Enhanced dissipation and temporal decay in the Euler-Poisson-Navier-Stokes equations

TL;DR

This paper studies the long-term behavior of solutions to a coupled fluid model involving Euler-Poisson and Navier-Stokes equations, revealing enhanced dissipation effects and specific decay rates for density and velocity in three dimensions.

Contribution

It demonstrates the global well-posedness and decay rates of solutions, utilizing dissipation from the Poisson equation and drag force effects, with new insights into the decay behavior of the coupled system.

Findings

Fluid density decays at rate (1+t)^{-11/4}

Velocity differences decay at rate (1+t)^{-7/4}

Velocities decay at rate (1+t)^{-3/4}

Abstract

This paper investigates the global well-posedness and large-time behavior of solutions for a coupled fluid model in $\mathbb{R}^3$ consisting of the isothermal compressible Euler-Poisson system and incompressible Navier-Stokes equations coupled through the drag force. Notably, we exploit the dissipation effects inherent in the Poisson equation to achieve a faster decay of fluid density compared to velocities. This strategic utilization of dissipation, together with the influence of the electric field and the damping structure induced by the drag force, leads to a remarkable decay behavior: the fluid density converges to equilibrium at a rate of $(1+t)^{-11/4}$, significantly faster than the decay rates of velocity differences $(1+t)^{-7/4}$ and velocities themselves $(1+t)^{-3/4}$ in the $L^2$ norm. Furthermore, under the condition of vanishing coupled incompressible flow, we demonstrate an exponential decay to a constant state for the solution of the corresponding system, the damped Euler-Poisson system.