Global strong solutions to a compressible fluid-particle interaction model with density-dependent friction force

Abstract

We investigate the Cauchy problem for a fluid-particle interaction model in $\mathbb{R}^3$. This model consists of the compressible barotropic Navier-Stokes equations and the Vlasov-Fokker-Planck equation coupled together via the density-dependent friction force. Due to the strong coupling caused by the friction force, it is a challenging problem to construct the global existence and optimal decay rates of strong solutions. In this paper, by assuming that the $H^2$-norm of the initial data is sufficiently small, we establish the global well-posedness of strong solutions. Furthermore, if the $L^1$-norm of initial data is bounded, then we achieve the optimal decay rates of strong solutions and their gradients in $L^2$-norm. The proofs rely on developing refined energy estimates and exploiting the frequency decomposition method. In addition, for the periodic domain case, our global strong solutions decay exponentially.