Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces

Abstract

In this paper, the compressible Navier-Stokes-Korteweg equations with a potential external force is considered in $\mathbb{R}^3$. Under the smallness assumption on both the external force and the initial perturbation of the stationary solution in some Sobolev spaces, we establish the existence theory of global solutions to the stationary profile. What's more, when the initial perturbation is bounded in $L^p$-norm with $1\leq p<2$, the optimal time decay rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm are shown. On the other hand, when the $\dot{H}^{-s}$ norm $(s\in(0,\frac{3}{2}])$ of the perturbation is finite, we obtain the optimal time decay rates of the solution and its first order derivative in $L^2$-norm.