A revisit to the pressureless Euler--Navier--Stokes system in the whole space and its optimal temporal decay

TL;DR

This paper establishes a refined global well-posedness framework and optimal decay rates for solutions to the pressureless Euler--Navier--Stokes system, revealing that velocities decay similarly to heat equations and differences decay even faster.

Contribution

It introduces a new approach to prove global existence and uniqueness without large-time estimates and identifies optimal decay rates for solutions and their differences.

Findings

Solutions decay at the rate of heat equations.

Velocity differences decay faster than velocities.

Global well-posedness is achieved without a priori large-time estimates.

Abstract

In this paper, we present a refined framework for the global-in-time well-posedness theory for the pressureless Euler--Navier--Stokes system and the optimal temporal decay rates of certain norms of solutions. Here the coupling of two systems, pressureless Euler system and incompressible Navier--Stoke system, is through the drag force. We construct the global-in-time existence and uniqueness of regular solutions for the pressureless Euler--Navier--Stokes system without using a priori large time behavior estimates. Moreover, we seek for the optimal Sobolev regularity for the solutions. Concerning the temporal decay for solutions, we show that the fluid velocities exhibit the same decay rate as that of the heat equations. In particular, our result provides that the temporal decay rate of difference between two velocities, which is faster than the fluid velocities themselves, are at least the same as the second-order derivatives of fluid velocities.