ON DEMAND Oxbridge PDE Conference 2022 - DAY THREE
Dr Matthew Schrecker - University College London
Self-similar gravitational collapse for Newtonian star
The Euler-Poisson equations give the classical model of a self-gravitating star under Newtonian gravity. It is widely expected that, in certain regimes, initially smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In this talk, I will present recent work with Yan Guo, Mahir Hadzic and Juhi Jang that demonstrates the existence of smooth, radially symmetric, self-similar blow-up solutions for this problem. At the heart of the analysis is the presence of a sonic point, a singularity in the self-similar model that poses serious analytical challenges in the search for a smooth solution.
Dr Zoe Wyatt - University of Cambridge
Coupled wave and Klein-Gordon equations in two and three spatial dimensions
Semilinear wave equations in three spatial dimensions with wave--wave nonlinearities exhibit interesting and well-studied phenomena: from John's famous blow-up examples, to the null condition of Christodoulou and Klainerman, and more recently to the weak null condition of Lindblad and Rodnianski. The study of coupled semilinear wave and Klein-Gordon equations is less well-developed, and interesting problems occur across the possible spectrum of wave--wave, wave--KG and KG--KG interactions. In this talk I will discuss some recent results, in collaboration with Shijie Dong (SUSTech), on such mixed systems. This includes a recent proof of small data global-existence and sharp asymptotics for a Dirac--Klein-Gordon system in two spatial dimensions.
Alexis Michelat - University of Oxford
Quantization of the Willmore Energy in Riemannian Manifolds
The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces.
Amongst the challenging problems raised by the associated fourth order non-linear elliptic partial differential equation, the question of compactness of sequences of critical immersions (called Willmore immersions) with uniformly bounded energy is one of the most natural ones. We show that Bernard-Rivière's energy quantization theorem generalises under similar assumptions to closed Riemannian manifolds, and we will explain during this presentation how (approximate) conservation laws and Lorentz (or Orlicz) spaces play a central role in the analysis of similar problems, ranging from harmonic maps to the Ginzburg-Landau functional.
Joint work with Andrea Mondino (University of Oxford).