JournalsjemsVol. 20, No. 6pp. 1303–1373

The onset of instability in first-order systems

  • Nicolas Lerner

    Université Pierre et Marie Curie, Paris, France
  • Toan Nguyen

    Pennsylvania State University, University Park, USA
  • Benjamin Texier

    Université Paris-Diderot, France
The onset of instability in first-order systems cover
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Abstract

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)].

Cite this article

Nicolas Lerner, Toan Nguyen, Benjamin Texier, The onset of instability in first-order systems. J. Eur. Math. Soc. 20 (2018), no. 6, pp. 1303–1373

DOI 10.4171/JEMS/788