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

TL;DR

This paper introduces a novel method using hybrid Besov spaces to analyze global strong solutions of partially dissipative hyperbolic systems in critical regularity, extending previous approaches and handling more general data.

Contribution

It develops a new analytical framework that surpasses the L^2 approach, enabling the study of more general initial data and precise dissipation dependency in one-dimensional hyperbolic systems.

Findings

Introduces hybrid Besov spaces with different regularity in low/high frequencies.

Extends analysis beyond the L^2 framework for low frequencies.

Applies method to toy models and more complex systems like Euler with damping.

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 exponent in low and high frequency. This allows 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 L^2 framework in the treatment of the low frequencies of the solution, which is totally new, to the best of our knowledge. Focus is on the one-dimensional setting (the multi-dimensional case will be considered in a forthcoming paper) and, for expository purpose, 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 elaborated systems (including the compressible Euler system with general increasing pressure law) are considered at the end of the paper.