Electron. J. Diff. Eqns., Vol. 2002(2002), No. 93, pp. 1-23.

On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient

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

Abstract:
We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation
$$ \partial_t u + \partial_x \bigl(\gamma(x)f(u)\bigr)=\partial_x^2 A(u), \quad A'(\cdot)\ge 0, $$
where the coefficient $\gamma(x)$ 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 $\varepsilon\downarrow 0$ in a suitable sequence $\{u_{\varepsilon}\}_{\varepsilon}$ of smooth approximations solving the problem above with the transport flux $\gamma(x)f(\cdot)$ replaced by $\gamma_{\varepsilon}(x)f(\cdot)$ and the diffusion function $A(\cdot)$ replaced by $A_{\varepsilon}(\cdot)$, where $\gamma_{\varepsilon}(\cdot)$ is smooth and $A_{\varepsilon}'(\cdot)>0$. The main technical challenge is to deal with the fact that the total variation $|u_{\varepsilon}|_{BV}$ cannot be bounded uniformly in $\varepsilon$, and hence one cannot derive directly strong convergence of $\{u_{\varepsilon}\}_{\varepsilon>0}$. In the purely hyperbolic case ($A'\equiv 0$), 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.

Submitted April 29, 2002. Published October 27, 2002.
Math Subject Classifications: 35K65, 35D05, 35R05, 35L80
Key Words: Degenerate parabolic equation, nonconvex flux, weak solution, discontinuous coefficient, viscosity method, a priori estimates, compensated compactness

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Kenneth H. Karlsen
Department of Mathematics
University of Bergen
Johs. Brunsgt. 12
N-5008 Bergen, Norway
e-mail: kennethk@mi.uib.no
http://www.mi.uib.no/~kennethk
Nils H. Risebro
Department of Mathematics
University of Oslo
P.O. Box 1053, Blindern
N-0316 Oslo, Norway
e-mail: nilshr@math.uio.no
http://www.math.uio.no/~nilshr
John D. Towers
MiraCosta College
3333 Manchester Avenue
Cardiff-by-the-Sea, CA 92007-1516, USA
e-mail: jtowers@cts.com
http://www.miracosta.cc.ca.us/home/jtowers/

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