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D. Matthes:
"Calculating entropies for higher-order diffusive PDEs";
Vortrag: Fakultätskolloquium der Universität Zagreb, Zagreb, Kroatien (eingeladen); 09.09.2008.

Siehe englischen Abstract.

Nonlinear parabolic equations of fourth (and higher) order are attracting a lot of attention recently. As prominent examples, we mention the thin film equation and the (logarithmic) quantum diffusion equation. Since maximum and comparision principles generally do not apply to the solutions, suitable Lyapunov functionals seemingly provide the only possibility to obtain a priori estimates, prove smoothness results and calculate long-time decay rates.

A systematic approach to determine Lyapunov functionals of entropy type has been recently developed by A. Juengel and the speaker. Assuming a special form of the potential entropy functionals, the proof of entropy dissipation reduces to verifying positivity of a certain integral expression, applying integration by parts. Thus the original analytical problem is translated into a real-algebraic one, which can -- in principle -- be solved algorithmically.

In this talk, the method is explained in detail for equations in one spatial variable, where the algebra can be carried out more or less explicitly. Then, an extension of the method to equations on multi- dimensional domains is presented. It will be shown how to formulate the algebraic problem in a fixed number of polynomial variables, independently of the dimension of the spatial domain. This allows to determine entropy functionals efficiently, using suitable software.

Entropy method, higher-order PDEs, diffusion equations

Projektleitung Ansgar Jüngel:
Entropie-Entropiedissipationsmethoden für nichtlineare partielle Differentialgleichungen höherer Ordnung (Postdoc-Stelle für Herrn Dr. Daniel Matthes)