The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

TL;DR

This paper introduces a coupled hydrodynamic model combining pressureless Euler and isentropic Navier-Stokes equations, establishing local and global classical solutions and demonstrating exponential alignment of velocities over time.

Contribution

The paper develops a new coupled system derived from particle-fluid equations, constructs classical solutions, and proves exponential velocity alignment in large-time behavior.

Findings

Existence of local-in-time classical solutions in Sobolev space

Global classical solutions with large-time behavior estimates

Exponential alignment of velocity functions over time

Abstract

We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate $L^2$ Sobolev space. We also establish the \emph{a priori} large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity.