On the Cauchy problem for the pressureless Euler-Navier-Stokes system in the whole space

TL;DR

This paper proves the global existence and uniqueness of classical solutions for a coupled pressureless Euler and incompressible Navier-Stokes system in the whole space, addressing challenges due to the absence of pressure.

Contribution

It establishes the first global-in-time well-posedness results for this coupled two-phase fluid model with pressureless Euler equations.

Findings

Global existence and uniqueness of classical solutions for small initial data

Effective handling of pressureless Euler equations without pressure term

Analysis of long-time behavior of solutions

Abstract

In this paper, we study the global Cauchy problem for a two-phase fluid model consisting of the pressureless Euler equations and the incompressible Navier-Stokes equations where the coupling of two equations is through the drag force. We establish the global-in-time existence and uniqueness of classical solutions for that system when the initial data are sufficiently small and regular. Main difficulties arise in the absence of pressure in the Euler equations. In order to resolve it, we properly combine the large-time behavior of classical solutions and the bootstrapping argument to construct the global-in-time unique classical solutions.