Fall 2024 Seminars

Zoom Link for the Fall Semester 2024

the physical seminar is either in EA295 on Tuesdays 1:50-2:50pm

Abstracts

Robin Ming Chen (University of Pittsburgh) Mar 27 3:00-4:00pm EA265

Title:

Abstract:

Jia-Wei Chu (Hong Kong Polytech. University) Feb 11, 2025

Title: Global dynamics of an SIS epidemic model with cross-diffusion: applications to quarantine measures

Abstract: We consider an SIS model with a cross-diffusion dispersal strategy for the infected individuals describing the public health intervention measures (like quarantine) during the outbreak of infectious diseases.

Edriss Titi

Title: On Recent Advances of the 3D Euler Equations by Means of Examples

Abstract: In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for regularizing and stabilizing certain evolution equations, such as the Euler, Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit enhanced dissipation mechanism due to large spatial average in the initial data — a phenomenon which is similar to the “Landau-damping” effect.

Daozhou Gao

Title: Effects of Human Movement on Disease Spread: Persistence versus Prevalence

Abstract: Human movement not only facilitates disease spread but also poses a serious challenge to disease control and eradication. It is common to use patch models to describe the spatial spread of infectious diseases in a discrete space. The basic reproduction number R0 usually serves as a threshold for disease extinction and persistence. Thus, it is desirable to control population dispersal such that R0 is reduced to less than 1 to achieve disease eradication. However, in reality, disease eradication is extremely difficult or even impossible for most infectious diseases. Reducing disease prevalence (proportion of people being infected) to a low level is a more feasible and cost-effective goal. In this talk, based on an SIS patch model initially proposed and analyzed by Allen et al. (SIAM J Appl Math, 2007), I will explore the influence of dispersal intensity and dispersal asymmetry on the disease persistence and disease prevalence. Our study highlights the necessity of evaluating control measures with other quantities besides the basic reproduction number.

Timur Akhunov

Title: Modified energy method for illposedness of dispersive equations

Abstract: Korteweg and de Vries in 1890s derived an equation that bears their name to elucidate unusual behavior of water waves. They discovered solitons that behave like billiard balls when interacting. Can solitons be made compactly supported? Rosenau-Hyman in 93 proposed partial differential equations with such solutions that they dubbed “compactons”. Wellposedness of compacton equations is poorly understood. A plausible avenue to prove illposedness results is a modified energy method.

Vlad Kobzar (Sep 17)

Title: PDE-Based Analysis of Prediction with Expert Advice and Two-Armed Bandit

Abstract: This talk addresses the classic online learning problem of prediction with expert advice (the expert problem): at each round until the final time, the predictor (player) uses guidance from a collection of experts with the goal of minimizing the difference (regret) between the player’s loss and that of the best performing expert in hindsight. The experts’ losses are determined by the environment (adversary). Using verification arguments from optimal control theory, we view the task of finding lower and upper bounds on the value of the expert problem (regret) as the problem of finding sub- and supersolutions of certain partial differential equations (PDEs). These sub- and supersolutions serve as the potentials for player and adversary strategies, which lead to the corresponding bounds. To get explicit bounds, we use closed-form solutions of specific PDEs. In certain regimes, these bounds improve upon the previous state of the art. We will also briefly discuss our subsequent work generalizing this analysis to the two-armed bandit problem, which is a partial information counterpart of the corresponding expert problem. This talk is based on joint work with Robert Kohn and Zhilei Wang.

Chang-Hong Wu

Title:  Spreading fronts arising from the singular limit of reaction-diffusion systems

Abstract:
In this talk, we will focus on the singular limit of reaction-diffusion systems to gain insight into the formation of spreading fronts of invasive species. We will derive some free boundary problems and provide interpretations for spreading fronts from a modeling perspective. Additionally, numerical examples will be presented to facilitate discussion on invasion speed. This talk is based on joint works with Hirofumi Izuhara and Harunori Monobe.

