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arXiv · 2105.08333

Partially dissipative hyperbolic systems in the critical regularity setting : The multi-dimensional case

Timothée Crin-Barat·Raphaël Danchin
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Abstract

We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space $\mathbb{R}^d$ with $d\geq2$. Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies the so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular apply to the compressible Euler system with damping in the velocity equation. Compared to the papers by Kawashima and Xu [27, 28] devoted to similar issues, our use of hybrid Besov norms with different regularity exponents in low and high frequency enable us to pinpoint optimal smallness conditions for global well-posedness and to get more accurate information on the qualitative properties of the constructed solutions. A great part of our analysis relies on the study of a Lyapunov functional in the spirit of that of Beauchard and Zuazua in [2]. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.

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Timothée Crin-Barat, Raphaël Danchin. 2021-05-18. Partially dissipative hyperbolic systems in the critical regularity setting : The multi-dimensional case. https://arxiv.org/abs/2105.08333

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