Global well-posedness and large time behavior of strong solution to a kinetic-fluid model

This paper is concerned with a kinetic-fluid model describing the evolutions of disperse two-phase flows. The model consists of the Vlasov-Fokker-Planck equations for the particles (disperse phase) coupled with the compressible Navier-Stokes equations for the fluid (fluid phase) through the friction force. Our friction force depends on the density, which is different from many previous studies on kinetic-fluid models. The global well-posedness of strong solution in the three-dimensional whole space is established when the initial data is a small perturbation of some given equilibrium. Moreover, the algebraic rate of convergence of solution toward the equilibrium state is obtained. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential.