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

Wenjun Wang; Weike Wang

doi:10.3934/dcds.2015.35.513

Article Contents

2015, Volume 35, Issue 1: 513-536. Doi: 10.3934/dcds.2015.35.513

This issue Previous Article The singular limit problem in a phase separation model with different diffusion rates $^*$ Next Article Spectrum and amplitude equations for scalar delay-differential equations with large delay

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

Wenjun Wang^1, and
Weike Wang^2,

1.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093
2.
Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai

Received: December 31, 2013

Revised: May 31, 2014

Published: December 2014

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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.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 35M10, 93D20.

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