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A. Jüngel, I. Violet:
"Mixed entropy estimates for the porous-medium equation with convection";
Discrete and Continuous Dynamical Systems B, 12 (2009), 4; 783 - 796.

In this paper, we answer the question under which conditions the
porous-medium 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.

Siehe englisches Abstract.

Entropy methods; slow diffusion; first-order entropies