Electron. J. Diff. Eqns., Vol. 2003(2003), No. 46, pp. 1-31.

On uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations

Helge Holden, Kenneth H. Karlsen, & Nils H. Risebro

We prove existence and uniqueness of entropy solutions for the Cauchy problem of weakly coupled systems of nonlinear degenerate parabolic equations. We prove existence of an entropy solution by demonstrating that the Engquist-Osher finite difference scheme is convergent and that any limit function satisfies the entropy condition. The convergence proof is based on deriving a series of a priori estimates and using a general $L^p$ compactness criterion. The uniqueness proof is an adaption of Kruzkov's ``doubling of variables'' proof. We also present a numerical example motivated by biodegradation in porous media.

Submitted October 18, 2002. Published April 22, 2003.
Math Subject Classifications: 35K65, 65M12, 35L65.
Key Words: Nonlinear degenerate parabolic equations, weakly coupled systems, entropy solution, uniqueness, existence, finite difference method

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Helge Holden
Department of Mathematical Sciences
Norwegian University of Science and Technology
NO--7491 Trondheim, Norway
email: holden@math.ntnu.no
Kenneth H. Karlsen
Department of Mathematics
University of Bergen
Johs. Brunsgt. 12
N-5008 Bergen, Norway
e-mail: kennethk@mi.uib.no
Nils H. Risebro
Department of Mathematics
University of Oslo
P.O. Box 1053, Blindern
N-0316 Oslo, Norway
e-mail: nilshr@math.uio.no

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