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.