Seminar

I will type lecture notes and post them every week on my webpage.

Description of the course 

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

Course syllabus
I. Introduction
a. The incompressible Euler equations and the water-wave problem
b. Darcy’s law and the Hele-Shaw and Muskat equations
II. Methods in PDEs
a. An introduction to paradifferential calculus
b. Sobolev embedding theorem and the fractional Laplacian
c. The multipliers method, from Rellich to Morawetz and J.-L. Lions
III. On the Cauchy problem
a. Study of the Dirichlet-to-Neumann operator
b. The Cauchy problem for the water wave and Hele-Shaw equations
c. The Cauchy problem for the Muskat equation in critical spaces
IV. Exact identities and nonlinear methods
a. Hamiltonian, Lagrangian, Momentum, Entropy. Conservation laws vs gradient flows
b. Morawetz estimates for the water-wave equations
c. Lyapounov functionals and entropies for the Hele-Shaw equations