Global classical solutions and large-time behavior of the two-phase fluid model

TL;DR

This paper proves the global existence and analyzes the large-time behavior of solutions for a coupled two-phase fluid model, demonstrating exponential alignment of fluid velocities under small initial data.

Contribution

It establishes the existence of unique global strong solutions for the two-phase fluid system and characterizes their exponential large-time alignment behavior.

Findings

Global strong solutions exist for small initial data.

Fluid velocities align exponentially fast over time.

A Lyapunov function quantifies the convergence.

Abstract

We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov-Fokker-Planck/isentropic Navier-Stokes equations with strong local alignment forces. When the initial data is sufficiently small and regular, we establish the unique existence of the global $H^s$-solutions in a perturbation framework. We also provide the large-time behavior of classical solutions showing the alignment between two fluid velocities exponentially fast as time evolves. For this, we construct a Lyapunov function measuring the fluctuations of momentum and mass from its averaged quantities.