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A. Jüngel:
"Nichtlineare partielle Differentialgleichungen: Analysis meets Algebra";
Talk: Wissenwertes aus der Mathematik, TU Wien; 2007-12-10.

Many dynamical problems from physics give rise to a Lyapunov functional
which is nonincreasing in time. If diffusion or dissipation is involved,
this leads to functional inequalities which often allow to conclude
convergence to equilibrium. Since the physical entropy is usually
a special case of such functionals, we call these inequalities
entropy--entropy dissipation inequalities. In recent years, they
have been much studied since they allow for a deep understanding of
the dynamics of the solutions of the partial differential equations.
Moreover, there are surprising connections to mass transportation
theory and kinetic equations. A wide range of diffusion equations
can be treated, for instance cross-diffusion systems and higher-order
equations like the thin-film equation.
In this talk it will be explained how entropy--entropy dissipation
techniques may be used in the existence analysis, the study of
the long-time behavior of the solutions, and for the derivation of
new inequalities of logarithmic Sobolev type. Furthermore, a new
method of deriving entropy inequalities in an algorithmic way will
be introduced. The idea is to reformulate the integration by parts
as a decision problem for polynomial systems which can be solved in
principle by computer algebra systems.

Siehe englisches Abstract.

Entropy, fourth-order partial differential equations, decision problems, quantifier elimination