The water waves equations were introduced by Euler in the 18th century to describe the motion of a mass of water under the influence of gravity and with a free surface. The unknown of the problem are two time dependent functions describing how the velocity field of the fluid and the profile of the free surface (giving the shape of the waves) evolve. The mathematical analysis of the water waves equations is particularly challenging due to their quasilinear nature, and it has been (and still is) a central research line in fluid dynamics. The goal of the course is to present recent advances in the mathematical analysis of the water waves equations. We will employ modern tools – e.g. paradifferential calculus – to study the Cauchy problem and (time permitting) the long time existence theory of small amplitude solutions.

**Course contents:**

**Part 1:** The water waves equations - The incompressible Euler equation - Zakharov’s variables - The Dirichlet-Neumann operator

**Part 2:** Paradifferential calculus - Bony’s paradifferential operators - Continuity and symbolic paradifferential calculus - Sobolev energy estimates for hyperbolic equations - Paralinearization of the Dirichlet-Neumann operator

**Part 3:** The Cauchy problem - Paralinearization of the equations - Symmetrization - A priori estimates

**Part 4:** Long-time existence of small smooth data (time permitting) - The Hamiltonian paradifferential Birkhoff normal form, after Berti-Maspero-Murgante

**References:**

[1] Alazard: Free surface flows in fluid dynamics

[2] Berti, Maspero: Lectures on periodic paradifferential calculus

[3] Berti, Maspero, Murgante: Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence