Analysis Seminar 2024/25

In this talk I will introduce Alexandrov immersed mean curvature flow and extend Andrew’s non-collapsing estimate to include Alexandrov immersed surfaces. This estimate implies an all-important gradient estimate for the flow and allows mean curvature flow with surgery to be extended beyond flows of embedded surfaces to the Alexandrov immersed case. This is joint work with Elena Maeder-Baumdicker.

The essential numerical range is a useful concept to describe spectral pollution when approximating a linear operator by projection or domain truncation methods. In this talk I will first give an introduction to the topic and then present some new results for differential operators for which a modified definition gives tighter bounds on the set of spectral pollution. This is joint work with Marco Marletta (Cardiff) and Christiane Tretter (Bern).

In this work we study the evolution of a passive scalar advected by a cellular flow on a two-dimensional periodic box, where the centre of the flow undergoes a random walk. We prove exponential decay of correlations for mean-free $H^s$ functions, uniformly in diffusivity, leading to almost sure exponential mixing and optimal enhanced dissipation rates. Despite the stochastic nature of the flow, our approach is entirely analytical. We introduce a PDE for the expectation of the two-point process and establish its hypocoercivity in a weighted $H^1$ norm that degenerates at the diagonal. We prove that hypocoercivity of the two-point PDE works as an $L^2$-analogue of geometric ergodicity of the two-point chain in the random dynamical systems framework. This is a joint work with Christian Seis, Univeristät Münster.

In this talk I am going to present a recent result in collaboration with Tarek Elgindi, Yupei Huang and Chujing Xie where we show that all analytic steady solutions to the Euler equations in a simply connected domain are either radial or global solution to a semi-linear elliptic equation of the $\Delta \psi= F(\psi)$.

We study the behavior of the scaling critical Lebesgue norm for blow-up solutions to the nonlinear heat equation (the Fujita equation). For the energy supercritical nonlinearity, we show that the critical norm also blows up at the blow-up time. Moreover, we give estimates of the blow-up rate for the critical norm. This is based on joint works with Tobias Barker (University of Bath) and Hideyuki Miura (Institute of Science Tokyo).

I will consider the singularity formation problem for compressible fluid dynamics in the spherically symmetric case. I will review some know results and open problems in the field. We will be drawn to the question of regularity of self similar solutions in the semi classical regime and will propose a new approach to treat this question.

It is known that Liouville-type theorems guarantee universal estimates of solutions to various superlinear elliptic and parabolic problems which are scale-invariant. We discuss several recent related results, particularly for parabolic problems without scale invariance. This is a joint work with Philippe Souplet.

One of the key objects in the study of scattering theory are the so-called resonances. I will briefly talk about classical pictures of obstacle scattering resonances and give a mathematical description of transmission problems. Then I will discuss how geometry and refraction indices influence the distribution of resonances. Obstacles with negative index of refraction exhibit a new type of resonances called (surface) plasmons. I will discuss when they are present for non-trapping obstacles and what properties a plasmon has. Finally, I will briefly go over how Jeffrey Galkowski and I capture plasmons.

ABSTRACT

Standard models for opto-electronic/photonic devices often involve wave-guides in dielectric media delineated by changes in the refractive index, but without a hard boundary. Such devices, called open wave-guides, are difficult to analyze and simulate because they typically extend to infinity, with perturbations that are not compactly supported. I will describe a new approach to solving this class of problems that reduces the scattering problem to a transmission problem across an infinite artificial interface, and then to a Fredholm system of integral equations on the interface. The infinite extent of the perturbations requires new types of radiation conditions in order to uniquely specify a solution, which I will explain. Finally, the method has been numerically implemented in a range of interesting cases by Tristan Goodwill, Shidong Jiang, Manas Rachh and Jeremy Hoskins. If time permits, I will explain how this is accomplished.

The Riemann problem is the IVP having simple piecewise constant initial data that is invariant under scaling. In 1D, the problem was originally considered by Riemann during the 19th century in the context of gas dynamics, and the general theory was more or less completed by Lax and Glimm in the mid-20th century. In 2D and MD, the situation is much more complicated, and very few analytic results are available. We discuss a shock reflection problem for the Euler equations for potential flow, with initial data that generates four interacting shockwaves. After reformulating the problem as a free boundary problem for a nonlinear PDE of mixed hyperbolic-elliptic type, the problem is solved via a sophisticated iteration procedure. The talk is based on joint work with G-Q Chen (Oxford) et. al. arXiv:2305.15224.

In this talk I will first discuss recent advances on stability of Sobolev inequalities from both functional and critical points. We obtain sharp quantitative estimates of Struwe's decomposition, solving an open conjecture of Figalli. Then I will discuss stability and instability of harmonic map inequalities, exhibiting a striking difference between degree one and higher degree. Finally I will present some most recent results on stability of Caffarelli–Kohn–Nirenberg inequalities along Felli-Schneider curves as well as degenerate stability of Q-curvature metrics.

