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Summer Graduate School

Introduction to water waves [Virtual Summer Graduate School] July 27, 2020 - August 07, 2020
Parent Program:
Location: MSRI: Simons Auditorium, Atrium
Organizers Mihaela Ifrim (University of Wisconsin-Madison), Daniel Tataru (University of California, Berkeley)
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Description
Img 6168
Overturning wave, artistic drawing by E. Ifrim

Registered participants can access additional informaiton about this summer school by clickiing HERE.

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.

Suggested prerequisites:
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. 

Required Reading:
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 Kleinā€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.
 

Bibliography

For eligibility and how to apply, see the Summer Graduate Schools homepage

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • water waves

  • incompressible Euler equations

  • fluid flows

  • free boundaries

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Jul 27, 2020
Monday
09:00 AM - 09:10 AM
  Welcome
09:10 AM - 10:20 AM
  Lecture 1
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 1
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Jul 28, 2020
Tuesday
09:00 AM - 10:20 AM
  Lecture 2
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 2
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Lecture by the guest speaker
Igor Kukavica (University of Southern California)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Jul 29, 2020
Wednesday
09:00 AM - 10:20 AM
  Lecture 3
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 3
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Jul 30, 2020
Thursday
09:00 AM - 10:20 AM
  Lecture 4
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 4
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Lecture by the guest speaker
Igor Kukavica (University of Southern California)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Jul 31, 2020
Friday
09:00 AM - 10:20 AM
  Lecture 5
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 5
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Aug 03, 2020
Monday
09:00 AM - 10:20 AM
  Lecture 6
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 6
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Aug 04, 2020
Tuesday
09:00 AM - 10:20 AM
  Lecture 7
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 7
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Lecture by the guest speaker
Igor Kukavica (University of Southern California)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Aug 05, 2020
Wednesday
09:00 AM - 10:20 AM
  Lecture 8
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 8
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Aug 06, 2020
Thursday
09:00 AM - 10:20 AM
  Lecture 9
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 9
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Lecture by the guest speaker
Igor Kukavica (University of Southern California)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.
Aug 07, 2020
Friday
09:00 AM - 10:20 AM
  Lecture 10
Mihaela Ifrim (University of Wisconsin-Madison)
10:30 AM - 11:50 AM
  Lecture 10
Daniel Tataru (University of California, Berkeley)
11:50 AM - 01:30 PM
  Lunch Break
01:30 PM - 02:30 PM
  Office hours (simultaneous office hours held by Mihaela and Daniel)
02:30 PM - 03:00 PM
  Break
03:00 PM - 05:00 PM
  Mandatory Work sessions supervised by TAs. You are also free to ask questions and treat this is as office hours if needed/if time permits.