Probabilistic Model Associated with the Pressureless Gas Dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we construct a solution to the Riemann problem for the pressureless gas dynamics describing sticky particles. As a bridging step we consider a medium consisting of noninteracting particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case. Moreover, we describe how starting from smooth data a δ singularity arises in one component of the solution. Introduction We propose a method for solving the Riemann problem as well as for describing the formation of singularities for the pressureless gas dynamics system and a natural extension of it. The system of pressureless gas dynamics is very important since it is believed to be the simplest model describing the formation of structures in the universe (e.g.[27]) and plays a significant role in the theory of cooling gases and granular materials [7]. It is a system consisting of two equations for the components of the density f and velocity u expressing the conservation of mass and momentum ∂tf + divx(fu) = 0, (1) ∂t(fu) +∇x(fu⊗ u) = 0. (2) It first appears to be very simple, however a closer analysis reveals that it has some peculiar features due to its non strict hyperbolicity. The system has attracted a significant interest in the last decades and has been investigated quite intensively. In particular, it is well known that the arising in the velocity component of unbounded space derivatives implies the generation of a δ singularity in the component of the density. Therefore for this system one needs to define a generalized or measurevalued solution of a special kind. This was done in [20], [6], [16],[15],[29], [24], [8], where the authors used different techniques (vanishing viscosity, weak asymptotics, variational principle, duality) to define the solution and prove its existence. Further, the Riemann problem for the pressureless gas dynamics was studied (e.g. [29], [16],[28]), including a singular Riemann problem with a δ singularity concentrated 1991 Mathematics Subject Classification. 35L65; 35L67.