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

Date

2002-10-27

Authors

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

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Publisher

Southwest Texas State University, Department of Mathematics

Abstract

We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation ∂tu + ∂x (γ(x)ƒ(u)) = ∂2x A(u), A'(·) ≥ 0, where the coefficient γ(x) is possibly discontinuous and ƒ(u) is genuinely non-linear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as ε ↓ 0 in a suitable sequence {uε}ε > 0 of smooth approximations solving the problem above with the transport flux γ(x)ƒ(·) replaced by γε(x)ƒ(·) and the diffusion function A(·) replaced by Aε(·), where γε(·) is smooth and A'ε(·) > 0. The main technical challenge is to deal with the fact that the total variation |uε|BV cannot be bounded uniformly in ε, and hence one cannot derive directly strong convergence of {uε}ε > 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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Keywords

Degenerate parabolic equation, Nonconvex flux, Weak solution, Discontinuous coefficient, Viscosity method, a priori estimates, Compensated compactness

Citation

Karlsen, K. H., Risebro, N. H., & Towers, J. D. (2002). On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. <i>Electronic Journal of Differential Equations, 2002</i>(93), pp. 1-23.

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