Global existence of weak solutions for Navier-Stokes-BGK system

TL;DR

This paper proves the global existence of weak solutions for a coupled Navier-Stokes-BGK system, combining kinetic and fluid models, using advanced compactness and moment estimates.

Contribution

It establishes the first rigorous proof of global weak solutions for the Navier-Stokes-BGK system with a novel combination of compactness techniques.

Findings

Proved global existence of weak solutions for the coupled system.

Developed new compactness methods for kinetic and fluid components.

Extended mathematical understanding of kinetic-fluid interactions.

Abstract

In this paper, we study the global well-posedness of a coupled system of kinetic and fluid equations. More precisely, we establish the global existence of weak solutions for Navier-Stokes-BGK system consisting of the BGK model of Boltzmann equation and incompressible Navier-Stokes equations coupled through a drag forcing term. This is achieved by combining weak compactness of the particle interaction operator based on Dunford-Pettis theorem, strong compactness of macroscopic fields of the kinetic part relied on velocity averaging lemma and a high order moment estimate, and strong compactness of the fluid part by Aubin-Lions lemma.