Classical solutions to a BGK-type model relaxing to the isentropic gas dynamics

In this paper, we consider a BGK-type kinetic model relaxing to the isentropic gas dynamics in the hydrodynamic limit. We introduce a linearization of the equation around the global equilibrium. Then we prove the global existence of classical solutions with an exponential convergence rate toward the equilibrium state in the periodic domain when the initial data is a small perturbation of the global equilibrium.