Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

The relative entropy method in hyperbolic and diffusive systems

Athanasios Tzavaras

University of Crete

The relative entropy method is an efficient tool for comparing thermomechanical theories, In this talk we will review various uses of the method in first a context of hyperbolic conservation laws and then in contexts where a diffusion theory is approximated. Some examples of applications will be highlighted. These include convergence of a chain of spring-mass systems to the equations of nonlinear elasticity in the smooth regime, convergence from the equations of gas-dynamics with friction to the porous media equation. I will also present a framework for convergence from gas dynamics with friction to gradient flows describing diffusion theories. The latter will be applied to the convergence from Euler-Poisson with attracting potential to Keller-Segel type systems. Also, for convergence from the Korteweg theory with friction to Cahn-Hilliard type equations.