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
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.
Received: 6 January 2003
Revised: 3 June 2003
First available in Project Euclid: 1 January 2019
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Zentralblatt MATH identifier
Secondary: 82C22: Interacting particle systems [See also 60K35]
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