The wonderful world of entropy-entropy dissipation techniques for nonlinear diffusion equations
by
Ansgar Jüngel
Vienna University of Technology
Coauthors: Daniel Matthes

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.

Date received: July 17, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-46.