# 2009 2010 Dispersive PDE Seminar

This page contains information about an informal seminar on Dispersive PDEs during 2009-2010. The seminar is organized by J. Colliander and Hiro Oh.

Of related interest in Toronto (and sometimes cross-listed):

## July 27, Zworski, 13:10-14:00 @BA6183

 Maciej Zworski [1] (Berkeley) Dispersive PDE Seminar Tuesday July 27 13:10-14:00 BA6183 Title: Quantized Poincare maps in chaotic scattering Abstract: ... arXiv 2010_07_27_Zworski_Notes 2010_07_27

## July 26, Pocovnicu, 14:10-15:00 @BA6183

 Oana Pocovnicu [2] (Université Paris-Sud, Orsay) Dispersive PDE Seminar Monday July 26 14:10-15:00 BA6183 Title: Soliton resolution and action-angle coordinates for the Szego equation in the case of rational fraction initial data Abstract: The Szego equation is a model of a non-dispersive Hamiltonian equation. Like the 1-d cubic Schrodinger equation and KdV, it is known to be completely integrable in the sense that it enjoys a Lax pair structure. (The main operator in this Lax pair is the Hankel operator.) It turns out that a whole class of finite dimensional manifolds, consisting of rational fractions, is invariant under the flow of the Szego equation. In this talk, we consider the Szego equation on the real line. First, we show that solutions with generic rational fraction initial data can be decomposed, as time tends to infinity, into a sum of solitons plus a remainder. ("Soliton resolution".) Unlike the case of KdV, this remainder does not disperse, confirming the non-dispersive character of the Szego equation. To prove this result, we solve the inverse spectral problem for the Hankel operator in the finite dimensional case and find an explicit formula for solutions. Then, we use this result to introduce action-angle coordinates on the manifolds of generic rational fractions. In particular, this shows that the trajectories live on Lagrangian toroidal cylinders which are non-compact generalization of the Lagrangian tori in the Liouville-Arnold theorem. [{{{arxiv}}} arXiv] 2010_07_26_Pocovnicu_Notes 2010_07_26

## July 26, Pocovnicu, 11:10-12:00 @BA6183

 Oana Pocovnicu [3] (Université Paris-Sud, Orsay) Dispersive PDE Seminar Monday July 26 11:10-12:00 BA6183 Title: Liouville-Arnold Theorem and action-angle coordinates (expository) Abstract: In this talk, we consider finite dimensional integrable systems. In particular, we discuss Liouville-Arnold Theorem, the construction of the action-angle coordinates, and their application in perturbation theory. [{{{arxiv}}} arXiv] 2010_07_26_Pocovnicu_Notes 2010_07_26

## June 22, Oh, 13:10-14:00 @BA6183

 Tadahiro Oh [4] (U. Toronto) Dispersive PDE Seminar Tuesday June 22 13:10-14:00 BA6183 Title: Normal form and I-method Abstract: In this talk, I will discuss the main ideas and steps in Bourgain's paper: A remark on normal forms and the "$I$-method" for periodic NLS, J. Anal. Math. 94 (2004), 125--157. [{{{arxiv}}} arXiv] 2010_06_22_Oh_Notes 2010_06_22

## June 18, Oh, 13:10-14:00 @BA6183

 Tadahiro Oh [5] (U. Toronto) Dispersive PDE Seminar Friday June 18 13:10-14:00 BA6183 Title: Unconditional Well-Posedness of mKdV on T Abstract: In 1993, by introducing what is now known as the Bourgain space, Bourgain proved local-in-time well-posedness of the periodic modified KdV (mKdV) in $H^s, s \geq 1/2$. In this result, the uniqueness holds in $C([0, T]; H^s)$ intersected with this Bourgain space, since solutions in his construction necessarily belong to this auxiliary function space. In this talk, we present a simple proof of local well-posedness of mKdV for $s \geq 1/2$ based on differentiation by parts, which shows that the uniqueness holds unconditionally in $C([0, T]; H^s)$ (i.e. without intersecting with any auxiliary function space.) This is a joint work with Soonsik Kwon. [{{{arxiv}}} arXiv] 2010_06_18_Oh_Notes 2010_06_18

