Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations

Abstract

We develop a well-posedness theory for solutions in $L^1$ to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that $L^1$ is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in $L^1$, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.

Résumé

Nous développons une théorie d’existence et unicité pour les solutions $L^1$ seulement du problème de Cauchy pour un problème de Cauchy hyperbolique-parabolique avec diffusion non-isotrope générale. Des notions de formulations entropique et cinétique sont introduites qui incorporent un nouvel ingrédient, une condition de type dérivation composée, qui montre la différence fondamentale avec le cas d’une diffusion isotrope. L’avantage de la notion de solution cinétique est de travailler directement dans l’espace naturel $L^1$.