  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.
