) $\displaystyle
R\rho^{-1}\,e^{\beta\varepsilon^{-\delta/2}}$ $\displaystyle\ll$
$\displaystyle 1$ (5.18)
$\displaystyle\varepsilon^{-\delta/2}\varepsilon^{1/2}\rho$ $\displaystyle\ll$
$\displaystyle 1.$ (5.19)

This implies first, as in [Mor16a], that $\beta$ as to be of size
$\varepsilon^{\delta/2}$. We then posit

$\beta=\varepsilon^{\delta/2}.$

We have then

$\displaystyle\varepsilon^{1/2}\varepsilon^{-3\delta/2}$ $\displaystyle\ll$
$\displaystyle 1$ (5.20) $\displaystyle\varepsilon^{-\delta}R^{-1}$
$\displaystyle\ll$ $\displaystyle 1$ (5.21)
$\displaystyle\varepsilon^{1/2-\delta/2}e^{M^{\prime}-M}$ $\displaystyle\ll$
$\displaystyle 1$ (5.22) $\displaystyle R\rho^{-1}$ $\displaystyle\ll$
$\displaystyle 1$ (5.23) $\displaystyle\varepsilon^{1/2-\delta/2}\rho$
$\displaystyle\ll$ $\displaystyle 1.$ (5.24)

Asymptotic inequality (5.20) is equivalent to the limitation $\delta<1/3$ on
the Gevrey index. Next, as $\delta<1$, asymptotic inequality (5.22) is
satisfies as soon as $M^{\prime}=M-|\ln(\varepsilon)|$.

Finally, inequalities (5.21), (5.23) and (5.24) are equivalent to

$\varepsilon^{1/2-\delta/2}\ll\rho^{-1}\ll R^{-1}\ll\varepsilon^{\delta}.$
(5.25)

This chain of asymptotic inequalities is satisfies as soon as
$\varepsilon^{1/2-\delta/2}\ll\varepsilon^{\delta}$, which is equivalent again
to the limitation $\delta<1/3$ of the Gevrey index. Then the choice
$R^{-1}=\varepsilon^{1/6+\delta/2}$ and $\rho^{-1}=\varepsilon^{1/3}$
satisfies the constraints.

∎

######  Remark 5.3.

Note that in the case where ${\rm R}^{{\rm S}}\equiv 0$, both (5.20) and
(5.21) disappear hence the limitation $\delta<1$ in place of $\delta<1/3$.

######  Proposition 5.4 (Estimate from below: Airy case).

With the limitation of the Gevrey index

$\sigma<\delta<2/13$ (5.26)

both constraints (5.8) and (5.11) are satisfied. Then the fixed point equation
(3.42) has a unique solution ${\mathbf{u}}$ in ${\mathbb{E}}$ and

$|{\mathbf{u}}(s,x,\theta)|\gtrsim
e^{-M(\varepsilon)}e^{\gamma_{0}\frac{2}{3}s^{3/2}}\quad,\quad\forall\,(s,x,\theta)\in\Omega_{R,\varepsilon^{2/3}\rho}(0)\times\mathbb{T}.$
(5.27)

There holds also

$\underline{s}\approx\varepsilon^{-2\delta/3}.$ (5.28)

###### Proof.

We follow here the same proof as the one of Proposition 5.2. Two differences
appear: the extra weight $\varepsilon^{-1/3}$ coming from the specificity of
the Airy propagator in Lemma 3.5 and the extra $t_{\star}(R^{-1})$ in the
remainder term of Corollary 4.14.

Taking those both differences into account, the following constraints hold

$\displaystyle\beta^{-1}\varepsilon^{2/3}\varepsilon^{-3\delta/2}\,e^{\beta\varepsilon^{-2\delta/3}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.29)
$\displaystyle\beta^{-1}\varepsilon^{-5\delta/6}R^{-1}\,e^{\beta\varepsilon^{-2\delta/3}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.30)
$\displaystyle\beta^{-1}\varepsilon^{-\delta/6}\varepsilon^{-2/3}t_{\star}(R^{-1})\,e^{\beta\varepsilon^{-2\delta/3}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.31)
$\displaystyle\beta^{-1}\varepsilon^{-\delta/6}\varepsilon^{1/3}e^{M^{\prime}-M}\,e^{\beta\varepsilon^{-2\delta/3}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.32)
$\displaystyle\varepsilon^{-1/3}\varepsilon^{-\delta/6}R\rho^{-1}\,e^{\beta\varepsilon^{-2\delta/3}}$
$\displaystyle\ll$ $\displaystyle 1$ (5.33)
$\displaystyle\varepsilon^{-2\delta/3}\varepsilon^{2/3}\rho$
$\displaystyle\ll$ $\displaystyle 1$ (5.34)

where there holds $\underline{s}\approx\varepsilon^{-2\delta/3}$. Again, those
asymptotic inequalities imply that $\beta=\varepsilon^{2\delta/3}$ and
inequality (5.29) is replaced by

