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Journal of the European Mathematical Society

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Volume 20, Issue 6, 2018, pp. 1303–1373
DOI: 10.4171/JEMS/788

Published online: 2018-04-16

The onset of instability in first-order systems

Nicolas Lerner[1], Toan Nguyen[2] and Benjamin Texier[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) Pennsylvania State University, University Park, USA
(3) Université Paris-Diderot, France

We study the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., hyperbolicity is violated at initial time, the Cauchy problem is strongly unstable in the sense of Hadamard. This phenomenon, which extends the linear Lax–Mizohata theorem, was explained by G. Métivier [Contemp. Math. 368, 2005]. In the present paper, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is, the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under that hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto and C.-J. Xu [Amer. J. Math. 132 (2010)] on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein–Gordon–Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, proved by B. Texier [Indiana Univ. Math. J. 65 (2016)].

Keywords: Cauchy problem, hyperbolicity, ellipticity

Lerner Nicolas, Nguyen Toan, Texier Benjamin: The onset of instability in first-order systems. J. Eur. Math. Soc. 20 (2018), 1303-1373. doi: 10.4171/JEMS/788