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

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 $P^{\prime }(\bar{\rho })+\gamma \bar{\rho }>0$. 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 $P^{\prime }(\bar{\rho })+\gamma \bar{\rho }<0$ in the Hadamard sense.