$\varepsilon^{2/3-13\delta/6}\ll 1$

which gives the limitation $\delta<4/13$ on the Gevrey index.

To find now $R$ and $\rho$, we first use (5.30) and (5.33) to get

$\varepsilon^{2/3-2\delta/3}\ll\rho^{-1}\ll\varepsilon^{1/3+\delta/6}R^{-1}.$
(5.35)

For an asymptotic upper bound for $R^{-1}$, we have in this case two
possibilities, thanks to (5.30) and (5.31). If we assume here that $t_{\star}$
is of order $k\geq 2$ in $x$, these two inequalities are equivalent to

$R^{-1}\ll\varepsilon^{3\delta/2}$ (5.36)

and

$R^{-1}\ll\varepsilon^{\frac{1}{k}(2/3+5\delta/6)}.$ (5.37)

The question is which one of (5.36) or (5.37) is a stronger constraint on $R$
and $\rho$. By simple computations, we prove

$3\delta/2<\frac{1}{k}(2/3+5\delta/6)\quad\Longleftrightarrow\quad\delta<\frac{1}{9k/4-5/4}.$

We are then reduced to study two cases:

  * •

If $\delta<\frac{1}{9k/4-5/4}$, then
$R^{-1}\ll\varepsilon^{\frac{1}{k}(2/3+5\delta/6)}\ll\varepsilon^{3\delta/2}$.
With (5.35), we get the constraint
$\varepsilon^{2/3-2\delta/3}\ll\varepsilon^{1/3+\frac{1}{k}(2/3+5\delta/6)}$,
which is equivalent to $\delta<\frac{1/2-1/k}{1+1/k}$. We note in particular
that the non degenerate Airy case $k=2$ is out of reach with our method. In
the degenerate case $k=4$, we have the limitation
$\delta<\frac{1/2-1/k}{1+1/k}=4/21$, compatible with
$\delta<\frac{1}{9k/4-5/4}=4/31$.

  * •

If $\delta>\frac{1}{9k/4-5/4}$, then
$R^{-1}\ll\varepsilon^{3\delta/2}\ll\varepsilon^{\frac{1}{k}(2/3+5\delta/6)}$.
With (5.35), we get the constraint
$\varepsilon^{2/3-2\delta/3}\ll\varepsilon^{1/3+3\delta/2}$, which is
equivalent to $\delta<2/13$. It is incompatible with
$\delta>\frac{1}{9k/4-5/4}$ when $k=2$.

In each of the previous cases, the case $k=2$ leads to a contradiction, hence
proving that the non-degenerate Airy transition is out of reach of our method.
In the degenerate case $k=4$, the previous analysis shows that the limiting
Gevrey index is $2/13$.

∎

######  Remark 5.5.

On the contrary of the smooth case and Remark 5.3, the limitation
$\delta<2/13$ still holds when ${\rm R}^{{\rm Ai}}\equiv 0$. This can be
explained as $t_{\star}$ represents the transition time from hyperbolicity to
ellipticity, and the domain of hyperbolicity is too large to be considered as
an elliptic region.

###  5.3 Conclusion: Hadamard instability in Gevrey spaces

To close the proofs of Theorem 2.11 and Theorem 2.12 we have now to get an
estimate of the ratio

$\frac{\displaystyle||u_{\varepsilon}||_{L^{2}(\Omega_{R,\rho}(0))}}{\displaystyle||h_{\varepsilon}^{\eta}||^{\alpha}_{\sigma,c,K}}.$

The previous Sections show the existence, in either the smooth or the Airy
case, of a family of solutions ${\mathbf{u}}$ starting from
${\mathbf{f}}^{\eta}$ of the fixed point equation (3.42). Thanks to the ansatz
(3.2) which we recall here

