# ON DEMAND Oxbridge PDE Conference 2022 - DAY ONE

**Prof. José A. Carrillo - University of Oxford**

**Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications**

1.JAC_.SLIDES.Oxbridge PDE22.pptx

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

**Renato Adolfo Velozo Ruiz - ****University of Cambridge**

**Stability of Schwarzschild for the spherically symmetric Einstein--massless Vlasov system **

2.RV_.SLIDES.Oxbridge PDE22.pdf

The Einstein--massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present an upcoming stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the normal hyperbolicity of the set of trapped null geodesics to obtain quantitative decay estimates for the stress energy momentum tensor of matter. The main result requires precise estimates of radial derivatives of the energy momentum tensor, which we study by using Jacobi fields on the mass-shell in terms of the Sasaki metric.

**Prof. Costante Bellettini - University College London**

**Existence of hypersurfaces with prescribed-mean-curvature**

3.CB_.SLIDES.Oxbridge PDE22.pdf

Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. We prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed and two-sided (it admits a global unit normal); the scalar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

The construction of M is carried out largely by means of PDE principles: (i) a mountain pass construction for an Allen--Cahn energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when sending to 0 the Allen-Cahn parameter.

**Prof. Manuel Del Pino - University of Bath**

**Interacting concentrated vorticities in incompressible Euler flows**

4.MDP_.SLIDES.Oxbridge PDE22.pdf

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in $2d$ or vortex lines in $3d$. We discuss some recent results on these solutions' existence and asymptotic behaviour. We describe, with precise asymptotics, interacting vortices, and travelling helices. We rigorously establish the law of motion of “leapfrogging vortex rings”, originally conjectured by Helmholtz in 1858.

**Prof. Felix Schulze – University of Warwick**

**Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds**

We investigate the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by c t^{-1} converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. This is joint work with A. Deruelle and M. Simon.

**Catalina Pesce Reyes ****- University of Oxford**

**How do degenerate mobilities determine singularity formation in Cahn-Hilliard equations?**

5. CPR.SLIDES.Oxbridge PDE22.pdf

Cahn-Hilliard models are central for describing the evoluNon of interfaces inphase separaNon processes and free boundary problems. In general, they have non-constant and o`en degenerate mobiliNes. However, in the la]er case, the spontaneous appearance of points of vanishing mobility and their impact on the soluNon are not well understood.

In this talk I will present the work done by myself and Andreas Münch on the axially symmetric two dimensional seang. We develop a singular perturbaNon theory to idenNfy a range of degeneracies for which the soluNon of the Cahn-Hilliard equaNon forms a singularity |u|-> 1 in infinite Nme. Moreover, we obtain two different approximate self-similar profiles which divide this range of degeneracies into 1/2 < n <2 and 2<n. We also draw a parallel with a related result for a one dimensional thin-film equaNon.

**Paul Minter - ****University of Cambridge**

**Branch points in stable varifolds**

Almgren introduced the notion of *varifold* in the 1960’s to find critical points of the area functional in general closed Riemannian manifolds. However, aside from Allard’s regularity theory --- which gives that the smoothly embedded part of a stationary integral varifold is open and dense in the support of the varifold --- very little is known regarding the regularity of these stationary integral varifolds. In particular, nothing is known regarding the dimension of the singular set. A key difficulty lies within understanding a certain type of degenerate singularity known as a *branch point*. Very few techniques are known for understanding branch points; Almgren was able to prove dimension bounds on the branch set for area minimisers in part by utilising his *frequency function*; almost all other results in the regularity theory of the area functional need to a priori rule out branch points. In this talk, I will discuss a novel new approach for understanding branch points using the frequency function instead as a means for proving an epsilon-regularity theorem for a certain class of stable varifolds. This result has applications to understanding branch points in area minimising hypercurrents mod p. This talk is based on work joint with N. Wickramasekera.