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

TL;DR

This paper investigates the transition from hyperbolicity to ellipticity in first-order PDE systems, proving strong instability results and demonstrating how Gevrey regularity is lost instantaneously under certain conditions.

Contribution

It establishes a new instability result for scalar and certain degenerate hyperbolic-to-elliptic transitions, extending previous elliptic case analyses.

Findings

Proves strong Hadamard instability under hyperbolic-elliptic transition assumptions.

Shows instantaneous loss of Gevrey regularity in the flow.

Extends instability results to scalar and flat boundary hyperbolic zones.

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 H\"older 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, \textit{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 [\textit{The onset of instability in first-order systems}, to appear in J. Eur. Math. Soc.].