New Trends in Quantum and Classical Kinetic Equations and Related PDEs

Relative entropy for the Euler-Korteweg system

Athanasios Tzavaras

University of Crete

We consider a class of abstract Euler flows generated by a variational structure induced by an energy functional. This model admits as examples the Euler-Korteweg system and the Euler-Poisson system. If the functional is convex, the second variation of the functional provides a natural means to measure the distance between two states. Exploiting the variational structure, we develop a relative energy identity. The latter is used to derive various applications like (a) stability in the case of monotone or even non-monotone pressure laws; (b) convergence in the high-friction limit from Euler-Korteweg to Cahn-Hilliard equations ; (c) convergence to smooth compressible Euler flows in the zero-capillarity limit.