# American Institute of Mathematical Sciences

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November  2009, 12(4): 783-796. doi: 10.3934/dcdsb.2009.12.783

## Mixed entropy estimates for the porous-medium equation with convection

 1 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien 2 Laboratoire Paul Painlevé, CNRS-UMR 8524, Université Lille 1, Cité scientifique, 59655 Villeneuve d'Ascq, France

Received  September 2008 Revised  March 2009 Published  August 2009

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.
Citation: Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783
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