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A. Jüngel:
"Entropy-dissipation methods for nonlinear parabolic equations";
Vortrag: Gastvortrag, University of Sussex; 09.01.2014.

Entropy-dissipation methods have been developed recently to investigate
the qualitative behavior of solutions to nonlinear parabolic equations
and to derive explicit or even optimal constants in convex Sobolev
inequalities. The strength of the method lies in its flexibility and
applicability to a large class of nonlinear equations. In this talk,
two aspects of entropy methods will be detailed: the Bakry-Emery
technique and systematic integration by parts.

The Bakry-Emery technique is used to derive estimates on the
exponential decay rate of the relative entropy from equilibrium of
solutions to linear and nonlinear Fokker-Planck equations. As a
by-product of the proof, convex Sobolev inequalities are obtained.
The key idea is to estimate the second time derivative of the
entropy, which makes skillful integrations by parts necessary.
These integrations can be made systematic by formulating the
derivatives as polynomial variables. This procedure leads to a decision
problem in real algebraic geometry, which can be solved in an
algorithmic way. The technique is illustrated for the thin-film and
quantum diffusion equation.