# 2011S MAT1061HS PDE2

### From TorontoMathWiki

MAT1061HS Partial Differential Equations 2 J. Colliander T-Th 13:00-14:30 BA 6183

NOTE: The last class presentations should take place soon. We will meet on Thursday 2011-04-14 and again sometime during the following week. Some of us went to BA6183 today to hear talks but no speakers came to class...... Colliand 14:01, 12 April 2011 (EDT)

## Contents |

## Textbooks

- L.C. Evans, Partial Differential Equations, AMS.
- G.B. Folland, Introduction to Partial Differential Equations, Princeton.
- L. Hörmander, The Analysis of Linear Partial Differential Operators I,II,III,IV, Springer.
- M.E. Taylor, Partial Differential Equations I,II,III,IV, Springer.

## Web Resources

- T. Tao, UCLA Math254A: Harmonic Analysis in the Phase Plane
- Dispersive PDE Wiki
- PDEs on Wikipedia
- 2006 Version of this course; 2006 PDE2 Lecture Notes
- Almut Burchard's MAT1060 PDE1 Course 2009F MAT1060HS PDE1
- 2010 Version of this course

## Course Participants

(Alphabetized Course Participants List; Link your name to your page on this wiki.)

- Yannis Angelopoulos
- James Colliander
- Ryan Donnelly
- Daniel Fusca
- Lukas Gonon
- Yoontae Jeon
- Yevgeny Liokumovich
- Cui Cui Luo
- Zhansaya Tleuliyeva
- James Lutley
- Adrian Poon
- Mustazee Rahman
- David Reiss
- Craig Sinnamon
- Andrew Stewart
- Xiao Liu
- Alex Weekes
- Yik Chau (Kry) Lui

## Guest Lecture Event

- On Tuesday 2011-03-29, Benjamin Dodson will give a seminar talk during our course time in BA6183. Sutdents in our course are encouraged to attend Dodson's talk.

## Reading Topics

Professor Nabutovsky suggested that I read "Convergence to a geodesic" by Norihito Koiso [1] as a starting point. The paper studies convergence of a closed curve on a manifold under the flow in the direction of and proportional to the covariant derivative of the tangent vector. This defines a Cauchy problem for a parabolic PDE. The paper proves that on a compact manifold the solution exists for all times and that if the manifold is real analytic the curve converges to a geodesic as $t \rightarrow \infty$, but if it's only $C^{\infty}$ it may not converge.--Yevgeny Liokumovich 17:47, 31 January 2011 (EST)

- Thanks Yevgeny! This sounds like a good plan. Colliand 12:55, 25 January 2011 (EST)
- I've written some notes on the paper and uploaded them onto Zotero. --Yevgeny Liokumovich 21:02, 15 February 2011 (EST)

- Thanks! I'll take a look. Colliand 22:52, 15 February 2011 (EST)

Professor Quastel proposed the article "Application of Brownian Motion to the Equation of Kolmogorov-Petrovskii-Piskunov" [2]. It shows how probabilistic methods can be used to analyze the limiting behaviour of the solution of the KPP equation with specific initial data. McKean shows that if $u(t,x)$ is the solution of \begin{equation}\label{eq:kpp}\frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{ \partial x^2} + u^2 - u \end{equation} (taking values in $[0,1]$) with initial values $f(x)=1_{\{x>0\}}$, then $ \lim_{t \to \infty}u(t,x+m(t)) = w(x)$ exists (where $m(t)$ is the median of $u(t,\cdot)$)and $w$ is a "wave solution" of the above, i.e. $w(x-\sqrt{2}t)$ solves KPP as well. For a more detailed discussion, read the notes on my presentation.

- I've uploaded the notes on my presentation on Zotero. Lukas 19:18, 9 April 2011 (EST)

Professor Jerrard suggested me to start from the book "Plateau's problem and the Calculus of Variations" by Struwe, Michael (1989). The first 30 pages of the book provides the solution to the Plateau's problem in two dimensions - briefly speaking it asks that for the existence of minimal surface given a boundary. The solution to the Plateau's problem was solved independently by Douglas, Jesse (1931). "Solution of the problem of Plateau". Trans. Amer. Math. Soc. and Radó, Tibor (1930). "On Plateau's problem". Ann. Of Math. I will first study from these resources where the topological type of the domain is a disc and the boundary being (rectfiable) simple closed curve. The subject is probably rather broad; it appears in Riemannian Geometry, the isoperimetric problem, the geometry of fully nonlinear elliptic equations, the topology of three-dimensional manifolds and Geometric Measure Theory. I will try to learn as much as possible and hopefully write expository essays on what I learn.

