# 2010 2011 Dispersive PDE Seminar

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

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

- Fields Institute Colloquium/Seminar in Applied Mathematics
- Fields Analysis Working Group
- Analysis and Applied Math Seminar
- Numerical PDE and Evolution Equations Working Group Seminar.

Archives:

## June 22, **Richards**, 11:10-12:00 @BA 6183

Geordie Richards (University of Toronto) | Dispersive_PDE_Seminar | Wednesday | June 22 | 11:10-12:00 | BA 6183 | ||||||||

Title: Function spaces for critical well-posedness theory
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Abstract: We will introduce the (closely related) function spaces of $U^p$ and $V^p$ type, due to Koch-Tataru ($U^p$) and Wiener ($V^p$), which can be useful in the analysis of nonlinear dispersive PDEs at critical regularity. Time permitting, we will discuss the proof of local well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili-II (KP-II) equation in a critical space, and highlight the role played by a nonlinear estimate in the $U^p, V^p$ spaces.
[HHK 2009] M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space. arXiv:0708.2011 | |||||||||||||

arXiv | 2011_06_22_Richards_Notes | 2011_06_22 |

## November 30, **Wang**, 1:30-3:00 @KP 113

Jing Wang (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | November 30 | 1:30-3:00 | KP 113 | ||||||||

Title: Bilinear estimates and applications to NLW
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Abstract: We will continue the paper of Klainerman and Selberg.This time, we prove local wellposedness for the wave map type equations by the Picard iteration.The nonlinear estimates we have proved last time play an important role.
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[{{{arxiv}}} arXiv] | 2010_11_30_Wang_Notes | 2010_11_30 |

## November 16, **Colliander**, 1:30-3:00 @KP 113

James Colliander [1] (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | November 16 | 1:30-3:00 | KP 113 | ||||||||

Title: Critical Element + Rigidity Strategy for Evolution PDEs
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Abstract: A robust strategy for studying the maximal-in-time behavior of evolution PDEs has emerged in the past decade. The strategy has led to definitive progress on certain nonlinear Schrödinger equations, nonlinear wave equations and recently on the Navier-Stokes system. This talk will describe some aspects of this strategy.
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[{{{arxiv}}} arXiv] | 2010_11_16_Colliander_Notes | 2010_11_16 |

## November 09, **Wang**, 1:30-3:00 @KP 113

Jing Wang (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | November 09 | 1:30-3:00 | KP 113 | ||||||||

Title: Bilinear estimates and applications to NLW
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Abstract: I will give a short introduction to the paper of Klainerman and Selburg: Bilinear estimates and applications to nonlinear wave equations. In this paper, Klainerman and Selburg considered the Cauchy problem for three types nonlinear wave equations. They construct the so called wave sobolev space to get the local wellposedness. We will focus on this space and the related estimates since the method to prove local wellposedness is standard. In fact, the wave sobolev space is the Bourgain space in the wave equation context. As the Bourgain space corresponding Schrodinger equation is well known, we will see there are similar properties between the two spaces.
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[{{{arxiv}}} arXiv] | 2010_11_09_Wang_Notes | 2010_11_09 |

## November 02, **Huang**, 1:30-3:00 @KP 113

Arthur Huang (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | November 02 | 1:30-3:00 | KP 113 | ||||||||

Title: Refinement of the Strichartz's inequality for non-elliptic Fourier restriction
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Abstract: (Continued from 19 October...) Among the several refinements of Strichartz's inequality, the type bounding the L^4-norm in space-time of the free evolution by the X^p-norm (1<p<2) is particularly interesting and useful in proving concentration results. In this talk we shall examine in details such a refinement [RV, 2006] arised from the non-elliptic free Schrodinger equation in 2-d. With the elliptic version of this refinement, Bourgain proved the L^2 mass concentration for 2-d NLS; time permitted, I may also talk about the corresponding mass concentration of the non-elliptic NLS given blow-up in finite time. (Although the talk must end in finite time, the evolution governed by non-elliptic NLS may not.)
[RV, 2006] K. Rogers & A. Vargas: A refinement of the Strichartz inequality on the saddle and applications, JFA, 2006 | |||||||||||||

[{{{arxiv}}} arXiv] | 2010_11_02_Huang_Notes | 2010_11_02 |

## October 19, **Huang**, 1:30-3:00 @KP 113

Arthur Huang (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | October 19 | 1:30-3:00 | KP 113 | ||||||||

Title: Refinement of the Strichartz's inequality for non-elliptic Fourier restriction
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Abstract: Among the several refinements of Strichartz's inequality, the type bounding the L^4-norm in space-time of the free evolution by the X^p-norm (1<p<2) is particularly interesting and useful in proving concentration results. In this talk we shall examine in details such a refinement [RV, 2006] arised from the non-elliptic free Schrodinger equation in 2-d. With the elliptic version of this refinement, Bourgain proved the L^2 mass concentration for 2-d NLS; time permitted, I may also talk about the corresponding mass concentration of the non-elliptic NLS given blow-up in finite time. (Although the talk must end in finite time, the evolution governed by non-elliptic NLS may not.)
[RV, 2006] K. Rogers & A. Vargas: A refinement of the Strichartz inequality on the saddle and applications, JFA, 2006 | |||||||||||||

[{{{arxiv}}} arXiv] | 2010_10_19_Huang_Notes | 2010_10_19 |

## October 12, **Pigott**, 1:30-3:00 @KP 113

Brian Pigott (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | October 12 | 1:30-3:00 | KP 113 | ||||||||

Title: Upper bounds on the orbital instability of the generalized KdV equations below the energy norm
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Abstract: In this talk I will discuss some of the details relevant in establishing polynomial-in-time upper bounds on the orbital instability of the generalized KdV equations below H^1. The talk will focus on the construction of an almost conserved quantity which can be used to establish these upper bounds.
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[{{{arxiv}}} arXiv] | 2010_10_12_Pigott_Notes | 2010_10_12 |

## October 5, **Richards**, 1:30-3:00 @KP 113

Geordie Richards (University of Toronto) | Dispersive_PDE_Seminar | Tuesday | October 5 | 1:30-3:00 | KP 113 | ||||||||

Title: Invariant Gibbs measures for periodic nonlinear Schrodinger equations, Part II
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Abstract: This talk is a continuation of the FAWG talk from last week. Last week, we discussed two papers of Bourgain [MR1309539 (95k:35185), MR1374420 (96m:35292)] on the invariance of (suitably normalized) Gibbs measures under the flow of some periodic nonlinear Schrodinger (NLS) equations. In this second talk, we will focus on Bourgains second paper, where he proves that the Wick-ordered Gibbs measure is invariant under the flow of the Wick-ordered 2-dimensional cubic NLS. Bourgain's proof of almost sure LWP in the support of the Wick-ordered Gibbs measure (an ingredient in the proof of the Gibbs measure invariance) combines techniques from dispersive PDE and probability theory; he establishes an effect of nonlinear smoothing under randomization of the initial data. We will discuss this LWP
argument. | |||||||||||||

[{{{arxiv}}} arXiv] | 2010_10_5_Richards_Notes | 2010_10_5 |