Stability and instability for compressible Navier-Stokes equations with Yukawa potential

TL;DR

This paper investigates the stability and instability of the compressible Navier-Stokes equations with Yukawa potential, establishing conditions for global well-posedness and analyzing the system's behavior under different stability criteria.

Contribution

It extends previous work by analyzing both the well-posedness and instability of the system under various stability conditions, including long-term behavior and blow-up criteria.

Findings

Proved global well-posedness under stability condition $P'(ar ho)+ ext{ extgamma}ar ho>0$.

Established instability results when $P'(ar ho)+ ext{ extgamma}ar ho<0$.

Connected stability analysis with previous local existence and blow-up criteria.

Abstract

In this paper, we first consider global well-posedness and long time behavior of compressible Navier-Stokes equations with Yukawa-type potential in $L^p$-framework under the stability condition $P'(\bar\rho)+\gamma\bar\rho>0$. Here $\bar\rho>0$ is the background density, P is the pressure and $\gamma\in\mathbb{R}$ is Yukawa coefficient. This is a continuity work of Chikami \cite{chikami1} concerning on local existence and blow-up criterion. On the other hand, we study the instability of the linear and nonlinear problem of the system when $P'(\bar\rho)+\gamma\bar\rho<0$ in the Hadamard sense.