## June 16, Czubak, 13:10-14:00 @BA6183

 Magdalena Czubak [6] ([7]) Dispersive PDE Seminar Wednesday June 16 13:10-14:00 BA6183 Title: Introduction to Gauge Theory Abstract: [{{{arxiv}}} arXiv] 2010_06_16_Czubak_Notes 2010_06_16

## June 02, Richards, 13:10-14:00 @BA6183

 Geordie Richards (TMW) ([8]) Dispersive PDE Seminar Wednesday June 02 13:10-14:00 BA6183 Title: Invariant Measures for Hamiltonian PDE Abstract: We discuss the existence of invariant measures for Hamiltonian PDE, by interpreting these measures as the weak limit of finite dimensional measures in frequency space. As an example, we prove invariance of the Gibbs measure for the KdV equation. [{{{arxiv}}} arXiv] 2010_06_02_Richards_Notes 2010_06_02

## May 25, Muñoz, 14:10-15:00 @BA6183

 Claudio Muñoz [9] (Versailles) Dispersive PDE Seminar Tuesday May 25 14:10-15:00 BA6183 Title: On the soliton dynamics under a slowly varying medium for generalized KdV equations Abstract: We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly varying media induce on the soliton solution large dispersive effects at large time. Moreover, unlike gKdV equations, we prove that there is no pure-soliton solution in this regime. arXiv 2010_05_25_Muñoz_Notes 2010_05_25

## May 19, Selvitella, 13:10-14:00 @BA6183

 Alessandro Selvitella (TMW) (SISSA) Dispersive PDE Seminar Wednesday May 19 13:10-14:00 BA6183 Title: Introduction to Birkhoff Normal Form Abstract: Some background and discussion of the Birkhoff Normal Form aimed at looking at recent developments to extend these ideas to the PDE setting. [ arXiv] 2010_05_19_Selvitella_Notes 2010_05_19

## May 12, Colliander, 13:10-14:00 @BA6183

 James Colliander (TMW) (Toronto) Dispersive PDE Seminar Wednesday May 12 13:10-14:00 BA6183 Title: Organizational meeting for Summer Abstract: We will meet to organize into groups and identify topics we want to learn more about this summer. [ arXiv] 2010_05_12_Colliander_Notes 2010_05_12

## December 10, Anapolitanos, 1:30pm-3:00pm

Speaker: Ioannis Anapolitanos
Title: A simple derivation for Mean-Field limits for many-body quantum Systems.
Abstract: In this talk we will discuss a new strategy for handling Mean-field limits of quantum mechanical systems. The strategy is developed in a series of papers by Pickl, Knowles-Pickl. The result is roughly saying that if the initial state of the many body system is a mutliple product of the same one particle state, then its evolution remains close to a mutliple product of a one-particle state that evolves according to the Hartree equation. The closeness is in the sense of expectation values of certain class of quantum observables. The method is based on "counting" the particles that fail to be in the state that evolves according to the Hartree equation. We will discuss a simple version of the general results where we will see how we can control the number of such particles. Also we will discuss how the latter implies closeness in the topology of expectations of observables.
Notes from the talk.

## October 29, Czubak, 1:30pm-3:00pm

Speaker: Magdalena Czubak
Title: Local conservation laws, virial identities, and interaction Morawetz estimates.
Abstract: Since the pioneering work of C.S.Morawetz on the Nonlinear Klein-Gordon equation, Morawetz type estimates have become an important tool in the study of NLS. In particular, the interaction Morawetz estimate in 3D is fundamental for the theory of GWP and scattering for both the energy critical and subcritical NLS. We will outline two proofs of this estimate: one based on the original averaging argument of CKSTT, and the second one based on the tensor product idea of A.Hassell. Local conservation laws and the generalized virial identity of Lin and Strauss are a starting point for the two methods. Time permitting we will discuss possible extensions to other equations.
Remarks:
Extensions of these ideas have been obtained by Colliander-Grillakis-Tzirakis and Planchon-Vega. Here are some notes from a talk on the CGTz work. A nice survey of these developments has been written by Ginibre-Velo.

