On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case.

Morisse, Baptiste

Annales Henri Lebesgue, Tome 3 (2020), pp. 1195-1239.

Dans cet article nous prouvons que le problème de Cauchy pour des systèmes d’équations aux dérivées partielles non-linéaires du premier ordre sont mal posées dans les espaces de Gevrey, sous la condition d’initiale ellipticité. L’hypothèse porte sur le symbole principal de l’opérateur du premier ordre. Le caractère mal-posé s’entend ici au sens d’Hadamard, en particulier par un défaut instantané de la continuité Hölder du flot depuis $G^{\sigma }$ vers $L^2$, où $\sigma \in (0,1)$ dépend du spectre initial. En suivant la construction proposée par G. Métivier dans [Remarks on the well-posedness of the nonlinear Cauchy problem, Contemp. Math. 2005], nous prouvons que le caractère mal-posé découle d’une construction à la Cauchy–Kovalevskaya en temps longs d’une famille de solutions exactes, rapidement oscillantes et analytiques proches initialement de la solution nulle, et qui croissent exponentiellement vite en temps. Une des difficultés ici réside dans le temps d’observation de l’instabilité. Alors qu’en régularité Sobolev, ce temps est logarithmique en fréquence, en régularité Gevrey il est une puissance en fréquence. En particulier, en régularité Gevrey l’instabilité est observée bien plus tard qu’en régularité Sobolev.

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 $\sigma \in (0,1)$ 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.

Classification : 35L40
Mots clés : Gevrey regularity, hyperbolic systems, ill-posedness

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