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.
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