Vlad Kobzar (Oct 29)

Title: A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit

Abstract: The multiarmed bandit is one of the oldest problems in machine learning and is a foundational problem in reinforcement learning, large language modeling, adaptive medical trials, digital heath, marketing and recommendation systems, to name a few.  At each round, the predictor (player) selects a probability distribution from a finite collection of distributions (arms) with the goal of minimizing the difference (regret) between the player’s rewards sampled from the selected arms and the rewards of the arm with the highest expected reward. The player’s choice of the arm and the reward sampled from that arm are revealed to the player, and this prediction process is repeated until the final round.

Peter Takac

Title: A p(x)-Laplacian Extension of a Convexity Result and Applications to Quasi-linear Elliptic BVPs

Abstract: The main result of this work is a new extension of the well-known inequality by Dı́az and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, ∆ p(x) u ≡ div(|∇u| p(x)−2 ∇u). Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Chris Henderson (Arizona) Jan 23, 2025. (Colloquium Talk)

Title
A new approach to regularity, well-posedness, and blow-up in the Boltzmann equation

Abstract

Zhuolun Yang (Brown) Dec 19, 2024. (Colloquium Talk)

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

Abstract
When two extreme conductors are located close to each other, the electric field, represented by the gradient of solution to an elliptic PDE, may blow up as the distance between the inclusions approaches zero. This is called field concentration phenomenon, a central topic in composite material research. In this talk, I will review key literature and highlight two directions of my work. First, for the insulated conductivity problem, we derived the optimal gradient estimate of solutions, which settled down a major open problem in this area. Second, for a model involving imperfectly bonded conductors, which has significant applications in physics and biology, we discovered a novel dichotomy in field concentration behavior driven by the bonding parameter.

This talk is based on joint work with Hongjie Dong (Brown U.), Yanyan Li (Rutgers U.), and Hanye Zhu (Duke U.).

Chen-Chih Lai (Columbia) Jan 7, 2025 (Colloquium Talk)

Title:
Free boundary problems of PDEs concerning thermal effects in bubble dynamics

Abstract:
Bubble dynamics plays a crucial role not only in fundamental and applied physics but also in various engineering and industrial applications. In this talk, we will discuss mathematical models describing the deformation of a gas bubble in a liquid. These models fall under the category of fluid interface problems, a subclass of free boundary problems.

To provide a comprehensive understanding, we will begin with a brief overview of fundamental PDEs governing fluid dynamics, the associated boundary conditions, and the broader context of bubble dynamics. Subsequently, attention will be directed towards the thermal decay of bubble oscillation, particularly examining the approximate isobaric model proposed by A. Prosperetti in [J. Fluid Mech. 1991], under which the gas pressure within the bubble is spatially uniform and follows the ideal gas law. This model exhibits a one-parameter family of spherical equilibria, parametrized by the bubble mass. We prove that this family forms an attracting centre manifold for small spherically symmetric perturbations, with solutions approaching the manifold at an exponential rate over time. Furthermore, we show that under either liquid viscosity or irrotational flow assumptions, any equilibrium bubble must be spherical. Additionally, the manifold of spherically symmetric equilibria captures all regular sphreically symmetric equilibrium.

We also explore the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic, spherically symmetric external sound field. For this periodically forced system, we establish the existence of a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations.

In the latter part of the talk, I will discuss some limitations of the isobaric model in a more general (nonspherically symmetric) irrotational setting. Specifically, I will address issues such as (1) the undamped oscillations of shape modes due to spatial uniformity of the gas pressure, and (2) the incompatibility between viscosity and irrotationality assumptions. Our results suggest that to accurately capture the effect of thermal damping on the dynamics of general deformations of a gas bubble, the model should be considered within a framework that includes either non-zero vorticity, corrections to the isobaric approximation, or both.

If time permits, I will present ongoing work on the existence of nonsphreically symmetric equilibrium bubbles in a rotational framework.

This talk is based on joint work with Michael I. Weinstein ([Arch. Ration. Mech. Anal. 2023], [Nonlinear Anal. 2024], [arXiv:2408.03787], and work in progress).

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract:

Title:

Abstract: