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

TL;DR

This paper proves the global existence, decay rates, and stability of strong solutions for a coupled fluid-particle model with density-dependent friction, using energy estimates and frequency analysis.

Contribution

It establishes the first global well-posedness and decay results for a strongly coupled compressible fluid-particle system with density-dependent friction.

Findings

Global strong solutions exist under small initial data in H^2 norm.

Optimal decay rates are achieved if initial data has bounded L^1 norm.

Solutions decay exponentially in the periodic domain case.

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.