Branching eigenvalues and defect of hyperbolicity

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Background: on Lax-Mizohata results

The question of the well-posedness of the Cauchy problem was rst introduced and studied by Hadamard in [Had02]. Hadamard proved, in the case of linear second-order elliptic equations, that the associated solution ow is not regular in the vicinity of any solution of the system. The case of linear evolution systems of the form (2.1.1), that is with Aj(t; x; u) Aj(t; x), f(t; x; u) f(t; x) was rst studied by Lax in [Lax05], where the proof was given that hyperbolicity of the system, i.e. reality of the spectrum of the principal symbol, was a necessary condition for (2.1.1) to be well-posed in the sense of Hadamard in Ck spaces. Lax’s proof relied on separation of the spectrum. Mizohata extended Lax’s result without this assumption in [Miz61]. Some cases of nonlinear systems were studied later by Wakabayashi in [Wak01] (here with stability also with respect to source term) and by Yagdjian in [Yag98] and [Yag02] (there in the special case of gauge invariant systems).
A rst statement of a precise Lax-Mizohata result for rst-order quasi-linear systems was given by Métivier in [Mét05], with a precise description of the lack of regularity of the ow. As we will adapt the methods used by Métivier, we want to take a close look at [Mét05].

On Métivier’s result in Sobolev spaces

In Section 3 of [Mét05] Guy Métivier proves Hölder ill-posedness in Sobolev spaces for the Cauchy problem (2.1.1), as soon as hyperbolicity fails at t = 0. The initial defect of hyperbolicity means here that there are some x0 2 Rd, ~u0 2 RN and 0 2 Rd such that the principal symbol evaluated at (0; x0; ~u0; 0): A0 := X j Aj(0; x0; ~u0)0;j (2.1.2) is supposed to have a couple of eigenvalues with non zero imaginary part, say i 0, with eigenvectors ~e. Hölder well-posedness, locally in time and space, would mean that initial data h1 and h2 in H(Br0(x0)), for some small r0 > 0, would generate solutions u1 and u2 such that  for some space-time domain , for some 0, some 2 (0; 1]. In order to disprove (2.1.3), Métivier chooses h1 ~u0, and lets u1 the Cauchy-Kovalevskaya solution issued from h1, the existence of which is granted, locally in space and time, by the analyticity assumption on the coecients Aj and f. Translating, Métivier is reduced to the case ~u0 = 0, u1 0, and the proof that (2.1.3) does not hold is reduced to the construction of a family (u ») »>0 of initially small, exact analytical solutions such that.

Statement of the results

In the statement below we use notations introduced in Denitions 2.2.1 and 2.2.2.
Theorem 1. Under Assumptions 2.2.4 and 2.2.10, the Cauchy problem (2.1.1) is not Hölder well-posed in Gevrey spaces G for all 2 (0; 1=(m+ 1)) where m is the algebraic multiplicity of 0. That is for all c > 0, K compact of Rd and 2 (0; 1], there are sequences R􀀀1  » ! 0 and 􀀀1  » ! 0, a family of initial conditions h » 2 G and corresponding solutions u » of the Cauchy problem on domains R »; »(x0) such that lim « !0 jju »jjL2( R »; » (x0))=jjh »jj ;c;K = +1: (2.2.12) The time of existence of the solutions u » is at least of order « 1􀀀. We prove the instability for a larger band of Gevrey indices under stronger assumptions. First, the semisimplicity and non-coalescing Assumption 2.2.6 allows for a critical index equal to 1=2: Theorem 2. Under Assumptions 2.2.6 and 2.2.10, the result of Theorem 1 holds for any Gevrey index in (0; 1=2).
Second, under Assumption 2.2.6, the null condition (i) and the sign condition (ii) in Assumption 2.2.8 allow for the critical index to go from 1=2 up to 2=3: Theorem 3. Under Assumptions 2.2.6, 2.2.8 and 2.2.10, the result of Theorem 1 holds for any Gevrey index in (0; 2=3).
The rest of the paper is devoted to the proof of Theorems 1, 2 and 3. Remark 2.2.11. Higher-order null and sign conditions allow for a greater critical index. Precisely, under Assumption 2.2.6, if (0; x0) is a local maximum for Im, and if there holds (« s; x0) 􀀀 (0; x0) = O(« s)2k􀀀1, then our proof implies ill-posedness with a critical Gevrey index equal to 2k=(2k + 1). These null and sign conditions can be expressed in terms of derivatives of A, the partial inverse A􀀀1 0 and the projector P0, see [Kat66], or Remark 2.7 of [Tex04]. See also Remark 2.6.5.

Highly oscillating solutions and reduction to a xed point equation

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We want to compare two solutions of (2.1.1) with initial data h1 and h2 satisfying both hi(x = 0) = 0 for i = 1; 2 to t with ~u0 = 0 in (2.2.9). We can choose h1 analytic, which lead by Cauchy-Kovalevskaya theorem to an analytic solution u1 in some small neighborhood of (0; 0) 2 Rt Rd x. Then changing u into u 􀀀 u1 in (2.1.1) we get a new Cauchy problem @tu = X j Aj(t; x; u)@xju + F(t; x; u)u ; u(0; x) = h(x) (2.3.1) with F(t; x; u) 2 RNN is also analytic, by analyticity of f and u1. We consider for h small analytical functions satisfying hjx=0 = 0, as perturbations of the trivial datum h 0.

