Vol. 4, No. 1, 2022

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Partially dissipative one-dimensional hyperbolic systems in the critical regularity setting, and applications

Timothée Crin-Barat and Raphaël Danchin

Vol. 4 (2022), No. 1, 85–125
Abstract

Here we develop a method for investigating global strong solutions of partially dissipative hyperbolic systems in the critical regularity setting. Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces with different regularity exponents in low and high frequencies. This allows us to consider more general data and to track the exact dependency on the dissipation parameter for the solution. Our approach enables us to go beyond the L2 framework in the treatment of the low frequencies of the solution, which is totally new, to the best of our knowledge.

The focus is on the one-dimensional setting (the multidimensional case will be considered in a forthcoming paper) and, for expository purposes, the first part of the paper is devoted to a toy model that may be seen as a simplification of the compressible Euler system with damping. More elaborate systems (including the compressible Euler system with general increasing pressure law) are considered at the end of the paper.

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Keywords
hyperbolic systems, critical regularity, time decay, partially dissipative
Mathematical Subject Classification
Primary: 35Q35, 76N10
Milestones
Received: 6 January 2021
Revised: 25 October 2021
Accepted: 5 January 2022
Published: 29 April 2022
Authors
Timothée Crin-Barat
LAMA, UMR 8050
Université Paris-Est Créteil Val de Marne
Créteil
France
Raphaël Danchin
LAMA, UMR 8050
Université Paris-Est Créteil Val de Marne
Créteil
France