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We study the following class of scalar hyperbolic conservation laws withdiscontinuous fluxes: \partial_t\rho+\partial_xF(x,\rho)=0. The main feature ofsuch a conservation law is the discontinuity of the flux function in the spacevariable x. Kruzkov's approach for the L1-contraction does not apply since itrequires the Lipschitz continuity of the flux function; and entropy solutionseven for the Riemann problem are not unique under the classical entropyconditions. On the other hand, it is known that, in statistical mechanics, somemicroscopic interacting particle systems with discontinuous speed parameterlambda(x), in the hydrodynamic limit, formally lead to scalar hyperbolicconservation laws with discontinuous fluxes of the form:\partial_t\rho+\partial_x(\lambda(x)h(\rho))=0. The natural question ariseswhich entropy solutions the hydrodynamic limit selects, thereby leading to asuitable, physical relevant notion of entropy solutions of this class ofconservation laws. This paper is a first step and provides an answer to thisquestion for a family of discontinuous flux functions. In particular, weidentify the entropy condition for our PDE and proceed to show thewell-posedness by combining our existence result with a uniqueness result ofAudusse-Perthame (2005) for the family of flux functions; we establish acompactness framework for the hydrodynamic limit of large particle systems andthe convergence of other approximate solutions to our PDE, which is based onthe notion and reduction of measure-valued entropy solutions; and we finallyestablish the hydrodynamic limit for a ZRP with discontinuous speed-parametergoverned by an entropy solution to our PDE.

Hyperbolic Conservation Laws and Hydrodynamic Limit for Particle Systems