Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves

Abstract

This note presents a paradifferential approach to the analysis of the water waves equations. Keywordsincompressible Euler equation, free surface, paradifferential calculus MSC(2010) 35B65, 35H20Citation: Alazard T. Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves.This note presents a paradifferential approach to the analysis of water wave equations. The main issue is that, thanks to this approach, one can conjugate nonlinear equations to linear equations, to the price of error terms which are smoother than the main terms (and thus presumed to be harmless in the derivation of estimates). In the first section I will recall the water wave equations for gravity waves and introduce the Dirichlet-Neumann operator. Then I will recall basic results about paradifferential operators. The main goal of this note is to state a result proved with Guy Métivier about the paralinearization of this operator, together with some extensions obtained in collaboration with Nicolas Burq and Claude Zuily. This strategy has a number of consequences since it reduces the proof of some difficult nonlinear estimates to easy symbolic calculus questions for symbols. In collaboration with Burq and Zuily, we have obtained sharp results about the well-posedness of the water wave system. These results include a local in time Cauchy theory for rough data corresponding to a natural threshold (cf. [1,3]), and the proofs of several dispersive estimates (cf. [1,2]).As was shown by Zakharov, the water wave system is a Hamiltonian one of the formwhere H is the total energy of the system. Denoting by H 0 the Hamiltonian associated with the linearized system at the origin, we have (for gravity waves)