Preprint 2001-013

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

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 {u_e}_{e > 0} of smooth approximations solving (P) with the transport flux g(x)f(·) replaced by g_e(x)f(·) and the diffusion function A(·) replaced by A_e(·), where g_e(·) is smooth and A_e'(·) > 0. The main technical challenge is to deal with the fact that the total variation |u_e|_BV cannot be bounded uniformly in e, and hence one cannot derive directly strong convergence of {u_e}_{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.

Paper:

Available as PostScript (287 Kbytes) or gzipped PostScript (89 Kbytes; uncompress using gunzip).

Author(s):

Kenneth H. Karlsen , <kennethk@mi.uib.no>

Nils H. Risebro, <nilshr@math.uio.no>

John D. Towers, <jtowers@cts.com>

Publishing information:

Comments:

Submitted by:

<nilshr@math.uio.no> April 5 2001.