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Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves by Alazard, Thomas; Métivier, Guy is a scholarly article available to read on EtoBox.

What is Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves about?

This paper is concerned with a priori $C^\infty$ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically $C^\infty$. In particular, we prove that the solutions defined by Iooss and Plotnikov are $C^\infty$. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.

Author

Alazard, Thomas; Métivier, Guy

Published

2009

Language

EN