Article contents
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
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.
Keywords
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 756 - 779
- Copyright
- Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220511133247770-0360:S0308210521000329:S0308210521000329_inline258.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220511133247770-0360:S0308210521000329:S0308210521000329_inline259.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220511133247770-0360:S0308210521000329:S0308210521000329_inline260.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220511133247770-0360:S0308210521000329:S0308210521000329_inline261.png?pub-status=live)
- 6
- Cited by