Electron. J. Diff. Eqns., Vol. 2002(2002), No. 93, pp. 123.
On a nonlinear degenerate parabolic transportdiffusion
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 transportdiffusion equation
where the coefficient
is possibly discontinuous and
is genuinely nonlinear, but not necessarily
convex or concave. Existence of a weak solution is proved by
passing to the limit as
in a suitable sequence
of
smooth approximations solving the problem above with the transport
flux
replaced by
and the diffusion function
replaced by
,
where
is smooth and
.
The main technical challenge is to deal with the fact
that the total variation
cannot be bounded uniformly in
,
and hence one cannot derive directly strong convergence of
.
In the purely hyperbolic case
(),
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
N5008 Bergen, Norway
email: kennethk@mi.uib.no
http://www.mi.uib.no/~kennethk 

Nils H. Risebro
Department of Mathematics
University of Oslo
P.O. Box 1053, Blindern
N0316 Oslo, Norway
email: nilshr@math.uio.no
http://www.math.uio.no/~nilshr 

John D. Towers
MiraCosta College
3333 Manchester Avenue
CardiffbytheSea, CA 920071516, USA
email: jtowers@cts.com
http://www.miracosta.cc.ca.us/home/jtowers/ 
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