Stochastic isentropic Euler equations

We study the stochastically forced system of isentropic Euler equations of gas dynamics with a $\gamma$-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section

Stochastic isentropic Euler equations

International audienceWe study the stochastically forced system of isentropic Euler equations of gas dynamics with a γ-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section

A BGK approximation to scalar conservation laws with discontinuous flux

We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem

A model for the evolution of traffic jams in multi-lane

In [7], Berthelin, Degond, Delitala and Rascle introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model consists of a Pressureless Gas Dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. In the present paper we propose an improvement of this model by allowing the road to be multi-lane piecewise. The idea is to use the maximal constraint to modelize the number of lanes. We also add in the model a parameter {\alpha} which modelize the various speed limitations according to the number of lanes. We present the dynamical behaviour of clusters (traffic jams) and by approximation with such solutions, we obtain an existence result of weak solutions for any initial data.Comment: 40 pages, 15 figure

Theoretical study of a multi-dimensional pressureless model with unilateral constraint

International audienceWe extend to multi-dimension the study of a pressureless model of gas system with unilateral constraint. Several difficulties are added with respect to the one-dimensional case. First, the geometry of the dynamics of blocks cannot be conserved and to solve this problem, a splitting with respect to the various directions is done. This leads to approximations of solutions for special initial data. Besides, the stability of the solutions is also quite different from the one-dimensional case. We finaly get the existence and the stability of solutions

Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure

International audienceIn this article, we study a one-dimensional hyperbolic quasi-linear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line \Er and on a bounded interval with no-flux boundary conditions. In the case of the whole line \Er, we find only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an infinite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations

Regularity results for the solutions of a non-local model of traffic

International audienceWe consider a non-local traffic model involving a convolution product. Unlike other studies, the considered kernel is discontinuous on R. We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, $C([0,T],L^2(\R))$, and smooth, $W^{2,2N}([0,T]\times\R)$, solutions for the non-local traffic model

Particle approximation of a constrained model for traffic flow

International audienceWe rigorously prove the convergence of the micro-macro limit for particle approximations of the Aw-Rascle-Zhang equations with a maximal density constraint. The lack of BV bounds on the density variable is supplied by a compensated compactness argument

Des lemmes de moyenne avec un terme de force dans l’équation de transport

International audienc