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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)∂tρ+∂xF(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 $L1$-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 λ(x), in the hydrodynamic limit, formally lead to scalar hyperbolic conservation laws with discontinuous fluxes of the form:

(0.2)∂tρ+∂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).

Paper:
Available as PDF (276 Kbytes).
Author(s):
Gui-Qiang Chen
Nadine Even
Christan Klingenberg,
Publishing information:
Comments:
Submitted by:
; 2007-11-23.