## MATH 595 Water Waves, Spring 2018

**Instructor** Vera Mikyoung Hur

**Time and Location** R 11:00-12:20am in 441 Altgeld Hall

**Office Hours** Make an appointment by email.

##### Course Notes and References

1. Introduction

2. Zakharov's Formulation and Hamiltonian (guest lecture by Sergey Dyachenko)

3. Shallow Water Theory (scribe: Derek Kielty)

- GB Whitham,
*Linear and Nonlinear Waves*, Sections 13.3-4, 10. - VM Hur, Shallow water models with constant vorticity in
*European Journal of Mechanics - B/Fluids*

4. KdV Theory, the Whitham Equation, and Wave Breaking (scribe: Gayana Jayasinghe)

- GB Whitham,
*Linear and Nonlinear Waves*, Sections 13.11 and 13.14. - RL Seliger, A note on the breaking of waves in
*Proceedings of the Royal Society A* - A Constantin and J Escher, Wave breaking for nonlinear nonlocal shallow water equations in
*Acta Mathematica*

- VM Hur, Wave breaking in the Whitham equation in
*Advances in Mathematics*

5. Hamiltonian Mechanics and FPU to KdV (guest lecture by Sergey Dyachenko)

6. Well-posedness vs. Wave Breaking for fKdV

- N Tzvetkov, Ill-posedness issues for nonlinear dispersive equations
- A Castro, D Cordoba and F Gancedo, Singularity formations for a surface wave model in
*Nonlinearity* - A Bressan and KT Nguyen, Global existence of weak solutions for the Burgers--Hilbert equation in
*SIAM Journal on Mathematical Analysis* - VM Hur, On the formation of singularities for surface water waves in
*Communications on Pure and Applied Analysis*

- H Dong, D Du and D Li, Finite time singularities and global well-posedness for fractal Burgers equations in
*Indiana University Mathematics Journal*

7. Ill-posedness for fKdV

- H Koch and N Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation in
*International Mathematics Research Notices* - A Himonas and G Misiolek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics in
*Communications in Mathematical Physics* - VM Hur, Norm inflation for equations of KdV type with fractional dispersion in
*Differential and Integral Equations*

8. Small Data and Long Time Existence (scribe: Haojian Li)

- JK Hunter and M Ifrim, Enhanced life span of smooth solutions of a Burgers--Hilbert equation in
*SIAM Journal on Mathematical Analysis* - JK Hunter, M Ifrim, D Tataru and TK Wong, Long time solutions for a Burgers--Hilbert equation via a modified energy method in
*Proceedings of the American Mathematical Society*

- S Wu, Wellposedness and singularities of the water wave equations in
*Lectures on the Theory of Water Waves*

10. Stability of KdV Solitary Waves (scribe: Matt Ellis)

- TB Benjamin, The stability of solitary waves in
*Proceedings of the Royal Society A* - JL Bona, On the stability theory of solitary waves in
*Proceedings of the Royal Society A*

11. Solitary Water Waves (scribe: Ashish Pandey)

- JT Beale, The existence of solitary water waves in
*Communications on Pure and Applied Mathematics* - VM Hur, No solitary waves exist on 2D deep water in
*Nonlinearity* - W Craig, Non-existence of solitary water waves in three dimensions in
*Philosophical Transactions of the Royal Society A*

- E Zehnder, Generalized implicit function theorems in
*Topics in Nonlinear Functional Analysis*

12. Global Bifurcation of Stokes waves (scribe: Sam Wester)

- B Buffoni and JF Toland, Analytic Theory of Global Bifurcation, Chapters 10--11

13. Modulational instability and variational structure (scribe: Peixue Wu)

- JC Bronski and VM Hur, Modulational instability and variational structure in
*Studies in Applied Mathematics*

14. Nondegenercy (scribe: Shunkun Wu)

- JC Bronski and VM Hur, Modulational instability and variational structure in
*Studies in Applied Mathematics*