Advanced Studies in Pure Mathematics

Entropy Pairs and Compensated Compactness for Weakly Asymmetric Systems

József Fritz

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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


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