Entropy methods for nonlinear parabolic systems: mathematical and numerical analysis

Entropy methods are well-known in the analysis of hyperbolic equations. Recently, so-called entropy dissipation methods have been introduced for the analysis of the qualitative behavior of solutions to parabolic equations. The entropy method gives additional a priori estimates for the solutions of the corresponding equations.

In this talk we present by examples the principal ideas of these entropy methods. In particular we consider strongly nonlinear parabolic systems and higher-order parabolic equations. For these problems maximum principles are not available. The entropy method will be essential to prove (i) existence of solutions, (ii) explicit long-time decay rates of solutions, (iii) stability of certain numerical schemes, (iv) asymptotic limits, and (v) uniqueness of solutions.