Global classical solutions for partially dissipative hyperbolic system of balance laws

TL;DR

This paper develops a comprehensive theory for the existence and uniqueness of classical solutions to partially dissipative hyperbolic systems of balance laws in arbitrary dimensions, using advanced functional space techniques.

Contribution

It introduces a new framework for well-posedness in Chemin-Lerner's spaces and applies it to the compressible Euler equations with damping.

Findings

Established local and global well-posedness results

Developed relations between homogeneous and inhomogeneous Chemin-Lerner spaces

Applied theory to fluid models like Euler equations with damping

Abstract

This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.