THE CAUCHY PROBLEM FOR A COUPLING SYSTEM OF VLASOV-FOKKER-PLANCK/COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper is concerned with the Cauchy problem for the compressible fluid-particle interaction system in R3, which is based on a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Navier-Stokes equations for a compressible fluid. Global well-posedness of the Cauchy problem is proved in H-N (R-3)-framework (N >= 2), and optimal decay rates of all order spatial derivatives of the solution toward equilibrium are established where we only assume that the H2-norm of initial data is small. The proof is accomplished by virtue of refined energy estimates and a new observation for decay properties of low-frequency and high-frequency quantities.