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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 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.

Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy