Entropy Pairs and Compensated Compactness for Weakly Asymmetric Systems

Abstract

The hyperbolic (Euler) scaling limit of weakly asymmetric Ginzburg–Landau models with a single conservation law is investigated, weak asymmetry means that the microscopic viscosity of the system tends to infinity in a prescribed way during the hydrodynamic limit. The system is not attractive, its potential is a bounded perturbation of a quadratic function. The macroscopic equation reads as $\partial_t \rho + \partial_x S'(\rho) = 0$, where $S$ is a convex function. The Tartar - Murat theory of compensated compactness is extended to microscopic systems, we prove weak convergence of the scaled density field to the set of weak solutions. In the attractive case of a convex potential this set consists of the unique entropy solution. Our main tool is the logarithmic Sobolev inequality of Landim, Panizo and Yau for continuous spins.