Some fairily nice descriptions of the topic online:

Encyclopaedia of Mathematics » P » Plateau problem [3]

Encyclopaedia of Mathematics » M » Minimal surface [4]

- This sounds great. I like to see that you will consult the classical works from the 1930s. I encourage you to reinterpret and streatmline those papers using the modern concepts in Struwe. Colliand 10:38, 1 February 2011 (EST)

Professor Catherine Sulem suggested me to study orbital stability of solitary wave solutions for the derivative nonlinear Schrodinger equation (DNLS):
\begin{equation}
i\partial_t u+\partial^2_x u+i\partial_x(|u|^{2\sigma}u)=0, (t,x)\in \mathbb{R}\times \mathbb{R}.
\end{equation}
When $\sigma=1$, it is known that DNLS has a two-parameter family of solitary wave solutions of the form:
$$u_{\omega,c}(t,x)=\phi_{\omega,c}(x-ct)\exp\{i\omega t+i\frac{c}{2}(x-ct)-\frac{3}{4}i\int^{x-ct}_{-\infty}|\phi_{\omega,c}(\eta)|^2d\eta\},$$
where $(\omega,c)\in\mathbb{R}^2,c^2<4\omega,$ and
$$\phi_{\omega,c}(x)=[\frac{\sqrt{\omega}}{4\omega-c^2}\{\cosh(\sqrt{4\omega-c^2}x)-\frac{c}{2\sqrt{\omega}}\}]^{-1/2}.$$
In the paper *Stability of solitary waves for derivative nonlinear Schrodinger equation* (2006), Professor Colin and Ohta show that $u_{\omega,c}(t)$ is orbitally stable.

We want to show that the solitary wave solutions are orbitally unstable when $\sigma>2.$

Here are some literature about stability theory of solitary waves:

C. Sulem and P.-L. Sulem,*The Nonlinear Schrodinger Equation*.

M. Grillakis, J. Shatah, W. Strauss, *Stability theory of solitary waves in the presence of symmetry, I*, J. Funct. Anal. **74**(1987) 160-197.

M. Grillakis, J. Shatah, W. Strauss, *Stability theory of solitary waves in the presence of symmetry, II*, J. Funct. Anal. **94**(1990) 308-348.

- Excellent! Is there any obstruction to applying the GSS machinery right away? Are there any quantitative aspects of the stability theory which you can show depends upon $\sigma$?
- We are studying a simpler equation right now:

\begin{equation} i\partial_t u+\partial^2_x u+i|u|^{2\sigma}\partial_x u=0, (t,x)\in \mathbb{R}\times \mathbb{R}. \end{equation}

- For the above equation, to use the GSS machinery to prove unsuitability, we need Hamiltonian operator only has 1 negative eigenvalue, but we can not prove it right now.
- Following the GSS method, we can get 3 conserved quantities which depend on $\sigma$: $E=\frac{1}{2}\int|\psi_x|^2dx+\frac{1}{2(\sigma+1)}\int\Im|\psi|^{2\sigma}\bar{\psi}$, $Q=\frac{1}{2}\int|\psi|^2 dx$ and $P=-\frac{1}{2}\Im\int\bar{\psi}\psi_x dx,$ and the Hamiltonian operator is $H=E"+w"+cP"$.

Professor Quastel suggested me to start from the following papers to study the connection between Optimal Stopping problem of the probability theory and Free-Boundary problem of PDE and Analysis (also called Stefan problem). Optimal stopping problems arise in many areas such as Mathematical Statistics and Mathematical Finance a lot. Surprisingly, it can be shown that the solution to these problems satisfy the Free-Boundary PDEs in the corresponding Stefan's problem (eg. Heat Equation with a free boundary problem corresponds to solving for the optimal stopping time to exercise the perpetual American put option). I plan to focus on the various solution method that has been developed including classic ones such as time & measure change and more recent ones such as nonlinear integral equations.

P.V. Moerbeke, "Optimal Stopping and Free Boundary Problems" (1974)

H.P. McKean, "A feee boundary problem for the heat equation arising from a problem in mathematical economics" (1965)

I am planning to use the following book as a main reference. It's a very comprehensive book that covers the recent development of this subject in great details with a lot of examples. I'll try to read the basic solution methods as much as possible.

Goran Peskir & Albert Shiryaev, "Optimal Stopping and Free-Boundary Problems" (2006)

- Thanks! This looks like an interesting topic. Colliand 18:04, 3 February 2011 (EST)

- Mustazee Rahman My notes on the Stein Tomas restriction theorem. I encourage others to have a look and then provide me with feedback. I would also be glad if someone is willing to discuss these topics with me in more depth. Perhaps then we can setup a talk to be presented to the class.

I am expanding on the introduction to Fourier Restriction Theorems that was given by Dr. Colliander. My study began with the Stein Tomas restriction theorem, but I am interested in learning how to deal with the boundary case $p=\frac{2n+2}{n+3}$ using Complex interpolation. Also, Knapp's example that shows sharpness of this result. I have been using Schlag's Harmonic Analysis notes and Stein's book titled Harmonic Analysis.

## Some Topics from Lectures

- 2011-01-26

- Mustazee Rahman has worked through Tomas' paper and is preparing a talk and report. Colliand 18:17, 3 February 2011 (EST)

- 2011-01-28 -- 2011-02-03
- Sobolev spaces Defined
- Discussion of pictures of Sobolev functions on log-frequency vs. log-fourier-tranform graphs. (Bode Plots) ( Xiao Liu prepared some MATLAB graphics illustrating the Bode Plots. )
- Proof of Sobolev embedding using high/low frequency cutting adapted to vertical level set decomposition. (I learned this proof from Chemin et. al. Mathematical Geophysics. Colliand 18:08, 3 February 2011 (EST))

- 2011-02-06 -- 2011-02-08
- Multilinear Operators in Harmonic Analysis (Coifman-Meyer Calculus)
- What is a Ginzburg-Landau Vortex?
- Navier-Stokes Initial Value Problem: First facts....