Hyperbolic Conservation Laws and Hydrodynamic Limit for Particle Systems

TL;DR

This paper investigates scalar hyperbolic conservation laws with discontinuous fluxes, establishing entropy conditions, well-posedness, and hydrodynamic limits for particle systems with discontinuous speeds, bridging microscopic models and macroscopic PDEs.

Contribution

It identifies entropy conditions for discontinuous flux PDEs, proves well-posedness, and establishes hydrodynamic limits for particle systems with discontinuous speed parameters.

Findings

Established entropy condition for discontinuous flux PDEs

Proved well-posedness combining existence and uniqueness results

Demonstrated hydrodynamic limit for zero-range process with discontinuous speeds

Abstract

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.