Analysis Seminar 2024/25

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.