Local well-posedness in the critical regularity setting for hyperbolic systems with partial diffusion

Abstract

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space $\mathbb{R}^d$ with $d\ge 1$. We address the question of well-posedness for large data having critical Besov regularity in the spirit of previous works by the second author on the compressible Navier-Stokes equations. Compared to the pioneering of Kawashima (S. Kawashima. Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics. PhD thesis, Kyoto University, 1983) and to the more recent work by Serre (D. Serre. Local existence for viscous system of conservation laws: $H^s$-data with $s > 1+d/2$. In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 339-358. Amer. Math. Soc., Providence, RI, 2010), we take advantage of the partial parabolicity of the system to consider data in functional spaces that need not be embedded in the set of Lipschitz functions. This is in sharp contrast with the classical well-posedness theory of (multi-dimensional) hyperbolic systems where it is mandatory. A leitmotiv of our analysis is to require less regularity for the components experiencing a direct diffusion, than for the hyperbolic components. We then use an energy method that is performed on the system after spectral localization and a suitable G\r{a}rding inequality. As an example, we consider the Navier-Stokes-Fourier equations.