Global existence and decay rates of solutions for Vlasov–Navier–Stokes–Fokker–Planck equations with magnetic field

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In this paper, we consider the two-phase flow model coupled by the incompressible magnetohydrodynamic equations and the discrete-phase (particle) dynamics equation Vlasov–Fokker–Planck in [Formula: see text]. When the initial value is a small perturbation of an equilibrium state, the global well-posedness of the strong solution in the three-dimensional global space is established by using the optimal decay estimation and energy estimation. In particular, we are able to show the algebraic convergence rate of the solution to the equilibrium state. While for periodic domain, the same result still holds, but the rate of convergence is exponential.