Global well-posedness and optimal time decay rates of solutions to the pressureless Euler-Navier-Stokes system

TL;DR

This paper establishes the global well-posedness and optimal decay rates for a coupled pressureless Euler and Navier-Stokes system, extending classical results to a two-phase flow model with novel decay analysis techniques.

Contribution

It introduces a new framework for analyzing the two-phase flow system, providing optimal decay rates and handling the absence of pressure in the Euler equations.

Findings

Proved global well-posedness of the system.

Derived optimal decay rates for solutions.

Generalized classical Navier-Stokes results to two-phase flows.

Abstract

In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force. To overcome the difficulties arising from the absence of the pressure term in the Euler equations, we establish the time decay estimates of the high-order derivative of the velocity to obtain uniform estimates of the fluid density. The upper bound decay rates are obtained by designing a new functional and the lower bound decay rates are achieved by selecting specific initial data. Moreover, the upper bound decay rates are the same order as the lower one. Therefore, the time decay rates are optimal. When the fluid density in the pressureless Euler flow vanishes, the system is reduced into an incompressible Navier-Stokes flow. In this case, our works coincide with the classical results by Schonbek \cite{M.S3} [JAMS,1991], which can be regarded as a generalization from a single fluid model to the two-phase fluid one.