Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 93, pp. 1-23.
Title: On a nonlinear degenerate parabolic transport-diffusion
equation with a discontinuous coefficient
Authors: Kenneth H. Karlsen (Univ. of Bergen, Norway)
Nils H. Risebro (Univ. of Oslo, Norway)
John D. Towers (MiraCosta College, CA, USA)
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 $\{u_{\varepsilon}\}_{\varepsilon>0}$ 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 ma\nolinebreak{}pping 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