Tue, 02 Sep 2025
15:00
L4

On a classification of steady solutions to two-dimensional Euler equations

Changfeng Gui
(University of Macau)
Abstract
In this talk,  I shall  provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature.  A  further classification  of this type of solutions will also be  discussed.    As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines.
 
This  talk is  based on  joint works with David Ruiz,  Chunjing Xie and  Huan Xu.

A consultation on the University’s REF 2029 Code of Practice will run from early July to mid-September. As part of the consultation, Research Services are holding a series of events, opportunities to find out more about REF, ask questions and provide feedback. 

Monday 21 July, 10-11am – Online information session 

Tue, 02 Sep 2025
14:00
L4

Uniqueness of critical points of the second Neumann eigenfunctions on triangles

Ruofei Yao
(South China University of Technology)
Abstract

The hot spots conjecture, proposed by Rauch in 1974, asserts that the second Neumann eigenfunction of the Laplacian achieves its global maximum (the hottest point) exclusively on the boundary of the domain. Notably, for triangular domains, the absence of interior critical points was recently established by Judge and Mondal in [Ann. Math., 2022]. Nevertheless, several important questions about the second Neumann eigenfunction in triangles remain open. In this talk, we address issues such as: (1) the uniqueness of non-vertex critical points; (2) the necessary and sufficient conditions for the existence of non-vertex critical points; (3) the precise location of the global extrema; (4) the position of the nodal line; among others. Our results not only confirm both the original theorem and Conjecture 13.6 proposed by Judge and Mondal in [Ann. Math., 2020], but also accomplish a key objective outlined in the Polymath 7 research thread 1 led by Terence Tao. Furthermore, we resolve an eigenvalue inequality conjectured by Siudeja [Proc. Amer. Math. Soc., 2016] concerning the ordering of mixed Dirichlet–Neumann Laplacian eigenvalues for triangles. Our approach employs the continuity method via domain deformation. 

 

Watch any Glastonbury? Go to Glastonbury? Maybe not your thing?  Well, controversy aside, the one thing you have to say about Glasto is that musicians have to be truly terrible to not be loved or get great reviews. The vibe is so positive that all faculties are suspended. And, you know, that's no bad thing.

Olivia Rodrigo closed Sunday night to great acclaim (unsurprisingly). So for the young and less young among you, here she is with Robert Smith of the Cure. 

Schism from exhibition

Since the autumn of 2022, Conrad Shawcross' artworks have lived and breathed in our building. It is perhaps their natural home given they are inspired by science and, in particular, mathematics.

Mon, 29 Sep 2025

14:00 - 15:00
Lecture Room 3

TBA

Wael Mattar
(Tel Aviv University)
Abstract

TBA

We couldn't really let it go and Conrad didn't really want to take it away so the artworks will stay for at least another year.  Look out for a repeat of this email in 12 months.

Fast randomized least-squares solvers can be just as accurate and stable as classical direct solvers
Epperly, E Meier, M Nakatsukasa, Y Communications on Pure and Applied Mathematics
Subscribe to