RIMS workshop "Well-posedness and Scattering for Nonlinear Dispersive and Wave Equations"

Contents

Outline

Dates
November 23 - 25, 2009
Venue
Hokkaido University, Conference Hall, Conference Room 1
Speakers
Takafumi Akahori (Ehime University, Japan)
James Colliander (University of Toronto, Canada)
Nobu Kishimoto (Kyoto University, Japan)
Satoshi Masaki (Tohoku University, Japan)
Kenji Nakanishi (Kyoto University, Japan)
Natasa Pavlovic (University of Texas at Austin, USA)
Akihiro Shimomura (Tokyo Metropolitan University, Japan)
Terence Tao (University of California, Los Angeles, USA)
Satoshi Tonegawa (Nihon University, Japan)
Program (PDF file)

There have been changes to workshop program on Monday and Tuesday afternoon. Lectures of third slot on Monday and third slot on Tuesday are switched.

November 23 (Mon.)

13:30 - 14:30
Akihiro Shimomura (Tokyo Metropolitan University, Japan)
Large time behavior of solutions to Schr\"{o}dinger equations with nonlinear dissipation for arbitrarily large initial data
We study the large time behavior of solutions to the initial value problem of the Schr\"{o}dinger equation with nonlinear dissipation of long-range type in one space dimension. We show the time decay estimate and the large time asymptotics of the solution for arbitrarily large initial data. This is a joint work with Naoyasu Kita (University of Miyazaki).

14:45 - 15:45
Satoshi Tonegawa (Nihon University, Japan)
Wave operators for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in two space dimensions
We study the asymptotic behavior of global solutions to the quadratic nonlinear Klein-Gordon equation in 2D with a small final data. The aim of this talk is to prove the existence of wave operators in a wider class of final data. (joint work with N. Hayashi and P. I. Naumkin)

16:15 - 17:15
Kenji Nakanishi (Kyoto University, Japan)
Sharp wellposedness of nonlinear Dirac and wave equations in one dimension

November 24 (Tue.)

9:45 - 10:45
Nobu Kishimoto (Kyoto University, Japan)
Well-posedness of the Cauchy problem for the KdV equation at the critical regularity
We prove the well-posedness for the nonperiodic Korteweg-de Vries (KdV) equation in H^{-3/4}. Our regularity s=-3/4 is critical in the sense that we have the global well-posedness in H^s with s>-3/4, while the solution map fails to be uniformly continuous below s=-3/4. Our main task is to construct a spacetime function space which yields the crucial bilinear estimate for the nonlinear term. It will be some modification of the Bourgain space X^{-3/4,1/2} and naturally defined through the analysis of several counterexamples to the bilinear estimate in the Bourgain spaces.

11:00 - 12:00
Terence Tao (University of California, Los Angeles, USA)
Wave maps
Wave maps are one of the fundamental geometric wave equations, being on the one hand the dynamic analogue of harmonic maps, and a simplified model for the Einstein equations and gauge field theories such as the Yang-Mills equations on the other. In recent years there has been substantial progress in understanding basic questions such as global regularity and singularity formation for this equation, using new tools such as the induction-on-energy strategy of Bourgain, the concentration-compactness technology of Kenig and Merle, a geometric gauge fixing arising from the harmonic map heat flow, and even some limiting arguments used by Perelman in his proof of the Poincare conjecture. We will survey some of these developments in this talk.

13:30 - 14:30
Natasa Pavlovic (University of Texas at Austin, USA)
The quintic NLS as the mean field limit of a Boson gas with three-body interactions
In this talk we will discuss joint work with Thomas Chen on the dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit as the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a limiting Gross-Pitaevskii (GP) hierarchy for which we prove existence and uniqueness of solutions. For factorized initial data, the solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schr\"{o}dinger equation.
Time permitting, we will briefly describe our new approach for studying well-posedness of the Cauchy problem for focusing and defocusing GP hierarchy.

14:45 - 15:45
James Colliander (University of Toronto, Canada)
Almost sure well-posedness for periodic cubic NLS below L^2
This talk reports on recent work with Tadahiro Oh. We consider the 1d periodic Wick ordered NLS equation. This equation is shown to be almost surely, with respect to a canonical Gaussian measure, locally and globally well-posed in certain negative Sobolev spaces. This talk reports on progress towards showing that white noise is invariant under the cubic NLS flow.

16:15 - 17:15
Takafumi Akahori (Ehime University, Japan)
Blowup and scattering problems for nonlinear Schr\"{o}dinger equations
We consider the mass-supercritical and energy-subcritical focusing nonlinear Schr\"{o}dinger equations. We introduce a subset PW of H^{1}. The set PW consists of two components PW_{+} and PW_{-}. We prove that any solution starting from a datum in PW_{+} behaves asymptotically free, and solution starting from a datum in PW_{-} blows up or grows up. (a joint work with H. Nawa, Osaka University)

18:00 - 20:00
Banquet

November 25 (Wed.)

10:00 - 11:00
Terence Tao (University of California, Los Angeles, USA)
Wave maps
Part II

11:15 - 12:15
Satoshi Masaki (Tohoku University, Japan)
Analysis of the Schr\"{o}dinger-Poisson system in the 2D whole space
We prove a local existence of a unique time-local solution to the Schr\"{o}digner-Poisson system in the 2D whole space. We use a new formula of the solution to the Poisson equation which is defined under a weaker assumption than that for the Newton potential. Though the nonlinear term may diverge at the spatial infinity, we obtain a time local solution for data in the Sobolev space. The key is a reduction of the system into a quantum hydrodynamical equation which is used in the study of the semiclassical limit.
Organizer
Hideo Takaoka (Hokkaido University)
Note
This workshop is held as part of the RIMS 2009 special project research "Qualitative Study on Nonlinear Partial Differential Equations of Dispersive Type". The site of the workshop is not the RIMS Kyoto University.