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arXiv · 2103.02401

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

Karim Ndoumajoud·Benjamin Texier
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Abstract

We continue our study of initial-value problems for fully nonlinear systems exhibiting strong or weak defects of hyperbolicity. We prove that, regardless of the initial Sobolev regularity, the initial-value problem has no local $H^s$ solutions, for $s > s_0 + d/2,$ if the principal symbol has a strong, or even weak, defect of hyperbolicity, and the purely imaginary eigenvalues of the principal symbol are semi-simple and have constant multiplicity. The index $s_0 > 0$ depends on the severity of the defect of hyperbolicity. These results recover and extend previous work from G. M\'etivier [{\it Remarks on the well posedness of the nonlinear Cauchy problem,} 2005], N.Lerner, Y. Morimoto, C.-J. Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, 2010] and N. Lerner, T. Nguyen, B. Texier, {\it The onset of instability in first-order systems}, 2018]

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Karim Ndoumajoud, Benjamin Texier. 2021-03-03. On M\'etivier's Lax-Mizohata theorem and extensions to weak defects of hyperbolicity. Part two. https://arxiv.org/abs/2103.02401

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