$u_{\varepsilon}(t,x)=\varepsilon^{2/(1+\eta)}{\mathbf{u}}(\varepsilon^{-1/(1+\eta)}\,t,x,x\cdot\xi_{0}/\varepsilon)$

we have then a family of solutions $u_{\varepsilon}$ existing in domains
$\Omega_{R,\rho}(0)$, for some well-chosen parameters described in the proof
of Proposition 5.2 or 5.4. In both cases we can verify that domains
$\Omega_{R,\rho}(0)$ contain the cube of size $\varepsilon$

$C_{\varepsilon}=\\{(t,x)\,|\,\underline{t}-\varepsilon<t<\underline{t},\quad|x|<\varepsilon\\}$

where we denote simply $\underline{t}=\varepsilon^{1/(1+\eta)}\underline{s}$.
The conclusion of the proof of Theorems 2.11 and 2.12 is the same as in
Section 7 in [Mor16a].

##  6 Appendix: on the Airy equation

The purpose of this Appendix is to bring some crucial elements on the Airy
equation, and to complete the proofs of Lemma 3.5 and 3.8. We recall here
equation (3.28), with $\eta=1/2$, that appears on Lemma 3.5:

$\partial_{s}\widetilde{V}_{n}(s^{\prime},s)-in\left(\begin{array}[]{cc}0&\varepsilon^{-1/3}\\\
-\varepsilon^{1/3}\gamma_{0}^{\,2}s&0\end{array}\right)\widetilde{V}_{n}(s^{\prime},s)=0\qquad\widetilde{V}_{n}(s^{\prime},s^{\prime})={\rm
Id}.$ (6.1)

The aim here is to get upper bounds for the matrix flow
$\widetilde{V}_{n}(s^{\prime},s)$ for all $0\leq s^{\prime}\leq s$ and all
$n\in\mathbb{Z}$, and hence to complete the proof of Lemma 3.5. For simplicity
we denote

$\widetilde{V}_{n}(s^{\prime},s)=\left(\begin{array}[]{cc}\widetilde{V}_{n,1,1}&\widetilde{V}_{n,1,2}\\\
\widetilde{V}_{n,2,1}&\widetilde{V}_{n,2,2}\end{array}\right)\qquad\widetilde{V}_{n,p,q}(s^{\prime},s^{\prime})=\delta(p,q).$
(6.2)

###  6.1 Reduction to the scalar Airy equation and resolution

The vector equation (6.1) becomes the system of scalar equations

$\left\\{\begin{array}[]{cclcc}\partial_{s}\widetilde{V}_{n,1,1}&=&\phantom{{}-{}}in\varepsilon^{-1/3}\,\widetilde{V}_{n,2,1}&&\widetilde{V}_{n,1,1}(s^{\prime},s^{\prime})=1\\\
\partial_{s}\widetilde{V}_{n,1,2}&=&\phantom{{}-{}}in\varepsilon^{-1/3}\,\widetilde{V}_{n,2,2}&&\widetilde{V}_{n,1,2}(s^{\prime},s^{\prime})=0\\\\[8.53581pt]
\partial_{s}\widetilde{V}_{n,2,1}&=&{}-in\varepsilon^{1/3}\gamma_{0}^{\,2}s\,\widetilde{V}_{n,1,1}&&\widetilde{V}_{n,2,1}(s^{\prime},s^{\prime})=0\\\
\partial_{s}\widetilde{V}_{n,2,2}&=&{}-in\varepsilon^{1/3}\gamma_{0}^{\,2}s\,\widetilde{V}_{n,1,2}&&\widetilde{V}_{n,2,2}(s^{\prime},s^{\prime})=1\end{array}\right.$
(6.3)

Differentiating the first equation and using next the third one, the entry
$\widetilde{V}_{n,1,1}$ solves the second order scalar differential equation

$\partial_{s}^{2}\widetilde{V}_{n,1,1}(s^{\prime},s)=(n\gamma_{0})^{2}\,s\,\widetilde{V}_{n,1,1}(s^{\prime},s)$
(6.4)

with the initial condition for $\widetilde{V}_{n,1,1}$:

$\widetilde{V}_{n,1,1}(s^{\prime},s^{\prime})=1.$ (6.5)

The initial condition for $\partial_{s}\widetilde{V}_{n,1,1}$ comes from the
first equation of the system (6.3) and the initial condition for
$\widetilde{V}_{n,2,1}$, as we have