## October 22, Richards, 1:30pm-3:00pm

Speaker: Geordie Richards
Title: Critical local well-posedness and perturbation theory.
Abstract: The proof of global well-posedness and scattering for the quintic defocusing NLS on $\mathbb{R}^{3}$ relies on the stability of $L^{10}_{t\in I,x\in\mathbb{R}^{3}}$ spacetime bounds (on the solution) under any perturbation of the initial data whose linear evolution is sufficiently small in $L^{10}_{t\in I}\dot{W}^{1,30/13}_{x\in\mathbb{R}^{3}}$. (Here I is a compact interval.) This stability holds even for large energy perturbations of so-called near solutions. We discuss these issues and establish this stability by proving two perturbation Lemmas.
Remarks: See Lemmas 3.9 and 3.10 from Section 3 of CKSTT Energy Critical Paper.

## October 15, Oh, 1:30pm-3:00pm

Speaker: Hiro Oh
Title: Basic H1-subcritical scattering theory: defocusing cubic NLS on $\mathbb{R}^3$.
Abstract: In this talk, we discuss the basic scattering theory for the H1-subcritical setting. First, we review the local-in-time Cauchy theory in $H^1(\mathbb{R}^3)$ in both subcritical and critical settings. Then, we discuss the issue of the existence of the wave operator as well as the asymptotic completeness for the defocusing cubic NLS. We prove the existence of the wave operator by considering the Cauchy problem from $t = +\infty$ to t = 0. Then, we prove the asymptotic completeness in several steps. We first show the asymptotic completeness from the strong space-time bound. Then, we show such strong bound follows from a weak space-time bound. Obtaining such weak space-time bound (called Morawetz inequality) is one of the main topics of the talk on Oct. 29 by M. Czubak.
Remarks: This talk will expose aspects of the paper posted here. Some related notes are posted here. Hiro's notes are posted here.

## October 8, Colliander, 1:30pm-3:00pm

Speaker: J. Colliander
Title: Overview of the energy critical defocusing NLS
Abstract: I will give an overview of the main pieces of the proof of scattering for the 3d energy critical quintic defocusing NLS. A pictorial overview of the topics is posted here. Subtopics will be developed in more detail later in the semester. Two talks on the same topic are posted here and here.

## October 1, Oh, 1:30pm-3:00pm

Speaker: Hiro Oh
Title: Cauchy problem of the cubic NLS on $\mathbb{T}$ (Part 2)
Abstract: I will discuss the ill-posedness issues as well as the weak continuity of the solution map.
Hiro's Lecture Notes

## September 24, Oh, 1:30pm-3:00pm

Speaker: Hiro Oh
Title: Cauchy problem of the cubic NLS on $\mathbb{T}$ (Part 1)
Abstract: In this (plus alpha) talk, I will discuss the basic well/ill-posedness issues on the cubic NLS on $\mathbb{T}$. The topics includes:
1.Basic properties of Xs,b spaces: definition, transference principle, linear estimates, time localization, etc.
2.Strichartz estimate: On $\mathbb{R}^d$: idea of proof, dimension counting for admissible pairs. On $\mathbb{T}$: L4-Strichartz (Zygmund, '74), L6 and L4-Strichartz (Bourgain '93.) This includes the number theoretic counting.
3.Well-posedness in $L^2(\mathbb{T})$.
4.Ill-posedness below $L^2(\mathbb{T})$: Failure of smoothness of the solution map (Bourgain '97), failure of uniform continuity (Burq-Gerard-Tzvetkov '02), discontinuity as a result of failure of weak continuity in $L^2(\mathbb{T})$ (Molinet '09.)
If time permits, I may give discussion on the weak continuity of the L2-subcritical NLS on $\mathbb{R}$ (Cui-Kenig '09) as well as well-posedness issue of the periodic (Wick ordered) cubic NLS outside $L^2(\mathbb{T})$.
Hiro's Lecture Notes