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Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas

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Abstract

The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, even beyond the appearance of shocks. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following [4, 5], the stochastic version of Tartar–Murat theory of compensated compactness is extended to two-component stochastic models.

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  1. Institute of Mathematics, Budapest University of Technology and Economics, 1111 Budapest, Hungary

    József Fritz & Bálint Tóth

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  1. József Fritz
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  2. Bálint Tóth
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Correspondence to József Fritz.

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Communicated by H.-T. Yau

Supported in part by the Hungarian Science Foundation (OTKA), grants T26176 and T037685.

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Fritz, J., Tóth, B. Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas. Commun. Math. Phys. 249, 1–27 (2004). https://doi.org/10.1007/s00220-004-1103-x

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  • DOI: https://doi.org/10.1007/s00220-004-1103-x

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