A kinetic model for approximately isentropic solutions of the Euler equations

The purpose of the paper is an investigation of compactness properties of a family of solutions of a BGK-type model for the system of the Euler equations for isentropic flows. The model involves a parameter that measures deviation of a kinetic density from the equilibrium distribution. It is shown that, in the vanishing relaxation time limit, the kinetic densities accumulate on a density, that corresponds an approximately isentropic solution of the Euler equations.