Global well-posedness and inviscid limit of the compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent friction force

TL;DR

This paper proves the global existence, uniform estimates, and inviscid limit of solutions for a coupled compressible Navier-Stokes and Vlasov-Fokker-Planck system, revealing a stabilizing kinetic effect and a new relaxation mechanism.

Contribution

It establishes the first global classical solutions for the compressible Euler-Vlasov-Fokker-Planck system with detailed decay rates and inviscid limit analysis, highlighting the impact of particle interactions.

Findings

Global classical solutions exist near equilibrium.

Inviscid limit with explicit convergence rate.

Microscopic components decay faster than macroscopic ones.

Abstract

This paper investigates the global dynamics of a three-dimensional fluid-particle interaction system that couples the compressible barotropic Navier-Stokes equations with the Vlasov-Fokker-Planck equation through a density-dependent friction force. The study establishes the global well-posedness, uniform-in-viscosity estimates, the global inviscid limit, and optimal large-time decay rates for classical solutions near equilibrium. First, for initial perturbations in $H^3$ sufficiently close to equilibrium, regularity estimates that are uniform in the viscosity coefficient are derived, and the existence of global classical solutions to the Cauchy problem is obtained. These uniform bounds enable us to rigorously justify the global-in-time inviscid limit as viscosity vanishes, with an explicit convergence rate proportional to the viscosity coefficient. This behavior differs significantly from that of the pure compressible Navier-Stokes system in the absence of particle interactions, emphasizing the stabilizing influence of kinetic coupling. Consequently, we establish for the first time the global existence of classical solutions to the compressible Euler-Vlasov-Fokker-Planck system. Moreover, under an additional mild assumption on the initial data, optimal time decay rates for both the solution and its spatial derivatives are obtained. Notably, the dissipative and microscopic components decay at a rate half an order faster than the macroscopic solution itself, indicating a novel relaxation mechanism induced by fluid-particle interactions. The analysis introduces new energy and dissipation structures for the coupled system, overcoming substantial difficulties arising from fluid-particle interactions.