# Preprint 2007-026

# Hyperbolic Conservation Laws with Discontinuous Fluxes and Hydrodynamic Limit for Particle Systems

## Gui-Qiang Chen, Nadine Even and Christan Klingenberg

**Abstract:**
We study the following class of scalar hyperbolic
conservation laws with discontinuous fluxes:

(0.1) ∂_{t}*ρ*+∂_{x}F(*x*,*ρ*)=0.

The main feature of such a conservation law is the
discontinuity of the flux function in the space
variable *x*.
Kruzkov's approach (1970) for the
$*L*^{1}$-contraction
does not apply since it requires the Lipschitz continuity
of the flux function;
and entropy solutions even for the Riemann problem are not
unique under the classical entropy conditions.
On the other hand, it is known that, in statistical
mechanics, some microscopic interacting particle systems
with discontinuous speed parameter
*λ*(*x*),
in the hydrodynamic limit,
formally lead to scalar hyperbolic conservation laws with
discontinuous fluxes of the form:

(0.2) ∂_{t}*ρ*+∂_{x}(*λ*(*x*)*h*(*ρ*))=0.

The natural question arises which entropy solutions the hydrodynamic limit selects, thereby leading to a suitable, physical relevant notion of entropy solutions of this class of conservation laws. This paper is a first step and provides an answer to this question for a family of discontinuous flux functions. In particular, we identify the entropy condition for (0.1) and proceed to show the well-posedness by combining our existence result with a uniqueness result of Audusse–Perthame (2005) for the family of flux functions; we establish a compactness framework for the hydrodynamic limit of large particle systems and the convergence of other approximate solutions to (0.1), which is based on the notion and reduction of measure-valued entropy solutions; and we finally establish the hydrodynamic limit for a ZRP with discontinuous speed-parameter governed by an entropy solution to (0.2).