Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain

TL;DR

This paper rigorously analyzes the hydrodynamic limit of a coupled kinetic-fluid system, showing convergence to a two-phase fluid model in a bounded domain using entropy methods and establishing solution existence and uniqueness.

Contribution

It provides the first rigorous derivation of a two-phase fluid model from a coupled Vlasov-Fokker-Planck and Navier-Stokes system in a bounded domain.

Findings

Established global-in-time weak solutions for the coupled system.

Proved convergence to the two-phase fluid model under strong noise.

Demonstrated existence and uniqueness of solutions for the limiting system.

Abstract

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier-Stokes system. Our main strategy relies on the relative entropy argument based on the weak-strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier-Stokes system.