Global existence and time decay rate of classical solutions to a hybrid Vlasov-Fokker-Planck-MHD equations

In this paper, we prove the existence of global classical solutions to a kinetic-fluid system when initial data is a small perturbation of some given equilibrium state in $R^3$. The system consists of the Vlasov-Fokker-Planck equation coupled with the compressible magnetohydrodynamics (MHD) equations via the nonlinear coupling terms of Lorenz force type. It describes the motion of energetic particles in a fluid with a magnetic field. The proof of global existence mainly relies on the energy method. Due to the complex nonlinear structure of Lorentz force, we need to establish a more refined uniform a prior estimates. Moreover, under additional conditions on initial data, the optimal time decay rate of solutions toward the equilibrium state can be obtained by using the Fourier analysis.