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Asymptotic stability of rarefaction wave for the compressible Navier‐Stokes‐Korteweg equations in the half space
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 3 / June 2022
- Published online by Cambridge University Press:
- 23 July 2021, pp. 756-779
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- June 2022
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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.
On the existence of global weak solutions to an integrable two-component Camassa–Holm shallow-water system
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 56 / Issue 3 / October 2013
- Published online by Cambridge University Press:
- 28 June 2013, pp. 755-775
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