  We study the following class of scalar hyperbolic conservation laws with
discontinuous fluxes: \partial_t\rho+\partial_xF(x,\rho)=0. The main feature of
such a conservation law is the discontinuity of the flux function in the space
variable x. Kruzkov's approach for the L1-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
lambda(x), in the hydrodynamic limit, formally lead to scalar hyperbolic
conservation laws with discontinuous fluxes of the form:
\partial_t\rho+\partial_x(\lambda(x)h(\rho))=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 our PDE 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 our PDE, 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 our PDE.
