Incompressible Navier–Stokes limit from nonlinear Vlasov–Fokker–Planck equation

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system.