Abstract:We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equationwhere 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

u _{t}+ (g(x)f(u))_{x}= A(u)_{xx}, A'(·) > = 0,_{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.

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