Preprint 2001-003

A Note on the Uniqueness of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

Kenneth Hvistendahl Karlsen and Mario Ohlberger

Abstract: Following the lead of Carrillo \cite{Carrillo}, recently several authors have used Kru\v{z}kov's device of ``doubling the variables'' to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result found in Karlsen and Risebro \cite{KR} to a class of degenerate parabolic equations with \textit{spatially} dependent second order differential operator. The class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation-consolidation processes.

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Kenneth Hvistendahl Karlsen, <>
Mario Ohlberger, <>
Publishing information:
UCLA Computational and Applied Mathematics Report.
Submitted by:
<> January 26 2001.

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