The incompressible inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations: global well-posedness and inviscid limit

TL;DR

This paper proves the global stability and well-posedness of a coupled fluid-particle system described by Navier-Stokes and Vlasov-Fokker-Planck equations, and rigorously justifies the inviscid limit with sharp convergence rates.

Contribution

It establishes the global well-posedness, uniform regularity, and optimal convergence rates for the inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck system, including the inviscid limit to Euler equations.

Findings

Proved global stability of equilibrium with Besov regularity.

Established uniform regularity estimates independent of viscosity.

Justified the inviscid limit with sharp convergence rates.

Abstract

The global well-posedness and inviscid limit are investigated for the fluid-particle interaction system, described by the Navier-Stokes equations for the inhomogeneous incompressible viscous flows coupled with the Vlasov-Fokker-Planck equation for particles through a density-dependent nonlinear friction force in three-dimensional space. It is challenging to establish the inviscid limit over large time periods for the incompressible Euler equations under the influence of the weak dissipative mechanism generated by the friction force. We first prove the global stability of the equilibrium, in the sense that initial perturbations with appropriate Besov spatial regularity lead to global well-posedness and uniform regularity estimates with respect to the viscosity coefficient for strong solutions of the inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations. In particular, we establish the optimal rates of convergence to equilibrium uniformly in Navier-Stokes. Then, we construct global solutions to the inhomogeneous Euler-Fokker-Planck equations via the vanishing viscosity limit. Furthermore, by capturing the dissipation arising from two-phase interactions, we rigorously justify the global-in-time strong convergence of the inviscid limit process, with a convergence rate that is in sharp contrast to that in the pure incompressible fluid case. To achieve this global convergence, novel ideas and new techniques are developed in the analysis and may be applied to other significant problems.