Clemson Analysis and PDE Seminar Schedule

Spring 2024

Insights into the Regularity of 3D Navier-Stokes Equations and the Impact of Rotation

Speaker:                Aseel Farhat - Florida State University
Time:                      Wednesday, February 7, 2024 - 11:15 am
Location:               Martin M103 and Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
In this talk, we will address the regularity challenges posed by the three-dimensional (3D) Navier-Stokes equations (NSE). Specifically, we demonstrate that the equations remain globally regular within a suitable functional framework, incorporating the intermittency observed in regions of intense vorticity/velocity gradients in computational simulations of 3D turbulence. Additionally, we will explore the impact of planetary rotation on fluid dynamics. Our recent findings shed light on the observed effects of rotation in numerical experiments, with the aim of providing a comprehensive understanding of the intricate interplay between regularity and rotation in complex fluid systems.

Singular integrals in Banach function spaces

Speaker:                Zoe Nieraeth (she/her) - Basque Center for Applied Mathematics
Time:                      Wednesday, March 27, 2024 - 11:15 am
Location:               Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
Singular integrals arise when studying properties of the solution of a given PDE by applying them to the initial data. Classically, this was studied for data belonging to weighted Lebesgue spaces. However, this is not always the right setting for every problem. For example, in recent years problems involving non-homogeneous data have been studied through weighted variable Lebesgue spaces, and elliptic boundary value problems through weighted Morrey spaces. These spaces are examples of the generalized notion of Banach function spaces. In this talk I will discuss a unification and extension to a limited range setting of some results classically studied for weighted Lebesgue spaces, such as the Rubio de Francia extrapolation theorem, through this general framework.

Concerning a nonstandard inf-sup formulation of a certain compressible flow -- incompressible fluid PDE system

Speaker:                George Avalos - University of Nebraska-Lincoln
Time:                      Wednesday, April 3, 2024 - 11:15 am
Location:               Martin M104 and Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
In this talk, we discuss our recent work concerning a nonstandard implementation of the Babuska-Brezzi Theorem, by way of ascertaining strongly continuous semigroup generation for a certain coupled compressible flow -- incompressible fluid dynamics. The coupling between these two disparate dynamics is enacted via a boundary interface. The modus operandi, entailed in this (continuous) mixed variational formulation, allows for the derivation of a (discrete) finite element method (FEM) with which to numerically approximate the fluid-flow solution variables. This is joint work with Paula Egging.

Application of the Enhanced Dissipation Phenomenon

Speaker:                Siming He - University of South Carolina
Time:                      Friday, April 5, 2024 - 11:15 am
Location:               Martin M103 and Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
The dissipation effect can be amplified when the fluid transport effect is apparent. This amplification is known as the enhanced dissipation phenomenon. In specific parameter regimes, this fluid mechanism can contribute to suppressing potential singularities in nonlinear chemotaxis systems, ensuring hydrodynamic stability of shear flows, and facilitating communication among agents in collective dynamics. In this talk, I will survey several results related to the enhanced dissipation phenomenon.

Approximation in Hölder spaces with applications to the compactness of the bi-commutator

Speaker:                Tuomas Oikari - Universitat Autònoma de Barcelona
Time:                      Wednesday, April 24, 2024 - 11:15 am
Location:               Martin M103 and Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
We discuss commutators of Calderón-Zygmund operators.It is classical that the Lp-to-Lp bounded commutators are those with symbols in BMO;that Lp-to-Lp compact commutators are those with symbols in VMO, a subspace of BMO;and that VMO coincides with those BMO symbols approximable by N := compactly supported smooth functions.

More recently, it was shown that all non-trivial Lp-to-Lq compact commutators, when p<q, are exactly those bounded commutators with symbols in Hölder classes approximable by N. A parallel holds in the case p>q. Thus, the high-level structure of the Lp-to-Lp case was fully recovered in the Lp-to-Lq cases, when p is not q. It was later noted that in fact all the sufficient conditions for compactness in the cases “p<q”, “p=q” and “p>q” follow from each other by a simple abstract principle coupled with the technically difficult fact that N constitutes a common shared dense subspace of all the symbol classes for boundedness. 

The following is morally true: to understand the Lp-to-Lq compactness of commutators, with p,q ordered anyhow, it is enough to completely understand:
1. approximability by nice functions N in ALL the different symbol classes that characterize boundedness,
2. compactness only in ONE of the ranges “p<q”, “p=q” and “p>q”.

In this talk, we discuss how the above two-step scheme can be realized in the study of the mixed Lp(Lq)-to-Ls(Lt) compactness of the bi-commutator [T2,[b,T1]]. Among other novel results, we have obtained a neat proof of the sufficiency of product VMO for bi-commutator compactness. 

Joint work with Carlos Mudarra and Henri Martikainen.

Nonlocal Games

Speaker:                Priyanga Ganesan - University of California San Diego
Time:                      Friday, April 26, 2024 - 11:15 am
Location:               Martin M103 and Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
A nonlocal game involves two non-communicating players who cooperatively strive to accomplish a task together by giving a correct pair of answers to questions posed by an external referee. These games provide a convenient mathematical framework for studying the advantages and limitations of quantum entanglement  and are widely studied in the context of quantum information theory. 

In recent years, nonlocal games have received significant attention in mathematics and resulted in highly fruitful interactions, including the recent resolution of the 50-year old Connes Embedding Problem in operator algebras. In this talk, I will provide an introduction to the theory of non-local games and quantum correlation classes. We will discuss the role of C*-algebras and operator systems in the study of winning strategies for nonlocal games. It will be shown that mathematical structures arising from entanglement-assisted strategies for nonlocal games can be naturally interpreted and studied using tools from analysis. I will then present recent work on a general operator algebraic framework for non local games where the questions or answers of the game are allowed to be quantum states.