Abstract:
We introduce a family of vector kinetic BGK equations
leading to isentropic gas dynamics in the relaxation limit, that have only
one entropy at the kinetic level. These models possess the generic structure
of kinetic relaxation models. By a sharp adaptation of averaging lemmas
to BGK models that have a dissipative entropy, we establish an estimate
in the inverse of the square root of the relaxation parameter on the $L^2$
norm of the gradient of the approximations. This estimation is new in the
context of kinetic equations, and it allows, by the method of DiPerna,
to establish the convergence towards weak solutions of isentropic gas dynamics
that satisfy a single entropy inequality.