## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Stochastic Analysis on Large Scale Interacting Systems, T. Funaki and H. Osada, eds. (Tokyo: Mathematical Society of Japan, 2004), 143 - 171

### 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.

#### Article information

**Source***Stochastic Analysis on Large Scale Interacting Systems*, T. Funaki and H. Osada, eds. (Tokyo: Mathematical Society of Japan, 2004), 143-171

**Dates**

Received: 6 January 2003

Revised: 3 June 2003

First available in Project Euclid:
1 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546369037

**Digital Object Identifier**

doi:10.2969/aspm/03910143

**Mathematical Reviews number (MathSciNet)**

MR2073333

**Zentralblatt MATH identifier**

1083.82001

**Subjects**

Primary: 60K31

Secondary: 82C22: Interacting particle systems [See also 60K35]

**Keywords**

Ginzburg–Landau models hyperbolic scaling Lax entropy pairs compensated compactness

#### Citation

Fritz, József. Entropy Pairs and Compensated Compactness for Weakly Asymmetric Systems. Stochastic Analysis on Large Scale Interacting Systems, 143--171, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03910143. https://projecteuclid.org/euclid.aspm/1546369037