Construction of BGK models with a family of kinetic entropies for a given system of conservation laws

We introduce a general framework for kinetic BGK models. We assume to be gi ven a system of hyperbolic conservation laws with a family of Lax entropies . and we characterize the BGK models that lead to this system in the hydro- dynamic limit, and that are compatible with the whole family of entropies. This is obtained by a new characterization of Maxwellians as entropy minimi zers that can take into account the simultaneous minimization problems corr esponding to the family of entropies. We deduce a general procedure to cons truct such BGK models. and we show how classical examples enter the framewo rk. We apply our theory to isentropic gas dynamics and full gas dynamics, a nd ill both cases we obtain new BGK models satisfying all entropy inequalit ies.