### 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)].

Original language | English (US) |
---|---|

Pages (from-to) | 1303-1373 |

Number of pages | 71 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*20*(6), 1303-1373. https://doi.org/10.4171/JEMS/788

}

*Journal of the European Mathematical Society*, vol. 20, no. 6, pp. 1303-1373. https://doi.org/10.4171/JEMS/788

**The onset of instability in first-order systems.** / Lerner, Nicolas; Nguyen, Toan; Texier, Benjamin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The onset of instability in first-order systems

AU - Lerner, Nicolas

AU - Nguyen, Toan

AU - Texier, Benjamin

PY - 2018/1/1

Y1 - 2018/1/1

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)].

UR - http://www.scopus.com/inward/record.url?scp=85046894686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046894686&partnerID=8YFLogxK

U2 - 10.4171/JEMS/788

DO - 10.4171/JEMS/788

M3 - Article

AN - SCOPUS:85046894686

VL - 20

SP - 1303

EP - 1373

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 6

ER -