Summer Graduate School
|Location:||MSRI: Simons Auditorium, Atrium|
Due to the COVID-19 pandemic, this summer school will be held online.
The goal is to present the main current research directions in water waves. We will begin with the physical derivation of the equations, and present some of the analytic tools needed in study. The final goal will be two-fold, namely (i) to understand the local solvability of the Cauchy problem for water waves, as well as (ii) to describe the long time behavior of solutions.
Through the lectures and associated problem sessions, students will learn about a number of new analysis tools which are not routinely taught in a graduate school curriculum. The goal is to help students acquire the knowledge needed in order to start research in water waves and Euler equations.
Students should have read thoroughly the the following references:
1. R. Strichartz: A guide to theory to distribution and Fourier transforms
2. L. C. Evans: Introduction to partial differential equations: Chapters 1, 2, 5, 6, 7.
3. Terence Tao: Nonlinear dispersive equations: local and global analysis Chapter 2, and 3
Solving the problems listed at the end of each of the chapters in each of the above books is strongly recommended.
Students should have read up on the following topics, suggested readings of the topics are also provided:
1. Theory of Distributions And Fourier transform
Suggested book: Introduction to the Theory of Distributions
By F. G. Friedlander and M. Joshi, Chapters 1-9
2. Introduction to PDE's, Sobolev spaces
Suggested book: Partial Differential Equations
By Lawrence C. Evans · 2010 Chapters 2,5,6,7
3. Complex Analysis, Including Riemann Mapping Theorem
Suggested book: Complex Analysis
By Elias M. Stein, Rami Shakarchi · 2010 Chapters 1,2,3,4,8
4. Dispersive PDE's, introduction
Suggested book: Nonlinear Dispersive Equations Local and Global Analysis
By Terence Tao · 2006, Up to section 3.6
Recommended reading (to be discussed during lectures), with emphasis on papers 1-7.
1. The lifespan of small data solutions in two dimensional capillary water waves, M. Ifrim and D. Tataru, Arch. Ration. Mech. Anal., 225, no. 3, 1279-1346, e-print available at arXiv, 2017
2. Two dimensional water waves in holomorphic coordinates II: global solutions, M. Ifrim and D. Tataru, Bull. Soc. Math. France, 144, no. 2, 369-394, e-print available at arXiv, 2016.
3. Global bounds for the cubic nonlinear Schrodinger equation (NLS) in one space dimension, M. Ifrim and D. Tataru, Nonlinearity, 28, no. 8, 2661-2675, e-print available at arXiv, 2015.
4. Two dimensional water waves in holomorphic coordinates, J. K. Hunter, M. Ifrim, and D. Tataru, Comm. Math. Phys., 346, no. 2, 483-552, e-print available at arXiv, 2016.
5. Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method, J. K. Hunter, M. Ifrim, D. Tataru, D. T. Wang, Proceedings of the AMS, Vol. 143(8), pp. 3407-3412, e-print available at arXiv, 2015.
6. The NLS approximation for two dimensional deep gravity waves, M. Ifrim, D. Tataru, e-print available on arXiv, Sci. China Math, 2019.
7. No solitary waves in 2-d gravity and capillary waves in deep water, M. Ifrim, D. Tataru, e-print available on arXiv, (18 pages), 2018.
8. The Water Waves Problem: Mathematical Analysis and Asymptotics (book), author David Lannes
9. Normal forms and quadratic nonlinear Klein‐Gordon equations, Jalal Shatah, Communications on Pure and Applied, Volume38, Issue 5, September 1985, pages 685-696.
10. Paralinearization of the Dirichlet to Neumann operator, and regularity of diamond waves, Thomas Alazard, Guy Métivier Comm. Partial Differential Equations, 34 (2009), no. 10-12, 1632-1704.
11. Global solutions and asymptotic behavior for two dimensional gravity water waves, Thomas Alazard and Jean-Marc Delort, Ann. Sci. Éc. Norm. Supér., 48, (2015), no. 5, 1149-1238.
12. Sobolev estimates for two dimensional gravity water waves, Thomas Alazard, and Jean-Marc Delort, Astérisque 374 (2015), viii+241 pages.
For eligibility and how to apply, see the Summer Graduate Schools homepage
incompressible Euler equations