On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part one
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].
Keywords
Cite
@article{arxiv.2012.08222,
title = {On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part one},
author = {Karim Ndoumajoud and Benjamin Texier},
journal= {arXiv preprint arXiv:2012.08222},
year = {2020}
}