Analysis Seminar 2019/20

The Analysis and Differential Equations Seminar takes place on Thursdays at 2.15 in 4W 1.7 (Wolfson Lecture Theatre).
If you have any queries, or if you would like to be on our e-mail list, please contact the organisers Karsten Matthies and Monica Musso.


Autumn 2019

Date Speaker Title/Abstract
3 Oct Yuxia Guo
Tsinghua
Non-degeneracy of Multi-bubbling Solutions
We consider the following prescribed scalar curvature equations in RN:-Δu = K(|y|))u 2*-2, u > 0 in RN, u ∈ D1,2(RN); where K(r) is a positive function, 2*=2N/(N-2). We first prove a non-degeneracy result for the positive multi-bubbling solutions for the above equation by using the local Pohozaev identities. Then we use this non-degeneracy result to glue together bubbles with different concentration rate to obatin new solutions. Joint works with M.Musso, S.Peng and S.Yan.
10 Oct Javier F Rosenblueth
UNAM
Necessity and Uniqueness of Multipliers in Constrained Optimization
In mathematical programming, first and second order necessary optimality conditions are strongly related to the uniqueness of Lagrange multipliers. In particular, it is well-known that a strict form of the Mangasarian-Fromovitz constraint qualification is equivalent to the uniqueness of Kuhn-Tucker multipliers and, moreover, it implies the satisfaction of second order necessary conditions at a local minimum. In this talk we shall explain how a surprising and entirely different situation occurs in infinite dimensional problems including problems in calculus of variations and optimal control.
17 Oct Miles Wheeler
University of Bath
New exact solutions to the steady 2D Euler equations
We present a large class of explicit "hybrid" equilibria for the 2D Euler equations, consisting of point vortices embedded in a smooth sea of "Stuart-type" vorticity. Mathematically, these are solutions of a singular Liouville equation with Dirac deltas on the right-hand side, together with an additional constraint at each singularity guaranteeing that corresponding point vortex is stationary. This is joint work with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin.
24 Oct Juan Diego Davila
University of Bath
Helicoidal vortex filaments in the 3-dimensional Ginzburg-Landau equation
We construct a family of entire solutions of the 3D Ginzburg-Landau equation with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. The existence of these solutions was conjectured by del Pino and Kowalczyk (2008), and answers negatively a question of Brezis analogous to the the Gibbons conjecture for the Allen-Cahn equation. This is joint work with Manuel del Pino, María Medina and Remy Rodiac.
31 Oct No seminar
No seminar
7 Nov Ben Pooley
Warwick/Bath
Asymptotics for a regularised local induction model and non-conservation of filaments in passive/active scalar systems
We begin by discussing asymptotics for a the evolution of vortex filaments under a regularised Biot-Savart law, in the context of the binormal curvature flow (a.k.a. the local induction approximation) for the 3D Euler equations. In the second part of the talk, we will construct explicit solutions to certain passive and active scalar systems that dramatically fail to conserve the dimension of filaments but where the velocity is nonetheless (weakly) divergence-free, continuous, and within the regime of Di Perna and Lions theory. This is based on recent works with Charles Fefferman and Jose Rodrigo.
14 Nov Ben Sharp
University of Leeds
Quantitative estimates on the index of a minimal hypersurface
A minimal (hyper)surface is a critical point of the area functional. The second variation of area (or the Hessian of this functional) at a minimal surface corresponds to a self-adjoint elliptic operator whose spectrum is bounded from below. The index of this operator (how many negative eigenvalues it has), thus indicates how many ways we can push a minimal surface to reduce its area. For example, an equator on a 2-sphere is a closed minimal curve (a geodesic), but its index is one since we can always roll the equator up the sphere to reduce its length. We will discuss various relationships between the topology, geometry and index of minimal hypersurfaces, with a focus on a linear estimate on the first Betti number in terms of index. The talk will include some joint works with L. Ambrozio, R. Buzano and A. Carlotto.
21 Nov Aram Karakhanyan
University of Edinburgh
Finite morse index solutions of the Alt-Caffarelli problem
In this talk I will introduce the stability operator for the solutions of Alt-Caffarelli problem with quasilinear elliptic operators. Using some ideas from the theory of varifolds and classical minimal surfaces we will classify the global solutions of finite Morse index in two and three dimensions.
28 Nov cancelled
cancelled
5 Dec Antonio Fernandez
University of Bath
Some remarks on a biharmonic NLS with mixed dispersion
Abstract in pdf
12 Dec No seminar
No seminar

Spring 2020

Date Speaker Title/Abstract
6 Feb TBA
TBA
13 Feb TBA
TBA
20 Feb TBA
TBA
27 Feb TBA
TBA
5 Mar TBA
TBA
12 Mar TBA
TBA
19 Mar TBA
TBA
26 Mar TBA
TBA
2 Apr TBA
TBA
23 Apr TBA
TBA
30 Apr TBA
TBA

Previous Analysis Seminars



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