On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case.
Annales Henri Lebesgue, Volume 3 (2020) , pp. 1195-1239.

KeywordsGevrey regularity, hyperbolic systems, ill-posedness

### Abstract

In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Hölder continuity of the flow from ${G}^{\sigma }$ to ${L}^{2}$, where $\sigma \in \left(0,1\right)$ depends on the initial spectrum. Building on the analysis carried out by G. Métivier [Remarks on the well-posedness of the nonlinear Cauchy problem, Contemp. Math. 2005], we show that ill-posedness follows from a long-time Cauchy–Kovalevskaya construction of a family of exact, highly oscillating, analytical solutions which are initially close to the null solution, and which grow exponentially fast in time. A specific difficulty resides in the observation time of instability. While in Sobolev spaces, this time is logarithmic in the frequency, in Gevrey spaces it is a power of the frequency. In particular, in Gevrey spaces the instability is recorded much later than in Sobolev spaces.

### References

[Bed16] Bedrossian, Jacob Nonlinear echoes and Landau damping with insufficient regularity (2016) (https://arxiv.org/abs/1605.06841)

[BMM16] Bedrossian, Jacob; Masmoudi, Nader; Mouhot, Clément Landau damping: paraproducts and Gevrey regularity, Ann. PDE, Volume 2 (2016) no. 1, 4, pp. 1-71 | MR 3489904 | Zbl 1402.35058

[Car61] Cartan, Henri Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Enseignement des sciences, Hermann, 1961 no. 26 | Zbl 0094.04401

[GVD10] Gérard-Varet, David; Dormy, Emmanuel On the ill-posedness of the Prandtl equation, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 591-609 | Article | MR 2601044 | Zbl 1197.35204

[GVM15] Gérard-Varet, David; Masmoudi, Nader Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 6, pp. 1273-1325 | Article | MR 3429469 | Zbl 1347.35201

[Had02] Hadamard, Jacques Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin, Volume 13 (1902) no. 28, pp. 49-52

[Kat66] Kato, Tosio Perturbation theory for linear operators Volume 132, Springer, 1966 | MR 203473 | Zbl 0148.12601

[Lax05] Lax, Peter D. Asymptotic solutions of oscillatory initial value problems, Selected Papers Volume I (Sarnak, Peter; Majda, Andrew, eds.), Springer, 2005, pp. 56-75

[LMX10] Lerner, Nicolas; Morimoto, Yoshinori; Xu, Chao-Jiang Instability of the Cauchy-Kovalevskaya solution for a class of nonlinear systems, Am. J. Math., Volume 132 (2010) no. 1, pp. 99-123 | Article | MR 2597507

[LNT18] Lerner, Nicolas; Nguyen, Toan; Texier, Benjamin The onset of instability in first-order systems, J. Eur. Math. Soc. (JEMS), Volume 20 (2018) no. 6, pp. 1303-1373 | Article | MR 3801816 | Zbl 1414.35047

[Miz61] Mizohata, Sigeru Some remarks on the Cauchy problem, J. Math. Kyoto Univ., Volume 1 (1961) no. 1, pp. 109-127 | MR 170112 | Zbl 0104.31903

[Mor18] Morisse, Baptiste On hyperbolicity and Gevrey well-posedness. Part two: scalar or degenerate transitions, J. Differ. Equations, Volume 264 (2018) no. 8, pp. 5221-5262 | Article | MR 3760173 | Zbl 1383.35051

[MV11] Mouhot, Clément; Villani, Cédric On Landau damping, Acta Math, Volume 207 (2011) no. 1, pp. 29-201 | Article | MR 2863910 | Zbl 1239.82017

[Mét05] Métivier, Guy Remarks on the well-posedness of the nonlinear Cauchy problem, Geometric analysis of PDE and several complex variables (Contemporary Mathematics) Volume 368, American Mathematical Society, 2005, pp. 337-356 | Article | MR 2127041 | Zbl 1071.35074

[Rod93] Rodino, Luigi Linear partial differential operators in Gevrey spaces, World Scientific, 1993 | Zbl 0869.35005

[Tex04] Texier, Benjamin The short-wave limit for nonlinear, symmetric, hyperbolic systems, Adv. Differ. Equ., Volume 9 (2004) no. 1-2, pp. 1-52 | MR 2099605 | Zbl 1108.35372

[Tex17] Texier, Benjamin Basic matrix perturbation theory (2017) (to appear in L’Enseignement Mathématique) | Zbl 07117757

[Uka01] Ukai, Seiji The Boltzmann–Grad limit and Cauchy–Kovalevskaya theorem, Japan J. Ind. Appl. Math., Volume 18 (2001) no. 2, pp. 383-392 | MR 1842918 | Zbl 0981.35004

[Wag79] Wagschal, Claude Le problème de Goursat non-linéaire, Séminaire Goulaouic-Schwartz 1978-1979, Équations aux dérivées partielles, Éditions de l’École polytechnique, 1979, pp. 1-11 | Numdam | MR 544256 | Zbl 0427.35021

[Wak01] Wakabayashi, Seiichiro The Lax–Mizohata theorem for nonlinear Cauchy Problems, Commun. Partial Differ. Equations, Volume 26 (2001) no. 7-8, pp. 1367-1384 | Article | MR 1855282 | Zbl 1001.35003

[Yag98] Yagdjian, Karen A note on Lax–Mizohata theorem for quasilinear equations, Commun. Partial Differ. Equations, Volume 23 (1998) no. 5-6, pp. 1-14 | Article

[Yag02] Yagdjian, Karen The Lax–Mizohata theorem for nonlinear gauge invariant equations, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 49 (2002) no. 2, pp. 159-175 | Article | MR 1885116 | Zbl 1008.35038