A. Jüngel, I. Violet:

"Mixed entropy eatimates for the porous-medium equation with convection";

in: "ASC Report 33/2008", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2008, ISBN: 978-3-902627-01-8.

In this paper, we answer the question under which conditions the porousmedium

equation with convection and with periodic boundary conditions possesses gradient-

type Lyapunov functionals (first-order entropies). It is shown that the weighted

sum of first-order and zeroth-order entropies are Lyapunov functionals if the weight for

the zeroth-order entropy is sufficiently large, depending on the strength of the convection.

This provides new a priori estimates for the convective porous-medium equation. The

proof is based on an extension of the algorithmic entropy construction method which is

based on systematic integration by parts, formulated as a polynomial decision problem.

http://www.asc.tuwien.ac.at/preprint/2008/asc33x2008.pdf

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