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Asymptotic stability of rarefaction wave for the compressible Navier‐Stokes‐Korteweg equations in the half space

Published online by Cambridge University Press:  23 July 2021

Yeping Li
Affiliation:
School of Sciences, Nantong University, Nantong 226019, P.R. China [email protected]
Jing Tang
Affiliation:
School of Sciences, Nantong University, Nantong 226019, P.R. China [email protected]
Shengqi Yu
Affiliation:
School of Sciences, Nantong University, Nantong 226019, P.R. China [email protected]

Abstract

In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$-energy method and some time-decay estimates in $L^{p}$-norm for the smoothed rarefaction wave.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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