Time Periodic Solution to the Two-phase Fluid Model With an External Force in a Periodic Domain

We consider the time periodic solution to a two–phase fluid system consisting 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 Choi [SIAM J. Math. Anal., 48(2016), pp. 3090–3122] by taking the hydrodynamic limit from the Vlasov– Fokker–Planck/isentropic Navier–Stokes equations with strong local alignment forces. In this article, we obtain the existence of time periodic solutions of the regularized problem under some smallness and structure assumptions imposed on the time periodic force. Meanwhile, by a limiting process, the existence of time periodic solutions for the two-phase fluid system can be obtained. Moreover, we also show the uniqueness of the time periodic solution. Our method is based on parabolic regularization, the topological degree theory and energy methods. AMS subject classification. 35B10; 35Q30; 75N10.