Virtual Analysis and PDE Seminar (VAPS)



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.

VAPS was started in Fall 2020. For talks and information of the previous years (2020-2023), see here or here.

For some recorded talks of VAPS, see VAPS Youtube channel.

Talks in academic year 2023-2024


Title: A mean curvature flow approach to density of minimal cones

Abstract: Minimal cones are models for singularities in minimal submanifolds, as well as stationary solutions to the mean curvature flow. In this talk, I will explain how to utilize mean curvature flow to yield near optimal estimates on density of topologically nontrivial minimal cones. This is joint with Jacob Bernstein.


Zhiwu Lin (Gatech), November 16. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title：Dynamical magneto-rotational instability                        


Abstract: Magneto-rotational instability (MRI) is an important instability mechanism for rotating flows with magnetic fields. When the strength of the magnetic field tends to zero, the stability criterion for rotating flow is different from the classical Rayleigh criterion for rotating flow without magnetic field. MRI had found many physical applications, for example in the explanation of turbulence and enhanced momentum transport in accretion disks around black holes. We gave a rigorous proof of linear MRI and a complete description of the spectra and semigroup growth. Moreover, we prove nonlinear stability and instability from the sharp linear stability/instability criteria. This is a joint work with Yucong Wang and Wenpei Wu.



Yuming Paul Zhang (Auburn), December 14. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth


Abstract: In this work, we investigate the general Porous Medium Equations used in tumor growth models, emphasizing their distinctive

free boundaries. While much literature focuses on the incompressible limit of solutions, we provide the first proof of the convergence of

free boundaries in Hausdorff distance in this limit. For this purpose, we establish uniform-in-$m$ strict propagation and stability properties

of the free boundaries. As another consequence of these, we provide an upper bound of the Hausdorff dimension of the free boundary and

show that the limiting free boundary has finite perimeter. This is an ongoing joint work with Jiajun Tong.


Zhuolun Yang (Brown), January 18. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: Gradient estimates for conductivity problems from high-contrast composite materials


Abstract: In this talk, I will describe the conductivity problem from composite materials. The electric field, represented by the gradient of solutions, may blow up as the distance between inclusions approaches to 0. When the current-electric field relation obeys Ohm's law, we obtained an optimal gradient estimate of solutions in terms of the distance between two insulators, which settled down a major open problem in this area. I will also present our recent results on both perfect and insulated conductivity problems when the current-electric field relation is a power law. The talk is based on joint work with Hongjie Dong (Brown), Yanyan Li (Rutgers), and Hanye Zhu (Brown).


Leonardo Abbrescia (Vanderbilt and Georgia Tech), February 29. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: The structure of the maximal development in 3D compressible Euler flow


Abstract: I will discuss my works with J. Speck on 3D compressible Euler flow, in which we reveal the structure of the maximal Cauchy (MCD) development for open sets of initial data without symmetry, irrotationality, or isentropicity assumptions. The MCD is roughly the largest spacetime region on which the solution exists classically and is uniquely determined by the initial data – the holy grail in the classical study of hyperbolic equations. In particular, we describe the full structure of the singular set, where the solution’s gradient blows up in a shock singularity, as well as the emergence of a Cauchy horizon from the singularity. It is known that MCD's might fail to be unique unless one constructs it in its entirety and proves that it enjoys some crucial properties. The portion of the MCDs we construct proves that a localized version of the crucial properties hold for shock-forming initial data. Time permitting, I will discuss some of the many open problems in the field.





Dennis Kriventsov (Rutgers), March 21. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: Stationary solutions to the Bernoulli free boundary problem


Abstract: Free boundary problems of Bernoulli type arise naturally in fluid dynamics, thermal models, shape optimization, and other contexts. We will focus on the simplest possible archetype problem, and consider many examples of solutions in one and two dimensions. Then we will look at the question of which kinds of solutions are closed under taking limits, and how those limits look like–this is a topic of practical importance for constructing solutions by any argument save for direct minimization. Thanks to a recent breakthrough in joint work with Georg Weiss, we can now give a very precise and descriptive answer to such questions.


Matias Delgadino (UT Austin), April 4. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: Generative Adversarial Networks: Dynamics and Mode Collapse

 

Abstract:  Generative Adversarial Networks (GANs) was one of the first Machine Learning algorithms to be able to generate remarkably realistic synthetic images. In this presentation, we delve into the mechanics of the GAN algorithm and its profound relationship with optimal transport theory. Through a detailed exploration, we illuminate how GAN approximates a system of PDE, particularly evident in shallow network architectures. Furthermore, we investigate the phenomenon of mode collapse, a well-known pathological behavior in GANs, and elucidate its connection to the underlying PDE framework through an illustrative example.


Diego Cordoba (ICMAT), May 23. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).


Title: Finite time blow-up for the hypodissipative Navier Stokes equations


Abstract: In this talk we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla|^{\alpha}$ for any $\alpha\in [0, \alpha_0)$ ($\alpha_0 = 0.101\cdots$).


Zoom link for this talk only.

https://ucsd.zoom.us/j/97964366052?pwd=alRhdWVScFZ0WXpDNjlkWkZSZW1WUT09

Meeting ID: 979 6436 6052

Password: VAPS24




Join Zoom Meeting for VAPS

https://uwmadison.zoom.us/j/97685819574?pwd=OGlucTJLWkJYb0d2bWlKN1lrRTRPUT09

Meeting ID: 976 8581 9574

Passcode: 216931