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

We establish the global-in-time existence of weak solutions to a variant of the BGK model proposed by Bouchut [J. Stat. Phys., 95, (1999), 113–170] which leads to the barotropic Euler equations in the hydrodynamic limit. 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 [SIAM J. Math. Anal., 36, (2005), 1807–1835] completely rigorous.