Global existence and large time behavior of weak solutions to the two-phase flow

TL;DR

This paper proves the global existence of weak solutions and their large-time behavior for a complex two-phase flow model involving coupled compressible Navier-Stokes systems with different viscosities.

Contribution

It establishes the existence of global weak solutions for a two-phase flow model with degenerate and constant viscosities and analyzes their convergence to equilibrium.

Findings

Global weak solutions exist for the model.

Solutions converge to equilibrium as time approaches infinity.

The results are valid for general initial data in a 3D periodic domain.

Abstract

In this paper, we consider a two-phase flow model consisting of the compressible Navier-Stokes systems with degenerate viscosity coupled with the compressible Navier-Stokes systems with constant viscosities via a drag force, which can be derived from Chapman-Enskog expansion for the compressible Navier-Stokes-Vlasov-Fokker-Planck system. For general initial data, we establish the global existence of weak solutions with finite energy to the initial value problem in the three-dimensional periodic domain, and prove the convergence of global weak solutions to its equilibrium state as the time tends to infinity.