This is a graduate course, not a general lecture. While all are welcome to attend, please keep in mind that the core audience of these courses is a cohort of students taking them for credit and being evaluated by course leaders, in part, on their participation.

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This class will give an overview of analytical tools and recent advances in the study of the water waves equations. We will begin with an introduction to the free boundary Euler equations in various physical settings and different formulations (e.g., Lagrangian and Eulerian). We will then discuss some analytical tools for the study of the Dirichlet-to-Neumann operator and the local well-posedness of the equations. Topics for this part of the class will include Littlewood-Paley theory, Paradifferential Calculus, and Energy methods for quasilinear equations. 

In the second part of the semester we will discuss the global well-posedness of the irrotational water waves equations in dimensions 3 and 2, along with various tools from asymptotic analysis (e.g., stationary phase and oscillatory Fourier integrals) and nonlinear harmonic analysis (e.g., multilinear estimates in weighted spaces).

We aim to tailor the pace and difficulty of the lectures to the background and interests of the audience. Various additional or alternative topics will be considered, including dispersive models related to water waves (such as the Korteweg- de Vries equation), nonlinear Schrödinger equations and modied scattering, spatially periodic motions and normal form techniques.