Main Purpose
This online seminar aims to provide a convenient platform for mathematicians to disseminate interesting and significant progress in the field of applied analysis and partial differential equations (PDE). In particular, it provides an opportunity for junior researchers to present their work.
Schedule
This seminar is held via zoom meeting on Thursdays at 12-1pm (PST). Each month, participating universities take turns in hosting the seminar. Meeting IDs and passwords will be sent to participants through the seminar mail list. If you wish to be added to the mail list, please contact Connor Mooney (mooneycr@math.uci.edu) or Yifeng Yu (yifengy@uci.edu).
With permission of the speakers, all talks will be recorded and uploaded to our Youtube Channel: Virtual Analysis and PDE Seminar.
Past Talks (2020-2021)
Past Talks (2021-2022)
Upcoming Talks (2022-2023)
1. Oct 6th, 12-1pm (PST) Hosted by UC Irvine
Speaker: Alessio Figalli, ETH Zurich
Title: Complete classification of global solutions to the obstacle problem
Abstract: The characterization of global solutions to the obstacle problems in R^n, or equivalently of null quadrature domains, has been studied for more than 90 years. In this talk, I will discuss a recent result with Eberle and Weiss, where we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
2. Nov 3rd, 12-1pm (PST) Hosted by UC San Diego
Speaker: Thomas Hou, Caltech
Title: A constructive proof of finite time blowup of 3D incompressible Euler equations with smooth data
Abstract: Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, we will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup results. We overcome these difficulties by establishing a constructive proof strategy. We first construct an approximate self-similar blowup profile using the dynamic rescaling formulation. To establish the stability of the approximate blowup profile, we decompose the linearized operator into a leading order operator plus a finite rank perturbation operator. We use sharp functional inequalities and optimal transport to establish the stability of the leading order operator. To estimate the finite rank operator, we use energy estimates and space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. This provides the first rigorous justification of the Hou-Luo blowup scenario.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
3. Dec 8th, 12-1pm (PST) Hosted by UW Madison
Speaker: Tuoc Phan, The University of Tennessee, Knoxville
Title: Some recent results on $L_p$-estimates of solutions to linear elliptic and parabolic equations with singular or degenerate coefficients
Abstract: We discuss several classes of linear elliptic and parabolic equations in which the coefficients can singular or degenerate. Generic weighted and mixed-normed Sobolev spaces will be derived in which the existence, uniqueness, and regularity estimates of solutions are proved under some optimal regularity conditions on the leading coefficients. Some key ideas in the proofs will be pointed out and several future research directions will be mentioned.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
4. Jan 5th, 12-1pm (PST) Hosted by Brown University
Speaker: Yu Yuan, University of Washington, Seattle
Title: A monotonicity approach to Pogorelov’s Hessian estimates for Monge-Ampere equation
Abstract: We present an integral approach to the classical Hessian estimates for the Monge-Ampere equation, originally obtained via a pointwise argument by Pogorelov. The monotonicity employed here results from a maximal surface interpretation of “gradient” graph of solutions in pseudo-Euclidean space.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
5. Feb 16th, 12-1pm (PST) Hosted by Georgia Tech
Speaker: Fabio Pusateri, University of Toronto
Title: Recent results on the stability of solitons, kinks, and radiation damping
Abstract: This talk will give an overview of some recent results on nonlinear evolution equations with potentials, and applications to the stability of solitons and kinks, as well as to the phenomenon of “radiation damping”. Our approach to this class of problems is based on the use of the distorted Fourier transform and the development of multilinear harmonic analysis in this setting; in particular, we use these tools to understand the interaction of localized waves on the background of a large potential, and to handle the singularities in (distorted) Fourier space that naturally arise in many situations. This talk is based on joint works with P. Germain (Imperial), A. Soffer (Rutgers), T. Leger (Princeton), Gong Chen (Georgia Tech), Zhiyuan Zhang (NYU) and Adilbek Kairzhan (U of Toronto).
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
6. March 2nd, 12-1pm (PST) Hosted by Columbia University
Speaker: Nicola Garofalo, University of Padova
Title: Some strong unique continuation results for nonlocal parabolic equations
Abstract: I will discuss two results of strong unique continuation for nonlocal parabolic equations. The former establishes strong uniqueness backwards for global solutions, and is inspired to a well-known result by Poon. The latter is a space-like uniqueness theorem for local solutions which extends previous works of Escauriaza, Fernandez and Vessella. The material presented is taken from joint works with A. Banerjee, and with V. Arya, A. Banerjee and D. Danielli.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
7. April 27th, 12-1pm (PST) Hosted by UCLA
Speaker: Antonio De Rosa，University of Maryland
Title: Min-max construction of anisotropic CMC surfaces
Abstract: We prove the existence of nontrivial closed surfaces with
constant anisotropic mean curvature with respect to elliptic integrands
in closed smooth 3-dimensional Riemannian manifolds. The constructed
min-max surfaces are smooth with at most one singular point. The
constant anisotropic mean curvature can be fixed to be any real number.
In particular, we partially solve a conjecture of Allard [Invent.
Math.,1983] in dimension 3. Joint work with G. De Philippis.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
8. May 4th, 12-1pm (PST) Hosted by Purdue University
Speaker: Dallas Albritton, Princeton University
Title: Kinetic shock profiles for the Landau equation
Abstract: Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Jacob Bedrossian (UCLA) and Matthew Novack (Purdue University).
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
9. June 8th, 12-1pm (PST) Hosted by UT Austin
Speaker: Serena Dipierro, University of Western Australia
Title: Long-range phase transitions
Abstract: Phase transitions are a classical topic of investigation. They represent a complex phenomenon which needs to be attacked with different methodologies and different perspectives. I will discuss some rigidity and symmetry results for a long-range phase coexistence equation, their close relation with surfaces of minimal perimeter and a famous conjecture by Ennio De Giorgi.
Zoom meeting ID: 969 5880 7716. https://uci.zoom.us/j/96958807716
Managing Organizers (2020-2023):
Connor Mooney, Yifeng Yu (UC Irvine)
Participating schools and organizers:
Hongjie Dong (Brown University)
Daniela De Silva, Ovidiu Savin (Columbia University)
Andrzej Swiech, Chongchun Zeng (Georgia Tech University)
Hung V. Tran (University of Wisconsin at Madison)
Changyou Wang (Purdue University)
Inwon Kim (UCLA)
Andrej Zlatos (UCSD)
Stefania Patrizi (UT Austin)