No abstract

We consider a two-phase model described by a pressureless gas system with unilateral constraint. We prove weak stability and the existence of weak solutions by passing to the limit in the sticky-blocks dynamics. We obtain the maximum principle on the velocity, the Oleinik entropy condition and local entropy inequalities. Initial data are taken in a very weak sense since the solution can jump initially in time.

This article is devoted to the proof of the hydrodynamical limit from kinetic equations (including BGK-like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method; hence the derivation is valid only before shocks appear on the limit system solution. However, no a priori knowledge on high velocity distributions for kinetic functions is needed. The case of the Saint-Venant system with topography (where a source term is added) is included.

Recently, Berthelin et al [5] introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper we propose a more realistic model namely the Second Order Model with Constraint (SOMC model), derived from the Aw & Rascle model [1] abd which takes into this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill & Whitham model [17]. We prove an existence result of weak solutions for this model by means of cluster dynamics in order to construct a sequence of approximations and we solve completely the associated Riemann problem.