FD - Mathematical Problems in Fluid Dynamics
Monday, Wednesday, and Friday at 10:00am Pacific Time
Chancellor Professor and Instructor: Thomas Alazard
Lecture notes will be posted every week at: http://talazard.perso.math.cnrs.fr/CL.pdf
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
- Introduction
- The incompressible Euler equations and the water-wave problem
- Darcy’s law and the Hele-Shaw and Muskat equations
- Methods in PDEs
- An introduction to paradifferential calculus
- Sobolev embedding theorem and the fractional Laplacian
- The multipliers method, from Rellich to Morawetz and J.-L. Lions
- On the Cauchy problem
- Study of the Dirichlet-to-Neumann operator
- The Cauchy problem for the water wave and Hele-Shaw equations
- The Cauchy problem for the Muskat equation in critical spaces
- Exact identities and nonlinear methods
- Hamiltonian, Lagrangian, Momentum, Entropy. Conservation laws vs gradient flows
- Morawetz estimates for the water-wave equations
- Lyapounov functionals and entropies for the Hele-Shaw equations
January 20, 2021 (10:00 AM PST - 11:00 AM PST)
January 22, 2021 (10:00 AM PST - 11:00 AM PST)
January 25, 2021 (10:00 AM PST - 11:00 AM PST)
January 27, 2021 (10:00 AM PST - 11:00 AM PST)
January 29, 2021 (10:00 AM PST - 11:00 AM PST)
February 01, 2021 (10:00 AM PST - 11:00 AM PST)
February 03, 2021 (10:00 AM PST - 11:00 AM PST)
February 05, 2021 (10:00 AM PST - 11:00 AM PST)
February 08, 2021 (10:00 AM PST - 11:00 AM PST)
February 10, 2021 (10:00 AM PST - 11:00 AM PST)
February 12, 2021 (10:00 AM PST - 11:00 AM PST)
February 15, 2021 (10:00 AM PST - 11:00 AM PST)
February 15, 2021 (10:00 AM PST - 11:00 AM PST)
February 17, 2021 (10:00 AM PST - 11:00 AM PST)
February 19, 2021 (10:00 AM PST - 11:00 AM PST)
February 22, 2021 (10:00 AM PST - 11:00 AM PST)
February 24, 2021 (10:00 AM PST - 11:00 AM PST)
February 26, 2021 (10:00 AM PST - 11:00 AM PST)
March 01, 2021 (10:00 AM PST - 11:00 AM PST)
March 03, 2021 (10:00 AM PST - 11:00 AM PST)
March 05, 2021 (10:00 AM PST - 11:00 AM PST)
March 08, 2021 (10:00 AM PST - 11:00 AM PST)
March 10, 2021 (10:00 AM PST - 11:00 AM PST)
March 12, 2021 (10:00 AM PST - 11:00 AM PST)
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March 26, 2021 (10:00 AM PST - 11:00 AM PST)
March 29, 2021 (10:00 AM PST - 11:00 AM PST)
March 31, 2021 (10:00 AM PST - 11:00 AM PST)
April 02, 2021 (10:00 AM PDT - 11:00 AM PDT)
April 05, 2021 (10:00 AM PDT - 11:00 AM PDT)
April 07, 2021 (10:00 AM PDT - 11:00 AM PDT)
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