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

TL;DR

This paper proves the global existence and decay of solutions for multi-dimensional partially dissipative hyperbolic systems under the (SK) condition, using hybrid Besov norms and Lyapunov functionals to improve previous results.

Contribution

It extends the analysis of partially dissipative hyperbolic systems to multiple dimensions with optimal smallness conditions using hybrid Besov norms.

Findings

Established global strong solutions in multi-dimensions

Derived decay estimates with optimal smallness conditions

Identified a damped mode with faster decay rate

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.