Preprint 2001-013

On a Nonlinear Degenerate Parabolic Transport-Diffusion Equation with a Discontinuous Coefficient

Kenneth H. Karlsen, Nils H. Risebro, and John D. Towers

Abstract: We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation
ut + (g(x)f(u))x = A(u)xx,        A'(·) > = 0,
where the coefficient g(x) is possibly discontinuous and f(u) is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as e->0 of a suitable sequence {ue}e > 0 of smooth approximations solving (P) with the transport flux g(x)f(·) replaced by ge(x)f(·) and the diffusion function A(·) replaced by Ae(·), where ge(·) is smooth and Ae'(·) > 0. The main technical challenge is to deal with the fact that the total variation |ue|BV cannot be bounded uniformly in e, and hence one cannot derive directly strong convergence of {ue}e > 0. In the purely hyperbolic case (A' = > 0), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term.

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Kenneth H. Karlsen , <>
Nils H. Risebro, <>
John D. Towers, <>
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<> April 5 2001.

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