Local existence for systems of conservation laws with partial diffusion

Abstract

This paper is dedicated to the study of the local existence theory of the Cauchy problem for symmetric hyperbolic partially diffusive systems (also known as hyperbolic-parabolic system) in dimension $d\ge 1$. The system under consideration is a coupling between a symmetric hyperbolic system and a parabolic system. We address the question of well-posedness for large data having critical Besov regularity. This improves the analysis of Serre \cite{Serr10} and Kawashima \cite{Kawashima83}. Our results allow for initial data whose components have different regularities and we enlarge the class of the components experiencing the diffusion to $H^s$, with $s>d/2$ (instead of $s>d/2+1$ in Serre's work and $s>d/2+2$ in Kawashima's one). Our results rely on G\r{a}rding's inequality, composition estimates and product laws. As an example, we consider the Navier-Stokes-Fourier equations.