Well-Posedness for Anisotropic Degenerate Parabolic-Hyperbolic Equations

Authors: Gui-Qiang Chen and Benoit Perthame

Title: Well-Posedness for Anisotropic 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 anisotropic nonlinearity. A new notion of entropic and kinetic solutions, and a corresponding kinetic formulation is 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.

This article has appeared in:
Annales de l'Institut Henri Poincare: Analyse Non Lineaire, vol. 20, pages 645-668 (2003)

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Author Address
    Gui-Qiang Chen
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    gqchen@math.nwu.edu