The Water-Wave Equations in Eulerian Coordinates

This chapter introduces the analysis of the water-wave equations in Eulerian coordinates using the Dirichlet-to-Neumann operator. We will present the equations and explain how to analyze the Dirichlet-to-Neumann operator using various techniques: either with the multiplier method or with paradifferential calculus. In simple terms, the goal is to find a method to transform the water-wave equations in such a way that they can be reduced to simpler equations. Applications will be provided for various topics, including the study of the Cauchy problem, the proof of dispersive estimates, and the controllability of water waves.