Global existence of weak solutions to a BGK model relaxing to the barotropic Euler equations

Abstract

We establish the global-in-time existence of weak solutions to a variant of the BGK model proposed by Bouchut (1999) which leads to the barotropic Euler equations, where the pressure is given by p(ρ)=ργ with γ∈(1,1+2/(d+2)], in the hydrodynamic limit. For the existence of solutions, we deal with γ∈(1,3] for the one-dimensional case, and γ∈(1,1+2/(d+2)]∪{1+2/d} for the multi-dimensional case. In particular, our existence theory makes the quantified estimates of hydrodynamic limit from the BGK-type equations to the multi-dimensional barotropic Euler system discussed by Berthelin and Vasseur (2005) completely rigorous.