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Wed Feb 26

Analysis and PDE Working Seminar

3:35pm - Vincent Hall 570
Local stability of the critical Fisher-KPP front via resolvent expansions near the essential spectrum
Montie Avery

We revisit the stability of the critical front in the Fisher-KPP equation, which travels with the linear spreading speed c = 2. We recover a celebrated result of Gallay with a new method, establishing stability of the critical front with optimal decay rate t^(-3/2) as well as an asymptotic description of the perturbation of the front. Our approach is based on studying detailed regularity properties of the resolvent for this problem in algebraically weighted spaces near the branch point in the absolute spectrum, and renders the nonlinear analysis much simpler. We briefly further explore the relationship between the localization of perturbations and their decay rate.

Mon Mar 30

Analysis and PDE Working Seminar

3:35pm - Zoom link. https://umn.zoom.us/j/881291791
Mathematical foundations of slender body theory
Laurel Ohm

Slender body theory (SBT) facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Despite the popularity of SBT in computational models, there had been no rigorous analysis of the error in using SBT to approximate the interaction of a thin fiber with fluid. In this talk, we develop a PDE framework for analyzing the error introduced by this approximation. In particular, given a 1D force along the fiber centerline, we define a notion of ‘true’ solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. We also present similar estimates in case of free-ended and rigid filaments.

Mon Apr 06

Analysis and PDE Working Seminar

3:35pm - VinH 301
Analysis & PDE Working Seminar
Timur Yastrzhembskiy
Mon Apr 13

Analysis and PDE Working Seminar

3:35pm - Zoom link. https://umn.zoom.us/j/538855343
The boundary value problems in higher codimension
Zanbing Dai

The boundary value problems have been studied for decades. People first studied boundary value problems for the Laplace operator on bounded Lipschitz domains. Using a change of variable argument, we can map Lipschitz domains onto the upper half plane $\mathbb{R}^{d+1}_+$ and converts Laplace operator into a second order elliptic divergence operator, whose coefficient satisfies a certain smoothness condition, the Carleson measure condition. Recently, David, Feneuil and Mayboroda developed an elliptic theory in higher co- dimension. They studied a particular degenerate second order elliptic operator $L={\rm div} A\nabla$. Now, the domain we are interested in has more than one non-tangential direction. In this talk, I will focus on flat domain $\mathbb{R}^n\setminus \mathbb{R}^d$ and introduce the Dirichlet results, which has been proved recently. Finally, I will introduce my project on the regularity problem in higher codimension.

Mon Apr 20

Analysis and PDE Working Seminar

3:35pm - Vincent Hall 6
Analysis & PDE Working Seminar
Dallas Albritton
Mon May 04

Analysis and PDE Working Seminar

3:35pm - Vincent Hall 6
Analysis & PDE Working Seminar
Ryan Matzke