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

• Main Authors: Helge Holden, Kenneth H. Karlsen, Nils H. Risebro
• Format: Article
• Language: English
• Published: Texas State University 2003-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Nonlinear degenerate parabolic equations; weakly coupled systems; entropy solution; uniqueness; existence; finite difference method
• Online Access: http://ejde.math.txstate.edu/Volumes/2003/46/abstr.html

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.