# Difference between revisions of "PDE Geometric Analysis seminar"

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|Longjie Zhang (University of Tokyo) | |Longjie Zhang (University of Tokyo) | ||

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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. | 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. | ||

+ | ===Longjie Zhang=== | ||

+ | |||

+ | On curvature flow with driving force starting as singular initial curve in the plane | ||

+ | |||

+ | We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle. | ||

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

## Revision as of 12:37, 8 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) | On curvature flow with driving force starting as singular initial curve in the plane | 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) | Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application. | 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.

### Longjie Zhang

On curvature flow with driving force starting as singular initial curve in the plane

We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.

### 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.

### Tuoc Phan

Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.

Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.