The Navier-Stokes-Vlasov-Fokker-Planck System in Bounded Domains

This paper is concerned with the initial boundary value problem of the Vlasov-Fokker-Planck equation coupled with either the incompressible or compressible Navier-Stokes equations in a bounded domain. The global existence of unique strong solution and its exponential convergence rate to the equilibrium state are proved under the Maxwell boundary condition for the incompressible case and specular reflection boundary condition for the compressible case, respectively. For the compressible model, to overcome the lack of regularity due to the coupling with the kinetic equation in a bounded domain, an essential L-10/3 estimate is analyzed so that the a priori estimate can be closed by applying the S-L(P) theory developed by Guo et al. for kinetic models, [Arch Ration Mech Anal 236(3): 1389-1454 (2020)].