On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part one

TL;DR

This paper proves that certain nonlinear PDE systems with elliptic principal symbols lack solutions in specific Sobolev spaces and extends these results to systems transitioning between hyperbolic and elliptic types, highlighting instability phenomena.

Contribution

It provides a detailed proof of Métivier's theorem and extends the analysis to systems with changing hyperbolic-elliptic behavior, advancing understanding of solution non-existence and instability.

Findings

No solutions exist in certain Sobolev spaces under ellipticity.

Extension of non-existence results to hyperbolic-elliptic transition systems.

Identification of instability mechanisms in nonlinear PDE systems.

Abstract

We prove that, for first-order, fully nonlinear systems of partial differential equations, under an hypothesis of ellipticity for the principal symbol, the Cauchy problem has no solution within a range of Sobolev indices depending on the regularity of the initial datum. This gives a new and greatly detailed proof of a result of G. M\'etivier [{\it Remarks on the Cauchy problem}, 2005]. We then extend this result to systems experiencing a transition from hyperbolicity and ellipticity, in the spirit of recent work by N. Lerner, Y. Morimoto, and C.-J. Xu, [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, 2010], and N. Lerner, T. Nguyen and B. Texier [{\it The onset of instability in first-order systems}, 2018].