Global existence and large time behavior of classical solutions to the incompressible inhomogeneous Euler–Vlasov–Fokker–Planck system

This paper is concerned with a kinetic-fluid model, which contains a Vlasov–Fokker–Planck equation for particles and an incompressible inhomogeneous Euler equations for a fluid. The two parts in the model twist together via the friction force which reveals the inhibition of the fluid on the motion of particles. Consequentially, the particles provide a damping effect on the motion of the fluid. The global existence and uniqueness classical solutions near the equilibrium are established in a [Formula: see text]-D torus. Moreover, the exponential convergence rate of the solution is given. The proofs are mainly based on the macro–micro decomposition and the classical energy estimates. And there are two main novelties in our arguments: on the one hand, our friction force depends on the fluid density [Formula: see text], which is more physical in modeling but more difficult in computations than the density-independent case. On the other hand, we introduce a new energy method to enclose the a priori estimates. More precisely, we derive the dissipation rate on the gradient of the pressure [Formula: see text] instead of the density [Formula: see text] in [Formula: see text] norm to enclose the estimates [Formula: see text]. We then apply the equation of [Formula: see text] and Gronwall’s inequality to obtain the estimates of [Formula: see text].