Global weak solutions for compressible Navier-Stokes-Vlasov-Fokker-Planck system

TL;DR

This paper proves the existence and uniqueness of global weak solutions for a one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent coefficients, and analyzes their long-term behavior.

Contribution

It establishes the first comprehensive results on global weak solutions and their asymptotic convergence for this coupled kinetic-fluid system with variable coefficients.

Findings

Distribution function converges to Maxwellian over time

Fluid and particle velocities synchronize asymptotically

Global weak solutions exist and are unique for general initial data

Abstract

The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper. The existence, uniqueness, and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain. Moreover, the long time behavior of the weak solution is analyzed. It is shown that as the time grows, the distribution function of the particles converges to the global Maxwellian, and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.