Global well-posedness and large time behavior of classical solutions to the Vlasov-Fokker-Planck and magnetohydrodynamics equations

We are concerned with the global well-posedness of the fluid-particle system which describes the evolutions of disperse two-phase flows. The system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the compressible magnetohydrodynamics equations modelling a dense phase (fluid) through the friction forcing. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution and decay rate of the solution are proved based on the classical energy estimates and Fourier multiplier technique, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between particle and fluid.