Apparaît dans la collection : Summer School 2023 - New trends in mathematical fluid dynamics

These lectures are based on a point of view introduced by Walter Craig, Catherine Sulem and Vladimir Zakharov, which consists in studying the water-wave equations in Eulerian coordinates, using the Dirichlet to Neumann operator. I will present the equations and explain how to study the Dirichlet to Neumann operator using different techniques: either global with the method of multipliers, or microlocal using paradifferential operators. Applications will be given to different topics: study of the Cauchy problem, equipartition of energy, control theory and to other free boundary problems.