$\displaystyle\partial_{s}\widetilde{V}_{n,1,1}(s^{\prime},s^{\prime})$
$\displaystyle=$ $\displaystyle
in\varepsilon^{-1/3}\,\widetilde{V}_{n,2,1}(s^{\prime},s^{\prime})$ (6.6)
$\displaystyle=$ $\displaystyle 0.$

Note also that we can retrieve $\widetilde{V}_{n,2,1}$ thanks to the first
line of (6.3), as

$\widetilde{V}_{n,2,1}(s^{\prime},s)=\frac{\varepsilon^{1/3}}{in}\,\partial_{s}\widetilde{V}_{n,1,1}(s^{\prime},s)$
(6.7)

Doing the same for the second and fourth equations, we obtain the same second
order scalar differential equation for $\widetilde{V}_{n,1,2}$

$\partial_{s}^{2}\widetilde{V}_{n,1,2}(s^{\prime},s)=(n\gamma_{0})^{2}\,s\,\widetilde{V}_{n,1,2}(s^{\prime},s)$
(6.8)

with initial conditions

$\widetilde{V}_{n,1,2}(s^{\prime},s^{\prime})=0$ (6.9)

and

$\partial_{s}\widetilde{V}_{n,1,2}(s^{\prime},s^{\prime})=in\varepsilon^{-1/3}.$
(6.10)

We have also the relation

$\widetilde{V}_{n,2,2}(s^{\prime},s)=\frac{\varepsilon^{1/3}}{in}\,\partial_{s}\widetilde{V}_{n,1,2}(s^{\prime},s).$
(6.11)

Equations (6.4) and (6.8) are exactly the $\varepsilon$-independent scalar
Airy equation

$y_{n}^{\prime\prime}(s)=(|n|\gamma_{0})^{2}\,s\,y_{n}(s)$ (6.12)

which is a second-order scalar differential equation. The solutions of (6.12)
are given by the following

######  Lemma 6.1 (Scalar Airy equation).

For $n\in\mathbb{Z}^{*}$, let ${\rm Ai_{n}}(z)$ be

${\rm Ai_{n}}(z)=(2\pi)^{-1}\int_{{\rm
Im}(\zeta)=a}\exp\left((|n|\gamma_{0})(i\zeta^{3}/3+i\zeta z)\right)d\zeta$
(6.13)

for $a>0$. Then for all $n\in{\mathbb{Z}}$, ${\rm Ai_{n}}$ is a holomorphic
function in ${\mathbb{C}}$ independent of $a$, and the couple $({\rm
Ai_{n}}(\cdot),{\rm Ai_{n}}(j\cdot))$ is a basis of solutions of (6.12), with
$j=e^{2i\pi/3}$.

To prove this Lemma, it suffices to adapt the proof following Definition 7.6.8
in [Hör83].

As $({\rm Ai_{n}}(\cdot),{\rm Ai_{n}}(j\cdot))$ is a basis of solutions of
equation (6.12), and as both entries $\widetilde{V}_{n,1,1}(s^{\prime},s)$ and
$\widetilde{V}_{n,1,2}(s^{\prime},s)$ solve equations (6.4) and (6.8), there
are $(\alpha_{1}(s^{\prime}),\beta_{1}(s^{\prime}))\in{\mathbb{R}}^{2}$ and
$(\alpha_{2}(s^{\prime}),\beta_{2}(s^{\prime}))\in{\mathbb{R}}^{2}$ such that

$\displaystyle\widetilde{V}_{n,1,1}(s^{\prime},s)$ $\displaystyle=$
$\displaystyle\alpha_{1}(s^{\prime}){\rm Ai_{n}}(s)+\beta_{1}(s^{\prime}){\rm
Ai_{n}}(js)$ (6.14) $\displaystyle\widetilde{V}_{n,1,2}(s^{\prime},s)$
$\displaystyle=$ $\displaystyle\alpha_{2}(s^{\prime}){\rm
Ai_{n}}(s)+\beta_{2}(s^{\prime}){\rm Ai_{n}}(js)$ (6.15)

By the relations (6.7) and (6.11), there holds also

$\displaystyle\widetilde{V}_{n,2,1}(s^{\prime},s)$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}\left(\alpha_{1}(s^{\prime}){\rm
Ai_{n}}^{\prime}(s)+j\beta_{1}(s^{\prime}){\rm Ai_{n}}^{\prime}(js)\right)$
(6.16) $\displaystyle\widetilde{V}_{n,2,2}(s^{\prime},s)$ $\displaystyle=$
$\displaystyle\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}\left(\alpha_{2}(s^{\prime}){\rm
Ai_{n}}^{\prime}(s)+j\beta_{2}(s^{\prime}){\rm Ai_{n}}^{\prime}(js)\right).$
(6.17)

