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

TL;DR

This paper proves that smooth three-dimensional doubly periodic gravity water waves with a symmetric diamond domain are infinitely differentiable, using paralinearization of the Dirichlet to Neumann operator and addressing small divisors without smallness assumptions.

Contribution

It provides an exact paralinearization formula for the Dirichlet to Neumann operator and establishes $C^ abla$ regularity for a class of three-dimensional water waves under diophantine conditions.

Findings

Proves $C^ abla$ regularity for symmetric diamond water waves.

Derives an exact paralinearization formula for the Dirichlet to Neumann operator.

Shows no smallness condition is needed for regularity.

Abstract

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.