Time periodic solution to the two-phase fluid model with an external force in a periodic domain

The time periodic solution to a two-phase fluid system is considered, which is consisted of the compressible isothermal Euler equations coupled with the compressible isentropic Navier-Stokes equations through a drag forcing term. This model was first derived by taking the hydrodynamic limit from the Vlasov-Fokker-Planck/isentropic Navier-Stokes equations with strong local alignment forces. Based on regularized approximation and the topological degree theory, we obtain the existence of time periodic solutions under some smallness and structure assumptions imposed on the time periodic force in the periodic domain. Moreover, by energy methods, The uniqueness of the time periodic solution is proved.