The onset of instability in first-order systems
Journal of the European Mathematical Society, Tome 20 (2018) no. 6, pp. 1303-1373
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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)].
@article{JEMS_2018_20_6_a0,
author = {Nicolas Lerner and Toan T. Nguyen and Benjamin Texier},
title = {The onset of instability in first-order systems},
journal = {Journal of the European Mathematical Society},
pages = {1303--1373},
publisher = {mathdoc},
volume = {20},
number = {6},
year = {2018},
doi = {10.4171/jems/788},
url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/788/}
}
TY - JOUR AU - Nicolas Lerner AU - Toan T. Nguyen AU - Benjamin Texier TI - The onset of instability in first-order systems JO - Journal of the European Mathematical Society PY - 2018 SP - 1303 EP - 1373 VL - 20 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4171/jems/788/ DO - 10.4171/jems/788 ID - JEMS_2018_20_6_a0 ER -
%0 Journal Article %A Nicolas Lerner %A Toan T. Nguyen %A Benjamin Texier %T The onset of instability in first-order systems %J Journal of the European Mathematical Society %D 2018 %P 1303-1373 %V 20 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4171/jems/788/ %R 10.4171/jems/788 %F JEMS_2018_20_6_a0
Nicolas Lerner; Toan T. Nguyen; Benjamin Texier. The onset of instability in first-order systems. Journal of the European Mathematical Society, Tome 20 (2018) no. 6, pp. 1303-1373. doi: 10.4171/jems/788
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