Strong Solutions to the Compressible Navier--Stokes--Vlasov--Fokker--Planck Equations: Global Existence Near the Equilibrium and Large Time Behavior

Abstract

A kinetic-fluid model describing the evolutions of disperse two-phase flows is considered. The model consists of the Vlasov-Fokker-Planck equation for the particles (disperse phase) coupled with the compressible Navier-Stokes equations for the fluid (fluid phase) through the friction force. The friction force depends on the density, which is different from many previous studies on kinetic-fluid models and is more physical in modeling but significantly more difficult in analysis. New approach and techniques are introduced to deal with the strong coupling of the fluid and the particles. 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.