2009FallDispersiveSeminar

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This page contains information about an informal seminar on Dispersive PDEs during Fall 2009. The seminar is organized by J. Colliander and Hiro Oh. We will meet on Thursdays, 1:30pm--3:00pm in SS 2106. (NOTE: It is in Sydney Smith Hall and not in Bahen Center.)

This semester, our main focus is Global well-posedness and scattering of the energy critical defocusing NLS.

  • J. Bourgain, "Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case," J. Amer. Math. Soc. 12 (1999), no. 1, 145--171.
  • T. Tao, Sec. 3.5-3.6, Chap 5 of "Nonlinear Dispersive Equations, Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp. ISBN: 0-8218-4143-2.
  • J. Bourgain, "Global solutions of nonlinear Schrödinger equations," American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999. viii+182 pp. ISBN: 0-8218-1919-4.


Contents

Calendar


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
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