##
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)

This paper is available in the following formats:

A closely related paper is Change me.

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