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Asymptotic stability of a nonlinear wave for the compressible Navier–Stokes–Korteweg equations in the half space

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Abstract

This paper is concerned with the time-asymptotic stability of a nonlinear wave for the outflow problem of the isentropic compressible Navier–Stokes–Korteweg equations in the half space. Under some suitable assumptions on the spatial-asymptotic states and boundary data, the time-asymptotic profile is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the \(L^{2}\)-energy method and the decay (in both time and space variables) estimates of each component of the nonlinear wave, we prove that this nonlinear wave is time asymptotically stable under a small initial perturbation.

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Acknowledgements

We are grateful to the anonymous referees for valuable comments which greatly improved our original manuscript. Li is supported in part by the National Natural Science Foundation of China (Grant No. 12171258). Chen is supported in part by the National Natural Science Foundation of China (Grant No. 12171001) and the Support Program for Outstanding Young Talents in Universities of Anhui Province (Grant No. gxyqZD2022007).

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Authors and Affiliations

  1. School of Sciences, Nantong University, Nantong, 226019, People’s Republic of China

    Yeping Li & Rui Xu

  2. School of Mathematical Sciences, Anhui University, Hefei, 230601, People’s Republic of China

    Zhengzheng Chen

  3. Centre of Pure Mathematics, School of Mathematical Sciences, Anhui University, Hefei, 230601, People’s Republic of China

    Zhengzheng Chen

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  1. Yeping Li
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Li, Y., Xu, R. & Chen, Z. Asymptotic stability of a nonlinear wave for the compressible Navier–Stokes–Korteweg equations in the half space. Z. Angew. Math. Phys. 74, 167 (2023). https://doi.org/10.1007/s00033-023-02064-z

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