This is equivalent to say that both vectors

$\begin{pmatrix}{\rm Ai_{n}}(s)\\\ -in^{-1}\varepsilon^{1/3}{\rm
Ai_{n}}^{\prime}(s)\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}{\rm
Ai_{n}}(js)\\\ -in^{-1}\varepsilon^{1/3}j{\rm
Ai_{n}}^{\prime}(js)\end{pmatrix}$ (6.18)

forms a basis of solutions of the system (6.3). The functions
$(\alpha_{k}(s^{\prime}),\beta_{k}(s^{\prime}))$ are determined by the initial
conditions (6.5) and (6.6) for $k=1$ and (6.9) and (6.10) for $k=2$. We obtain
the matrix representation of the
$(\alpha_{k}(s^{\prime}),\beta_{k}(s^{\prime}))$:

$\left(\begin{array}[]{cc}\alpha_{1}(s^{\prime})&\alpha_{2}(s^{\prime})\\\\[8.53581pt]
\beta_{1}(s^{\prime})&\beta_{2}(s^{\prime})\end{array}\right)=\frac{\displaystyle
1}{\displaystyle
D_{n}(s^{\prime})}\left(\begin{array}[]{cc}\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}j{\rm Ai_{n}}^{\prime}(js^{\prime})&-{\rm
Ai_{n}}(js^{\prime})\\\\[8.53581pt]
-\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle in}{\rm
Ai_{n}}^{\prime}(s^{\prime})&{\rm Ai_{n}}(s^{\prime})\end{array}\right)$
(6.19)

where $D_{n}(s^{\prime})$ is the determinant of the basis (6.18), that is

$D_{n}(s^{\prime}):=\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}{\rm Ai_{n}}(s^{\prime})j{\rm
Ai_{n}}^{\prime}(js^{\prime})-\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}{\rm Ai_{n}}(js^{\prime}){\rm Ai_{n}}^{\prime}(s^{\prime})$

which is in fact independent of $s^{\prime}$:

$D_{n}(s^{\prime})\equiv
D_{n}(0)=\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle in}(j-1){\rm
Ai_{n}}(0){\rm Ai_{n}}^{\prime}(0).$

For simplicity we denote

$C=\left((j-1){\rm Ai_{n}}(0){\rm Ai_{n}}^{\prime}(0)\right)^{-1}.$ (6.20)

Putting altogether equalities (6.14) to (6.19), we obtain

$\left\\{\begin{array}[]{ccl}\widetilde{V}_{n,1,1}(s^{\prime},s)&=&\quad\quad\,\,C\Big{(}j{\rm
Ai_{n}}^{\prime}(js^{\prime}){\rm Ai_{n}}(s)-{\rm
Ai_{n}}^{\prime}(s^{\prime}){\rm Ai_{n}}(js)\Big{)}\\\\[8.53581pt]
\widetilde{V}_{n,2,1}(s^{\prime},s)&=&C\frac{\displaystyle\varepsilon^{1/3}}{\displaystyle
in}\,\Big{(}j{\rm Ai_{n}}^{\prime}(js^{\prime}){\rm Ai_{n}}^{\prime}(s)-j{\rm
Ai_{n}}^{\prime}(s^{\prime}){\rm Ai_{n}}^{\prime}(js)\Big{)}\\\\[8.53581pt]
\widetilde{V}_{n,1,2}(s^{\prime},s)&=&C\frac{\displaystyle
in}{\displaystyle\varepsilon^{1/3}}\,\,\,\Big{(}-{\rm Ai_{n}}(js^{\prime}){\rm
Ai_{n}}(s)+{\rm Ai_{n}}(s^{\prime}){\rm Ai_{n}}(js)\Big{)}\\\\[8.53581pt]
\widetilde{V}_{n,2,2}(s^{\prime},s)&=&\quad\quad-C\Big{(}{\rm
Ai_{n}}(js^{\prime}){\rm Ai_{n}}^{\prime}(s)-j{\rm Ai_{n}}(s^{\prime}){\rm
Ai_{n}}^{\prime}(js)\Big{)}.\end{array}\right.$ (6.21)

###  6.2 Upper bounds for the propagator: proof of Lemma (3.4)

In order to prove Lemma 3.5, we derive asymptotic estimates of ${\rm
Ai_{n}}(s)$ and ${\rm Ai_{n}}(js)$ when $s$ real and $s\to{}+\infty$.

