TY - JOUR

T1 - The onset of instability in first-order systems

AU - Lerner, Nicolas

AU - Nguyen, Toan

AU - Texier, Benjamin

N1 - Funding Information:
Acknowledgments. T.N. was supported by the Fondation Sciences Mathématiques de Paris through a postdoctoral grant. B.T. thanks Yong Lu and Baptiste Morisse for their remarks on an earlier version of the manuscript. The authors thank the referees for detailed and useful remarks.
Publisher Copyright:
© 2018 European Mathematical Society.

PY - 2018

Y1 - 2018

N2 - 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)].

AB - 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)].

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U2 - 10.4171/JEMS/788

DO - 10.4171/JEMS/788

M3 - Article

AN - SCOPUS:85046894686

SN - 1435-9855

VL - 20

SP - 1303

EP - 1373

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 6

ER -