First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation

Ansgar Jüngel; Ingrid Violet

doi:10.3934/dcdsb.2007.8.861

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2007, Volume 8, Issue 4: 861-877. Doi: 10.3934/dcdsb.2007.8.861

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First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation

Ansgar Jüngel^1, and
Ingrid Violet^2,

1.
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien
2.
Laboratoire de Mathématiques Appliquées, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière

Received: December 31, 2006

Revised: June 30, 2007

Published: October 2007

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Abstract

A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is analyzed. Using a new semi-discrete approximation in time, a first-order entropy–entropy dissipation inequality is proved. Passing to the limit of vanishing time discretization parameter, some regularity results are deduced. Moreover, it is shown that the solution is strictly positive for large time if it does so initially.

Keywords:

Entropy--entropy dissipation inequality,
existence of weak solutions,
regularity of solutions,
implicit Euler scheme,
entropy construction method,
long-time behavior of solutions.

Mathematics Subject Classification: Primary: 35G25, 35B40, 35B65, 35Q40.

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First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation

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