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 2024-2025

1. Hosted by UC Irvine - October 2024
   Aria Halavati (Courant), October 3. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: Decay of excess for the abelian Higgs model

Abstract: Entire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type large-scale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blow-downs and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of Savin's theorem in codimension two. This is a joint work with Guido de Philippis and Alessandro Pigati.

2. Hosted by Purdue - November 2024
   Armin Schikorra (Pittsburgh),  November 21. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: On Calderon-Zygmund Theory for the p-Laplacian

Abstract: Calderon-Zygmund theory for the Laplace equation is among the most classical results in Harmonic Analysis. It was conjectured by Iwaniec in 1983 that an analogue theory holds for the p-Laplace. I will show how you can disprove this conjecture using one-dimensional calculus and Linear Algebra.

3. Hosted by UW - December 2024
   Jiwoong  Jang (Maryland),  December 12. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: Periodic homogenization of geometric equations without perturbed correctors

Abstract: Proving homogenization is a subtle issue for geometric equations due to the discontinuity when the gradient vanishes. To conclude homogenization, the work of Caffarelli-Monneau provides a sufficient condition, namely that perturbed correctors exist. However, some noncoercive equations recently studied do not satisfy this condition. In this talk, we present the homogenization result of geometric equations without using perturbed correctors. For coercive equations, a quantitative result is derived by the fact that they remain coercive under perturbation. We present an example that homogenizes with a rate slower than O(\varepilon) in the last part.

4. Hosted by Brown - January 2025
   Hanye Zhu (Duke University),  January 23. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: Field concentration for the conductivity problem of closely spaced perfect conductors with imperfect bonding interfaces

Abstract: When two conducting or insulating inclusions are closely located in a composite, the gradient of the solution to the conductivity problem may become arbitrarily large in the narrow region in between them as the distance between the inclusions tends to zero. This phenomenon is known as field concentration, a central topic in the theory of composite material. We study the field concentration between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type, where the potential is allowed to be discontinuous across the interfacial boundaries. The imperfect bonding interface condition holds significant practical relevance, as it approximates the membrane structure in biological systems. We discover a new dichotomy for the field concentration depending on the bonding parameter of the interfaces. The results are surprisingly different from the perfect interface setting. Based on joint work with Hongjie Dong and Zhuolun Yang.

5. Hosted by Gatech - February 2025
   Edgard Pimentel (University of Coimbra, Portugal), February 13. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).
   
   

Title: Fully nonlinear free transmission problems

Abstract: We examine fully nonlinear free transmission problems in uniformly elliptic and degenerate settings. We start with previous

results concerning the existence of (viscosity) solutions and their interior regularity estimates. Then, we continue with the discussion of boundary regularity results; more precisely, we are interested in the regularity of the solutions at the intersection of the free and fixed boundaries. We conclude with a recent excursion into the realm of numerical schemes. If time permits, we close the talk with related open problems.

6. Hosted by Columbia - March 2025
   Daniel Restrepo (JHU), March 6. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: Plateau’s laws for surfaces of minimal capacity

Abstract: We will discuss a novel family of free boundary problems that arise in the study of variational problems with topological constraints in their level sets. Our two main examples are the classical problem surfaces of minimal capacity spanning a given wire frame and the Allen-Cahn approximation to solutions of the Plateau problem.  The heart of the matter in this reformulation of classical variational problems is the notion of spanning, which corresponds to a measure theoretical version of the so-called homotopic spanning condition introduced by Harrison and Pugh. Based on joint works with Anna Skoborogatova, Michael Novack, and Francesco Maggi.

7. Hosted by UCLA - April 2025
   Chris Henderson (Arizona), April 24. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Title: Nash inequalities and boundary behavior of kinetic equations

Abstract: Kinetic equations model systems, such as a gas, where particles move through space according to a velocity that is diffusing (due to, say, collisions with other particles).  The presence of spatial boundaries in these models causes technical issues because they are first order in the spatial variable and therefore cannot be defined everywhere on the boundary.  In this talk, I will present $L^1-L^\infty$ estimates that yield sharp bounds on the behavior at the spatial boundary.  The main estimate is a kinetic version of the Nash inequality.  This is a joint work with Giacomo Lucertini and Weinan Wang.

8. Hosted by UCSD - May 2025

Jiajie Chen (NYU), May 29. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Zoom: https://ucsd.zoom.us/j/96692922445?pwd=C4iLvAiDV0D3IaFlTXbOzLZJ0Wvbbq.1

Meeting ID: 966 9292 2445. Password: VAPS0525.

Title: Axisymmetry and vorticity blowup in compressible Euler equations 

Abstract: While it is known that a pre-shock or implosion singularity can form in finite time for the compressible Euler equations from smooth initial data, whether the vorticity blows up in finite time has remained an open problem. In this talk, we will explore the Euler equations with axisymmetry. Using (nearly) axisymmetric flow, we construct vorticity blowup in the compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity and is accompanied by an axisymmetric implosion, where the swirl velocity exhibits full stability as opposed to finite co-dimension stability. 

9. Hosted by UT Austin - June 2025

Maria Soria-Carro (Rutgers University), June 5. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

Zoom: https://utexas.zoom.us/j/91451327745

Meeting ID: 914 5132 7745

Title: A parabolic free boundary problem arising from the jump of conductivity 

Abstract: We will introduce a parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Our main goal is to establish regularity properties of both the solutions and the free boundary (the transition surface). First, we will see that, in the parabolic setting, solutions are not necessarily Lipschitz due to the anisotropic nature of the problem. Then, we will show that flat free boundaries are smooth, by adapting the nice geometric techniques developed by Daniela De Silva (2011) to our context. This is a joint work with Dennis Kriventsov (Rutgers University).

Talks in academic year 2023-2024

1. Hosted by UC Irvine - October 2023
   Lu Wang (Yale), October 12. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET).

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.

2. Hosted by Purdue - November 2023

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.

3. Hosted by UW - December 2023

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.

4. Hosted by Brown - January 2024

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).

5. Hosted by Gatech - February 2024

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.

6. Hosted by Columbia - March 2024

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.

7. Hosted by UCLA - April 2024

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.

8. Hosted by UCSD - May 2024

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

9. Hosted by UT Austin - June 2024

Nestor Guillen (Texas State), June 6. Time: 12PM-1PM (PT), 1-2PM (MT), 2-3PM (CT), 3-4PM (ET) - cancelled. 

Title:  The Landau equation does not blow up

Abstract: We consider solutions to the space-homogeneous Landau equation with a general family of interaction potentials — including the Coulomb potential. The question of finite time blow up for the equation in the space-homogeneous regime has been a well-known open problem for many decades. In a recent preprint with Luis Silvestre we answer this question by showing the Fisher information is a Lyapunov functional for the equation, a fact that rules out blow ups. In this talk I will review (some) of the history of the Landau equation and discuss the main ideas of the proof, these include the introduction of a related equation in double the number of variables, and a log-Sobolev inequality for functions on the sphere.