Topics in Geometric Analysis: The PDEs of incompressible fluid dynamics

This course first covers some classical topics about the Euler and Navier-Stokes equations, such as classical existence theories of smooth solutions and weak solutions, global well-posedness in two dimensions and local well-posedness in three dimensions, the Ladyzenskaya-Prodi-Serrin conditional regularity theorem, Caffarelli-Kohn-Nirenberg local regularity theory for Navier-Stokes and various a-priori estimates. We then get to more recent results, such as the Escauriaza-Seregin-Sverak theorem, more recent conditional regularity results, Liouville theorems and results under special symmetry assumptions.