Highly oscillating solutions

As in [Mét05] we look for high oscillating solutions of (2.3.1) with the aim of seeing the expected growth. In this view we posit the following ansatz u »(t; x) = « u(t= »; x; x = ») (2.3.2) where the function u(s; x; ) is 2-periodic in . We introduce for any analytical function H(t; x; u) the compact notation H(s; x; u) = H (« s; x; « u) :

Table of contents :

1 Introduction 
1.1 Préliminaire sur le problème de Cauchy
1.1.1 Étude du symbole principal, avec coecients constants
1.1.2 Stabilité du spectre par perturbation linéaire d’ordre 0
1.1.3 Inuence des termes non-linéaires
1.1.4 Régularité Gevrey
1.1.5 Instabilité
1.2 Caractère mal-posé pour un système non-hyperbolique
1.2.1 Présentation
1.2.2 Formes normales
1.2.3 Solutions à oscillations rapides
1.2.4 Croissance des modes de Fourier et équation de point xe
1.2.5 Méthode des séries majorantes, théorème de Cauchy-Kovalevskaya
1.2.6 Point xe et borne sur les indices Gevrey
1.3 Caractère bien-posé de systèmes faiblement hyperboliques
1.3.1 Le travail fondateur de Colombini, Janelli et Spagnolo
1.3.2 Sur l’inégalité de Glaeser
1.3.3 Au delà de l’article de 1983 de Colombini, Janelli et Spagnolo
1.3.4 Caractère bien-posé en Gevrey pour des transitions faiblement hyperboliques
1.4 Opérateurs pseudo-diérentiels sur les espaces de Gevrey
2 The elliptic case 
2.1 Introduction
2.1.1 Background: on Lax-Mizohata results
2.1.2 On Métivier’s result in Sobolev spaces
2.1.3 Extension to Gevrey spaces
2.2 Main assumptions and results
2.2.1 Denitions: Hölder well-posedness in Gevrey spaces
2.2.2 Assumptions
2.2.3 Statement of the results
2.3 Highly oscillating solutions
2.3.1 Preparation of the equation
2.3.2 Highly oscillating solutions
2.3.3 Upper bounds for the propagator
2.3.4 Free solutions
2.3.5 Fixed point equation
2.3.6 Sketch of the proof
2.4 Majoring series and functional spaces
2.4.1 Properties of majoring series
2.4.2 Denitions of functional spaces
2.4.3 Some properties of spaces E
2.4.4 Action of U(s0; s) on E
2.4.5 Norm of the free solution
2.5 Regularization by integration in time
2.5.1 Lack of boundedness of derivation operators
2.5.2 Integration in time and regularization of @
2.5.3 Integration in time and regularization of @xj
2.5.4 Integration in time and product
2.5.5 Contraction estimates
2.6 Existence of solutions and estimates from below
2.6.1 Existence of solutions
2.6.2 Bounds from below for the solutions
2.7 Conclusion: Hadamard instability in Gevrey spaces
3 Scalar or degenerate transitions 
3.1 Introduction
3.1.1 Background
3.1.2 Overview of the paper
3.2 Main assumptions and results
3.2.1 Branching eigenvalues and defect of hyperbolicity
3.2.2 The case of a smooth transition
3.2.3 The case of a sti transition
3.2.4 Statement of the results
3.3 Highly oscillating solutions
3.3.1 Highly oscillating solutions
3.3.2 Remainder terms
3.3.3 Upper bounds for the propagators
3.3.4 Free solutions
3.3.5 Fixed point equation
3.4 Contraction estimates
3.4.1 Functional spaces: denitions
3.4.2 Functional spaces: properties
3.4.3 Estimates of remainder terms
3.4.4 Contraction estimates
3.5 Estimates from below and Hadamard instability
3.5.1 Existence of solutions
3.5.2 Bounds from below
3.5.3 Conclusion: Hadamard instability in Gevrey spaces
3.6 Appendix: on the Airy equation
3.6.1 Reduction to the scalar Airy equation and resolution
3.6.2 Upper bounds for the propagator: proof of Lemma (3.4)
3.6.3 Growth of the free solution: proof of Lemma (3.6)
4 A class of weakly hyperbolic systems 
4.1 Introduction
4.1.1 Background: on weakly hyperbolic systems
4.1.2 Background: on systems transitioning away from hyperbolicity
4.1.3 Generic time transitions
4.1.4 Current result
4.2 Main assumptions and results
4.3 Proof of the energy estimate
4.3.1 Key preparatory Lemmas
4.3.2 Time derivative of the energy
4.3.3 Energy estimate
4.4 Appendices
4.4.1 Glaeser-type inequalities
4.4.2 Metrics in the phase space and pseudodierential calculus
5 Action of pseudo-dierential operators 
5.1 Introduction
5.2 Classes of Gevrey regular symbols
5.2.1 Gevrey spaces
5.2.2 Classes of symbols
5.3 Conjugation of a Gevrey function
5.4 Action of pseudo-dierential operators on Gevrey spaces
5.5 A conjugation Lemma for operators .


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