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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