# Difference between revisions of "PDE Geometric Analysis seminar"

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===Mihaela Ifrim=== | ===Mihaela Ifrim=== | ||

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+ | Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation | ||

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+ | Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru. | ||

===Jaeyoung Byeon=== | ===Jaeyoung Byeon=== |

## Revision as of 14:04, 7 September 2017

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

### Tentative schedule for Spring 2018

## PDE GA Seminar Schedule Fall 2017

date | speaker | title | host(s) |
---|---|---|---|

September 11 | Mihaela Ifrim (UW) | Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation | Kim & Tran |

September 18 | Longjie Zhang (University of Tokyo) | TBD | Angenent |

September 22,
VV B239 4:00pm |
Jaeyoung Byeon (KAIST) | Colloquium: Patterns formation for elliptic systems with large interaction forces | Rabinowitz |

September 25 | Tuoc Phan (UTK) | TBD | Tran |

September 26,
VV B139 4:00pm |
Hiroyoshi Mitake (Hiroshima University) | Joint Analysis/PDE seminar | Tran |

September 29,
VV901 2:25pm |
Dongnam Ko (CMU & SNU) | a joint seminar with ACMS: TBD | Shi Jin & Kim |

October 2 | No seminar due to a KI-Net conference | ||

October 9 | Sameer Iyer (Brown University) | TBD | Kim |

October 16 | Jingrui Cheng (UW) | TBD | Kim & Tran |

October 23 | Donghyun Lee (UW) | TBD | Kim & Tran |

October 30 | Myoungjean Bae (POSTECH) | TBD | Feldman |

November 6 | Jingchen Hu (USTC and UW) | TBD | Kim & Tran |

## Abstracts

### Mihaela Ifrim

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.

### Jaeyoung Byeon

Title: Patterns formation for elliptic systems with large interaction forces

Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.