# PDE Geometric Analysis seminar

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

# Seminar Schedule Fall 2012

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

September 17 | Bing Wang (UW Madison) |
On the regularity of limit space |
local |

October 15 | Peter Polacik (University of Minnesota) |
Exponential separation between positive and sign-changing solutions and its applications |
Zlatos |

November 26 | Kyudong Choi (UW Madison) |
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations |
local |

December 10 | Yao Yao (UW Madison) |
Confinement for nonlocal interaction equation with repulsive-attractive kernels |
local |

February 4 | reserved | Feldman |

# Abstracts

### Bing Wang (UW Madison)

*On the regularity of limit space*

This is a joint work with Gang Tian. In this talk, we will discuss how to improve regularity of the limit space by Ricci flow. We study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L1-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of K¨ahler manifolds.

### Peter Polacik (University of Minnesota)

* Exponential separation between positive and sign-changing solutions and its applications*

In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.

### Kyudong Choi (UW Madison)

*Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations*

We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We prove that k-th derivative of weak solutions is locally integrable in space-time for any real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of the initial data and on the domain of integration. Moreover, they are valid even for k >= 3 as long as we have a smooth solution. The proof uses a standard approximation of Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. Vasseur.

### Yao Yao (UW Madison)

*Confinement for nonlocal interaction equation with repulsive-attractive kernels*

In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.