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Seminar In the Analysis and Methods of PDE (SIAM PDE)
The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

More information can be found here: https://sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

Date: December 2, 2021, 11:30 am ET

Speaker: Yao Yao, National University of Singapore

Title: Symmetry and uniqueness via a variational approach

Abstract: For some nonlocal PDEs, their steady states can be seen as critical points of some associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progress in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan). I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Moderator: Michele Coti Zelati, Imperial College London
Jan 6, 2022 11:30 AM
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Speakers

Yao Yao
@National University of Singapore
Yao Yao is currently an Associate Professor of Mathematics at the National University of Singapore. She received her BS degree from Peking University in 2007, and PhD degree in 2012 from UCLA. She was a Van Vleck Visiting Assistant Professor at University of Wisconsin-Madison in 2012-2015, and an Assistant Professor at Georgia Institute of Technology in 2015-2021. Her research focuses on the analysis of partial differential equations arising in mathematical biology and fluid dynamics, especially on the equations with a nonlocal transport term. She was a recipient of the NSF CAREER Award in 2018 and Sloan Research Fellowship in 2020.