Compensated compactness and relaxation at the microscopic level

Fritz, József (2012) Compensated compactness and relaxation at the microscopic level Annales Mathematicae et Informaticae. 39. pp. 83-108. ISSN 1787-5021 (Print), 1787-6117 (Online)

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This is a survey of some recent results on hyperbolic scaling limits. In contract to diffusive models, the resulting Euler equations of hydrodynamics develop shocks in a finite time. That is why the derivation of the macroscopic equations from a microscopic model requires a synthesis of probabilistic and PDE methods. In the case of two-component stochastic models with a hyperbolic scaling law the method of compensated compactness seems to be the only tool that we can apply. Since the associated Lax entropies are not preserved by the microscopic dynamics, a logarithmic Sobolev inequality is needed to evaluate entropy production. Extending the arguments of Shearer (1994) and Serre–Shearer (1994) to stochastic systems, the nonlin- ear wave equation of isentropic elastodynamics is derived as the hyperbolic scaling limit of the anharmonic chain with Ginzburg–Landau type random perturbations. The model of interacting exclusion of charged particles results in the Leroux system in a similar way. In the presence of an additional creation-annihilation mechanism the missing logarithmic Sobolev inequality is replaced by an associated relaxation scheme. In this case the uniqueness of the limit is also known.

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Megjegyzés: Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics University of Debrecen August 22–24, 2011 - Debrecen, Hungary
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Kulcsszavak: Anharmonic chain, Ginzburg–Landau model, interacting exclusions, creation and annihilation, hyperbolic scaling, vanishing viscosity limit, logarithmic Sobolev inequalities, Lax entropy pairs, compensated compactness, relaxation schemes
Nyelv: angol
Kötetszám: 39.
ISSN: 1787-5021 (Print), 1787-6117 (Online)
Felhasználó: Tibor Gál
Dátum: 07 Már 2019 16:19
Utolsó módosítás: 07 Már 2019 16:19
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