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

Abstract

We develop a well-posedness theory for solutions in L1 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 L1 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 L1, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.