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On the water-wave equations with surface tension. (English) Zbl 1258.35043

Summary: The purpose of this article is to clarify the Cauchy theory of the water-wave equations in terms of regularity indexes for the initial conditions, as well as for the smoothness of the bottom of the domain. (Namely, no regularity assumption is assumed on the bottom.) Our main result is that, after suitable paralinearization, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one-dimensional) surface-tension water waves is in fact a rather direct consequence of this reduction, and following this approach, we are able to obtain a sharp result in terms of regularity of the indexes of the initial data and weights in the estimates.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
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