Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves

Thomas Alazard (1) , Guy Métivier (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IMB - Institut de Mathématiques de Bordeaux

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.

Domaines: Equations aux dérivées partielles [math.AP]

Thomas Alazard, Guy Métivier. Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves. Communications in Partial Differential Equations, 2009, 34 (12), pp.1632-1704. ⟨hal-00354473v2⟩