Abstract
We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C 2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C 2,μ.
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Acknowledgements
The authors are grateful to the referees for their valuable suggestions which improved the manuscript substantially. The work is supported by NSFC (11001207, 11201355), RFDP (20100141120064) and Hubei Province Key Laboratory of SSMP (WUST,Y201313).
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Communicated by G. Iooss.
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Chen, H., Li, WX. & Wang, LJ. Regularity of Traveling Free Surface Water Waves with Vorticity. J Nonlinear Sci 23, 1111–1142 (2013). https://doi.org/10.1007/s00332-013-9181-6
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DOI: https://doi.org/10.1007/s00332-013-9181-6