RIMS workshop "Wellposedness and Scattering for Nonlinear Dispersive and Wave Equations"
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 longrange 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 KleinGordon equation with a quadratic nonlinearity in two space dimensions
 We study the asymptotic behavior of global solutions to the quadratic nonlinear KleinGordon 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)
 Wellposedness of the Cauchy problem for the KdV equation at the critical regularity
 We prove the wellposedness for the nonperiodic Kortewegde Vries (KdV) equation in
H^{3/4}. Our regularity s=3/4 is critical in the sense that we have the global
wellposedness 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 YangMills 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 inductiononenergy strategy of Bourgain, the
concentrationcompactness 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 threebody interactions
 In this talk we will discuss joint work with Thomas Chen on the
dynamics of a boson gas with threebody interactions in dimensions d=1,2.
We prove that in the limit as the particle number N tends to infinity, the
BBGKY hierarchy of kparticle marginals converges to a limiting
GrossPitaevskii (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
wellposedness of the Cauchy problem for focusing and defocusing GP
hierarchy.
 14:45  15:45
 James Colliander (University of Toronto, Canada)
 Almost sure wellposedness 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 wellposed 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 masssupercritical and energysubcritical 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}dingerPoisson system in the 2D whole space
 We prove a local existence of a unique timelocal solution
to the Schr\"{o}dignerPoisson 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.