######  Lemma 6.2 (Asymptotic estimates for the Airy function).

There holds for all $n\in{\mathbb{Z}}^{*}$ and $s\geq 1$, up to some complex
constants:

$\displaystyle{\rm Ai_{n}}(s)$ $\displaystyle\approx$ $\displaystyle
s^{-1/4}|n|^{-1/2}\,\exp(-|n|\gamma_{0}(2/3)s^{3/2})$ (6.22)
$\displaystyle{\rm Ai_{n}}(js)$ $\displaystyle\approx$ $\displaystyle
s^{-1/4}|n|^{-1/2}\,\exp(|n|\gamma_{0}(2/3)s^{3/2})$ (6.23) $\displaystyle{\rm
Ai_{n}}^{\prime}(s)$ $\displaystyle\approx$ $\displaystyle
s^{1/4}|n|^{1/2}\,\exp(-|n|\gamma_{0}(2/3)s^{3/2})$ (6.24) $\displaystyle{\rm
Ai_{n}}^{\prime}(js)$ $\displaystyle\approx$ $\displaystyle
s^{1/4}|n|^{1/2}\,\exp(|n|\gamma_{0}(2/3)s^{3/2}).$ (6.25)

In particular, the Airy function ${\rm Ai_{n}}$ and its derivative satisfy the
upper bounds

$e^{|n|\gamma_{0}(2/3)s^{3/2}}\left|{\rm
Ai_{n}}(s)\right|+e^{-|n|\gamma_{0}(2/3)s^{3/2}}\left|{\rm
Ai_{n}}(js)\right|\lesssim|n|^{-1/2}(1+s)^{-1/4}\quad\forall 0\leq s,\;\forall
n\in{\mathbb{Z}}^{*}$ (6.26)

and

$e^{|n|\gamma_{0}(2/3)s^{3/2}}\left|{\rm
Ai_{n}}^{\prime}(s)\right|+e^{-|n|\gamma_{0}(2/3)s^{3/2}}\left|{\rm
Ai_{n}}^{\prime}(js)\right|\lesssim|n|^{1/2}s^{1/4}\quad\forall 0\leq
s,\;\forall n\in{\mathbb{Z}}^{*}$ (6.27)

###### Proof.

For $s\geq 1$, we put $a=is^{1/2}$ into the definition (6.13) to obtain

$\displaystyle{\rm Ai_{n}}(s)$ $\displaystyle=$
$\displaystyle(2\pi)^{-1}\int_{{\rm
Im}\zeta=is^{1/2}}\exp\left((|n|\gamma_{0})(i\zeta^{3}/3+i\zeta
z)\right)d\zeta$ $\displaystyle=$
$\displaystyle(2\pi)^{-1}\int_{\mathbb{R}}\exp\left((|n|\gamma_{0})(i(\xi+is^{1/2})^{3}/3+i(\xi+is^{1/2})s)\right)d\xi$
$\displaystyle=$
$\displaystyle(2\pi)^{-1}\,e^{-|n|\gamma_{0}(2/3)s^{3/2}}\,\int_{\mathbb{R}}\exp\left((|n|\gamma_{0})(i\xi^{3}/3-\xi^{2}s^{1/2})\right)d\xi.$

By the change of variables $\xi\mapsto(|n|\gamma_{0}s^{1/2})^{-1/2}\xi$ in the
integral, there holds

${\rm
Ai_{n}}(s)=\frac{|n|^{-1/2}s^{-1/4}}{2\pi\gamma_{0}^{1/2}}\,e^{-|n|\gamma_{0}(2/3)s^{3/2}}\,\int_{\mathbb{R}}\exp\left(i(|n|\gamma_{0})^{{}-1/2}s^{{}-3/4}\xi^{3}/3-\xi^{2}\right)d\xi.$

As the last integral satisfies the asymptotic development, for
$s\to{}+\infty$:

$\displaystyle\int_{\mathbb{R}}\exp\left(i(|n|\gamma_{0})^{{}-1/2}s^{{}-3/4}\xi^{3}/3-\xi^{2}\right)d\xi=\sqrt{2\pi}+O\left(|n|^{-1/2}s^{-3/4}\right)$

we obtain (6.22). By an analog computation we have (6.23), (6.24) and (6.25).

