Jozsef Fritz, Balint Toth
Derivation of the Leroux system as the hydrodynamic limit
of a two-component lattice gas
(476K, pdf)
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, \emph{ even
beyond the appearence 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 earlier
work of the first author the stochastic version of
Tartar--Murat theory of compensated compactness is
extended to two-component stochastic models.