STICKY PARTICLES AND SCALAR CONSERVATION LAWS ∗ YANN BRENIER † AND EMMANUEL GRENIER † SIAM J. NUMER. ANAL. c  1998 Society for Industrial and Applied Mathematics Vol. 35, No. 6, pp. 2317–2328, December 1998 010 In Memory of Ami Harten Abstract. One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum. Key words. gas dynamics, conservation laws AMS subject classifications. 35L65, 35L67 PII. S0036142997317353 1. Introduction. There has been a recent interest in the one-dimensional model of pressureless gases with sticky particles. This model can be described at a discrete level by a finite collection of particles that get stuck together right after they collide with conservation of mass and momentum. At a continuous level, the gas can be described by a density field and a velocity field ρ(t, x), u(t, x) that satisfy the mass and momentum conservation laws ∂ t ρ + ∂ x (ρu)=0, (1) ∂ t ρu + ∂ x (ρu 2 )=0. (2) This system can be formally obtained from the usual Euler equations for ideal com- pressible fluids by letting the pressure go to zero or from the Boltzmann equation by letting the temperature go to zero. This model of adhesion dynamics is connected to the sticky particle model of Zeldovich [18], [16], which also includes gravitational interactions and has interesting statistical properties. (See [6] as the most complete and recent reference.) For smooth solutions and positive densities, (2) is equivalent to the inviscid Burgers equation ∂ t u + ∂ x  1 2 u 2  =0, (3) which has been studied from a statistical point of view in [15], [14]. However, the reduction to the inviscid Burgers equation is not correct for general data. Bouchut [2] pointed out some mathematical difficulties to get a rigorous derivation of the sticky particle model (1), (2). First, at the continuous level, ρ must be considered as a nonnegative measure, with possible singular parts, and not as a function. Second, * Received by the editors February 24, 1997; accepted for publication (in revised form) October 7, 1997; published electronically November 13, 1998. This work was partly done while the first author was visiting the University of California, Los Angeles, with the support of ARPA/ONR grant N00014-92-J-1980. The research of both authors was partially supported by TMR project HCL ERBFMRXCT960033 of the European Community. http://www.siam.org/journals/sinum/35-6/31735.html † Universit´ e Paris 6 and ENS, DMI, 45 rue d’ULM, 75230 Paris Cedex, France (brenier@dmi.ens.fr, grenier@dmi.ens.fr). 2317