Global Classical Solutions and Large-Time Behavior of the Two-Phase Fluid Model

Abstract

We study the global existence of a unique strong solution, and its large-time behavior, of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier--Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with strong local alignment forces. When the initial data are sufficiently small and regular, we establish the unique existence of the global $H^s$-solutions in a perturbation framework. We also provide the large-time behavior of classical solutions showing the alignment between two fluid velocities exponentially fast as time evolves. For this, we construct a Lyapunov function measuring the fluctuations of momentum and mass from its averaged quantities.