From those asymptotic estimates, we deduce immediately uniform bounds for
${\rm Ai_{n}}$ and the time derivative ${\rm Ai_{n}}^{\prime}$.

∎

Thanks to the previous Lemma, we end the proof of Lemma 3.5 by getting the
upper bound of the propagator $\widetilde{V}_{n}(s^{\prime},s)$. Combining the
expression of $\widetilde{V}_{n}$ in function of ${\rm Ai_{n}}$ given by
(6.21) with the estimates (6.26) and (6.27), we obtain the upper bounds for
the coefficients of the matrix flow $\widetilde{V}_{n}(s^{\prime},s)$, for
$0\leq s^{\prime}\leq s\leq\varepsilon^{-2/3}$:

$\left\\{\begin{array}[]{ccl}\left|\widetilde{V}_{n,1,1}(s^{\prime},s)\right|&\approx&s^{\prime
1/4}(1+s)^{-1/4}\exp\left(|n|\gamma_{0}\frac{2}{3}(s^{3/2}-s^{\prime
3/2})\right)\\\\[8.53581pt]
\left|\widetilde{V}_{n,2,1}(s^{\prime},s)\right|&\approx&\varepsilon^{1/3}s^{\prime
1/4}s^{1/4}\exp\left(|n|\gamma_{0}\frac{2}{3}(s^{3/2}-s^{\prime
3/2})\right)\\\\[8.53581pt]
\left|\widetilde{V}_{n,1,2}(s^{\prime},s)\right|&\approx&\varepsilon^{-1/3}(1+s^{\prime})^{-1/4}(1+s)^{-1/4}\exp\left(|n|\gamma_{0}\frac{2}{3}(s^{3/2}-s^{\prime
3/2})\right)\\\\[8.53581pt]
\left|\widetilde{V}_{n,2,2}(s^{\prime},s)\right|&\approx&(1+s^{\prime})^{-1/4}s^{1/4}\exp\left(|n|\gamma_{0}\frac{2}{3}(s^{3/2}-s^{\prime
3/2})\right)\end{array}\right.$ (6.28)

As

$\varepsilon^{-1/3}(1+s)^{-1/4}>s^{1/4}\qquad\forall 0\leq
s<\varepsilon^{-2/3}$

we obtain the upper bound for the propagator

$|\widetilde{V}_{n}(s^{\prime},s)|\lesssim\varepsilon^{-1/3}(1+s^{\prime})^{-1/4}(1+s)^{-1/4}\exp(|n|\gamma_{0}(2/3)(s^{3/2}-s^{\prime
3/2}))\quad\forall 0\leq s^{\prime}\leq s<\varepsilon^{-2/3}$

which implies (3.26) and ends the proof of Lemma 3.5.

###  6.3 Growth of the free solution: proof of Lemma (3.6)

We prove here Lemma 3.8, following the proof of Lemma 3.7. We showed in it
that it suffices to prove the lower bound for $\widetilde{V}_{n}$. Thanks to
the equalities (6.21), a simple computation gives us

$\widetilde{V}_{+1}(0,s)\begin{pmatrix}{\rm Ai_{1}}(0)\\\
-i\varepsilon^{1/3}j{\rm Ai_{1}}^{\prime}(0)\end{pmatrix}=\begin{pmatrix}{\rm
Ai_{1}}(js)\\\ -i\varepsilon^{1/3}j{\rm Ai_{1}}^{\prime}(js)\end{pmatrix}$

and also

$\widetilde{V}_{-1}(0,s)\begin{pmatrix}{\rm Ai_{1}}(0)\\\
i\varepsilon^{1/3}j{\rm Ai_{1}}^{\prime}(0)\end{pmatrix}=\begin{pmatrix}{\rm
Ai_{1}}(js)\\\ i\varepsilon^{1/3}j{\rm Ai_{1}}^{\prime}(js)\end{pmatrix}.$

