Posted February 10, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
Hongki Jung, Louisiana State University
$\Lambda(p)$--subsets of manifolds
In 1989, Bourgain proved the existence of maximal $\Lambda(p)$--subsets within the collection of mutual orthogonal functions. We shall explore the Euclidean analogue of $\Lambda(p)$—sets through localization. As a result, we construct maximal $\Lambda(p)$--subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $p$. This is joint work with C. Demeter and D. Ryou.
Posted November 21, 2024
Last modified March 5, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 233
Christoph Fischbacher, Baylor University
Non-selfadjoint operators with non-local point interactions
In this talk, I will discuss non-selfadjoint differential operators of the form $i\frac{d}{dx}+V+k\langle \delta,\cdot\rangle$ and $-\frac{d^2}{dx^2}+V+k\langle \delta,\cdot\rangle$, where $V$ is a bounded complex potential. The additional term, formally given by $k\langle \delta,\cdot\rangle$, is referred to as ``non-local point interaction" and has been studied in the selfadjoint context by Albeverio, Cojuhari, Debowska, I.L. Nizhnik, and L.P. Nizhnik. I will begin with a discussion of the spectrum of the first-order operators on the interval and give precise estimates on the location of the eigenvalues. Moreover, we will show that the root vectors of these operators form a Riesz basis. If the initial operator is dissipative (all eigenvalues have nonnegative imaginary part), I will discuss the possibility of choosing the non-local point interaction in such a way that it generates a real eigenvalue even if the potential is very dissipative. After this, I will focus on the dissipative second order-case and show similar results on constructing realizations with a real eigenvalue. Based on previous and ongoing collaborations with Matthias Hofmann, Andrés Lopez Patiño, Sergey Naboko, Danie Paraiso, Chloe Povey-Rowe, Monika Winklmeier, Ian Wood, and Brady Zimmerman.
Posted October 14, 2024
Last modified February 28, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm – 4:30 pm Thursday, March 6, 2025 Lockett 232
Alexandru Hening, Texas A&M University
Stochastic Population Dynamics in Discrete Time
I will present a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time stochastic difference equations that can include population structure, eco-environmental feedback or other internal or external factors. Using the general theory, I will showcase some interesting examples. I will end my talk by explaining how the population size at equilibrium is influenced by environmental fluctuations.
Posted February 21, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
John Jairo Lopez, Tulane University
Title TBA
Abstract TBA (Host: Stephen Shipman)
Posted March 25, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm – 4:20 pm Lockett 232
Robert Lipton, Mathematics Department, LSU
Dynamic Fast Crack Growth
Nonlocal modeleling for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. The material displacement field is uniquely determined by the initial boundary value problem. The theory naturally satisfies energy balance, with positive energy dissipation rate in accord with the Clausius-Duhem inequality. Notably, these properties are not imposed but follow directly from the constitutive law and evolution equation. The limit of vanishing non-locality is analized using simple arguments from geometric measure theory to identify the limit damage energy and weak convergence methods of pde to identify the limit solution. The limiting energy is the Grifith fracture energy. The limit evolution is seen to be a weak solution for the wave equation on a time dependent domain. The exsistence theory for such solutions was recently developed in Dal Maso and Toader, J. Differ. Equ. 266, 3209–3246 (2019).
Posted December 10, 2024
Last modified January 5, 2025
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett
Yuanzhen Shao, University of Alabama
Some recent developments in the study of magnetoviscoelastic fluids
In this talk, we consider the motion of a magnetoviscoelastic fluid in a nonisothermal environment. When the deformation tensor field is governed by a regularized transport equation, the motion of the fluid can be described by a quasilinear parabolic system. We will establish the local existence and uniqueness of a strong solution. Then it will be shown that a solution initially close to a constant equilibrium exists globally and converges to a (possibly different) constant equilibrium. Further, we will show that that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria. If time permits, we will discuss some recent advancements regarding the scenario where the deformation tensor is modeled by a transport equation. In particular, we will discuss the local existence and uniqueness of a strong solution as well as global existence for small initial data.