Abstract
In this paper, we prove the local well-posedness of the viscous surface wave equation in low regularity Sobolev spaces. The key points are to establish several new Stokes estimates depending only on the optimal boundary regularity and to construct a new iteration scheme on a known moving domain. Our method could be applied to some other fluid models with free boundary.
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Acknowledgements
The first author was supported by National Postdoctoral Program for Innovative Talents of China (Grants No. BX201700039). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11571063 and 11771045). The third author was supported by National Natural Science Foundation of China (Grant No. 11425103). The authors are very grateful to the referees for their detailed comments and helpful suggestions, which greatly improved the manuscript.
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Ren, X., Xiang, Z. & Zhang, Z. Low regularity well-posedness for the viscous surface wave equation. Sci. China Math. 62, 1887–1924 (2019). https://doi.org/10.1007/s11425-018-9410-3
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DOI: https://doi.org/10.1007/s11425-018-9410-3