We denote

$\widetilde{{\mathbf{f}}}(s,\theta)={\rm
Re}\left(\widetilde{V}_{+1}(0,s)\begin{pmatrix}{\rm Ai_{1}}(0)\\\
-i\varepsilon^{1/3}j{\rm
Ai_{1}}^{\prime}(0)\end{pmatrix}e^{i\theta}+\widetilde{V}_{-1}(0,s)\begin{pmatrix}{\rm
Ai_{1}}(0)\\\ i\varepsilon^{1/3}j{\rm
Ai_{1}}^{\prime}(0)\end{pmatrix}e^{-i\theta}\right).$ (6.29)

and we compute

$\widetilde{{\mathbf{f}}}(s,\theta)=2{\rm Re}\begin{pmatrix}{\rm
Ai_{1}}(js)\cos(\theta)\\\ -\varepsilon^{1/3}j{\rm
Ai_{1}}^{\prime}(js)\sin(\theta)\end{pmatrix}.$ (6.30)

Next we denote $\widetilde{{\mathbf{f}}}_{1}(s)$ and
$\widetilde{{\mathbf{f}}}_{2}(s)$ the two components of the vector
$\widetilde{{\mathbf{f}}}$ defined by (6.29). Thanks to Lemma 6.2, we have

$|\widetilde{{\mathbf{f}}}_{1}(s)|\approx{\rm Ai_{1}}(js)\approx
s^{-1/4}\exp(\gamma_{0}2/3s^{3/2})$

and

$|\widetilde{{\mathbf{f}}}_{2}(s)|\approx\varepsilon^{1/3}{\rm
Ai_{1}}^{\prime}(js)\approx\varepsilon^{1/3}s^{1/4}\exp(\gamma_{0}2/3s^{3/2})$

for $1\leq s<\varepsilon^{-2/3}$. As $\varepsilon^{1/3}s^{1/4}<s^{-1/4}$ for
$s<\varepsilon^{-2/3}$, the $L^{\infty}$ norm of the vector
$\widetilde{{\mathbf{f}}}(s,\theta)$ satisfies

$|\widetilde{{\mathbf{f}}}(s,\theta)|_{\infty}=|\widetilde{{\mathbf{f}}}_{1}(s,\theta)|\approx
s^{-1/4}\exp(\gamma_{0}2/3s^{3/2})$

for all $1\leq s<\varepsilon^{-2/3}$. Using the same steps as in the proof of
Lemma 3.7, this suffices to end the proof of Lemma 3.8.

## References

  * [Gla63] Georges Glaeser.  Racine carrée d’une fonction différentiable.  In Annales de l’Institut Fourier, volume 13, pages 203–210, 1963.
  * [Hör83] Lars Hörmander.  The Analysis of Linear Partial Differential Operators: Vol.: 1.: Distribution Theory and Fourier Analysis.  Springer-Verlag, 1983.
  * [Leb02] Gilles Lebeau.  Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2d.  ESAIM: Control, Optimisation and Calculus of Variations, 8:801–825, 2002.
  * [LMX10] Nicolas Lerner, Yoshinori Morimoto, and Chao-Jiang Xu.  Instability of the Cauchy-Kovalevskaya solution for a class of nonlinear systems.  American Journal of Mathematics, 132(1):99–123, 2010.
  * [LNT17] Nicolas Lerner, Toan T. Nguyen, and Benjamin Texier.  The onset of instability in first-order systems.  Journal of the European Mathematical Society (to appear), 2017.
  * [LT15] Yong Lu and Benjamin Texier.  A stability criterion for high-frequency oscillations.  Mém. Soc. Math. Fr., 142:138 pp., 2015.
  * [Lu16] Yong Lu.  Higher-order resonances and instability of high-frequency WKB solutions.  Journal of Differential Equations, 260(3):2296–2353, 2016.
  * [Mét85] Guy Métivier.  Uniqueness and approximation of solutions of first order non linear equations.  Inventiones Mathematicae, 82(2):263–282, 1985.
  * [Mét05] Guy Métivier.  Remarks on the well-posedness of the nonlinear Cauchy problem.  Contemporary Mathematics, 368:337–356, 2005.
  * [Mor16a] Baptiste Morisse.  On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case.  arXiv preprint arXiv:1611.07225, 2016.
  * [Mor16b] Baptiste Morisse.  On hyperbolicity and Gevrey well-posedness. Part three: a class of weakly hyperbolic systems.  In preparation, 2016.
  * [Mor17] Baptiste Morisse.  On hyperbolicity and Gevrey well-posedness. Part four: generic non-scalar transitions.  In preparation, 2017.
  * [Tex16] Benjamin Texier.  Approximation of pseudo-differential flows.  Indiana Univ. Math. J., 65:243–272, 2016.