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

2020/12/15 by Karim Ndoumajoud, Ndoumajoud, Karim, Benjamin Texier +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2012.08222

openalex publication_date 2020/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. Métivier [\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].

Related

• On Métivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part two
• The onset of instability in first-order systems
• On hyperbolicity and Gevrey well-posedness. Part two: Scalar or\n degenerate transitions
• On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case
• Weak versus D-solutions to linear hyperbolic first order systems with constant coefficients