Higher order problems are very novel in the Calculus of Variations in $L^\infty$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. In this talk I will discuss how a complete theory can be developed for second order functionals. Under appropriate conditions, “localised” minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation; the latter is only a necessary, but not a sufficient condition for minimality. I will also discuss the existence and uniqueness of localised minimisers subject to Dirichlet boundary conditions, and also their partial regularity outside a singular set of codimension one, which may be non-empty even in 1D. The talk will not assume any previous knowledge on the topic, and is based on recent work (arXiv:2403.12625v1) with Roger Moser.

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a~deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett. 

The critical SQG (surface quasi-geostrophic) equation is widely studied in relation to rapid formation of small scales in fluids. In the whole space or on the torus, this dissipative equation has been shown to have global smooth solutions some fifteen years ago by Caffarelli–Vasseur and, independently, by Kiselev–Nazarov–Volberg.

The problem of existence and uniqueness of global smooth solution in bounded domains remained open until now. I will present a proof of global regularity obtained recently with Ignatova and Q-H. Nguyen. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

Relaxation of the Cartesian area functional allows to extend in a natural way the area of a graph to non smooth maps. The use of $L^1$-topology in the relaxation has been very successful for scalar valued functions mainly for coercivity reasons. However, in the vectorial case, an integral representation of the $L^1$-relaxed functional is not possible since, as suggested by De Giorgi and proven by Acerbi and Dal Maso in the '90s, the subadditivity property is very frequently lost, even in very simple cases, e.g. for the vortex map.

In this talk, we will treat the case of multivortex maps: interestingly, whether there is an integral representation will depend not only on the distance between the vortices and from the boundary (as already expected), but also on the sign of their topological degree.

We study a nonlinear eigenvalue problem associated with the Rayleigh quotient $|u|_{\mathrm{Lip}}/|u|_C$, where $|u|_{\mathrm{Lip}}$ is the Lipschitz constant of a function $u$ defined on a bounded domain in $\mathbb R^n$ and $|u|_C$ is its supremum norm. The problem of minimising this Rayleigh quotient is closely related to the infinity Laplacian: minimisers include infinity-harmonic potentials and so-called infinity ground states defined as solutions of a certain limiting PDE obtained by taking the limit $p\to \infty$ in the $p$-Laplace eigenvalue problem. Another notable minimiser is the distance function to the boundary of the domain. Unlike existing literature that studies $L^\infty$ problems as limits of $L^p$ problems, we study the limiting problem directly using tools from convex analysis. This allows us to obtain results that hold for all minimisers of the Rayleigh quotient. We obtain optimality conditions in form of a divergence PDE using a novel characterisation of the subdifferential of the Lipschitz seminorm $u \mapsto |u|_{\mathrm{Lip}}$ as a functional on $C$. We also study a minimisation problem for the dual Rayleigh quotient involving Radon measures and a variant of the Kantorovich–Rubinstein norm, and relate minimisers of the $L^\infty$ Rayleigh quotient to solutions of an optimal transport problem.

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. This is joint work with Ansgar Jüngel (TU Wien) and Li Chen (University of Mannheim).

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary. We also construct large amplitudes in the periodic case using global bifurcation techniques. These results are corroborated by numerical solutions.

In this talk, we will consider the problem of minimising the Dirichlet energy \[ \mathbb D(v) = \int_\Omega |\nabla v|^2\, \mathrm dx \] subject to a pointwise Jacobian constraint \[ \det \nabla v = \det \nabla u \qquad \text{a.e.} \] with Dirichlet boundary condition $v|_{\partial\Omega} = u|_{\partial\Omega}$. Both the Dirichlet energy and pointwise Jacobian constraints appear in a wide range of physical problems, such as elasticity, which we will briefly discuss. This talk will focus on one particuar method for showing that $u$ is a minimiser, which makes use of a so-called excess functional, taking the form \[ \mathbb E_{\mathsf p}(\varphi) = \int_\Omega |\nabla\varphi|^2 + \mathsf p \det \nabla \varphi(x)\, \mathrm dx. \] This functional is interesting in its own right as it serves as an example of a simple (but non-convex) functional that is not always bounded below (depending on the choice of $\mathsf p$) and so the direct method may not be of use.

The bulk of the talk will focus mainly on results relating to the excess functional, investigating qualitative changes that occur as $\mathsf p$ is varied. In particular, we will discuss methods for establishing sufficient and necessary conditions for the excess to be bounded below. We will then apply these methods to a particular one parameter family of pressure functions, deriving sufficient and necessary bounds on the parameter. Time allowing, we will then derive the maps $u$, associated with these pressure functions $\mathsf p$, that minimize the Dirichlet energy suject to a pointwise Jacobian constraint.