Analysis Seminar 2023/24

This is based on a joint work with Angeliki Menegaki. I will talk about the BGK equation which is a simplification of Boltzmann's equation for dilute gasses. We look at the situation where we couple the equation to a thermostat meaning that heat can enter and leave the system. This produces a non-equilibrium steady state (NESS). I will discuss the challenges in showing convergence towards a NESS and maybe how this is connected to derivation of Fourier's law.

The Brascamp–Lieb inequalities are a generalisation of the Hölder, Loomis–Whitney and Young convolution inequalities, and have found many applications in harmonic analysis and elsewhere. In this talk we present an “adjoint” version of these inequalities which may be viewed as an $L^p$ version of the entropic Brascamp–Lieb inequalities of Carlen and Cordero–Erausquin. As an application we establish some lower bounds on a range of tomographic transforms, such as the classical X-ray and Radon transforms. This is joint work with Terence Tao.

Systems that involve many elements, be it a gas of particles or a herd of animals, are ubiquitous in our day to day lives. Their investigation, however, is hindered by the complexity of such systems and the amount of (usually coupled) equations that are needed to be solved.

The late 50’s has seen the birth of the so-called mean field limit approach as an attempt to circumvent some of the difficulties arising in treating such systems. Conceived by Kac as a way to give justification to the validity of the Boltzmann equation, the mean field limit approach attempts to find the behaviour of a limiting “average” element in a many element system and relies on two ingredients: an average model of the system (i.e. an evolution equation for the probability density of the ensemble), and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity.

Mean field limits of average models, originally applied to particle models, have permeated to fields beyond mathematical physics in recent decades. Examples include models that pertain to biological, chemical, and even societal phenomena. However, to date we use only one asymptotic correlation relation – chaos, the idea that the elements become more and more independent. While suitable when considering particles in a certain cavity, this assumptions doesn’t seem reasonable in models that pertain to biological and societal phenomena.

In our talk we will introduce Kac’s particle model and the notions of chaos and mean field limits. We will discuss the problem of having chaos as the sole asymptotic correlation relation and define a new asymptotic relation of order. We show that this is the right relation for a recent animal based model suggested by Carlen, Degond, and Wennberg, and highlight the importance of appropriate scaling in its investigation.

In this talk, I will describe a result showing that solutions to an inhomogeneous Allen–Cahn equation with appropriate bounds on its generalised Allen–Cahn mean curvature are quantised in the sharp interface limit.

We study the existence of normalized solutions $u\in H^1(\mathbf R^N)$ for the following nonlinear Schrödinger equation: \[ \left\{ \begin{aligned} -&\Delta u + \mu u = g(u) \quad \hbox{in}\ \mathbf R^N, \\ &{1\over 2}\int_{\mathbf R^N} u^2\, dx =m, \end{aligned} \right. \] where $N\geq 2$, $m>0$ is a fixed mass, and $\mu>0$ is a unknown Lagrange multiplier. We consider the case where $g(s)$ has $L^2$ critical growth asymptotically as $s\sim 0$ and $s\sim\pm\infty$, that is, \[ g(s)= |s|^{p-1}s + h(s), \quad p=1+{4\over N}, \] where $h(s)$ satisfies \[ {h(s)\over |s|^{p-1}s}\to 0 \quad \hbox{as}\ s\to 0 \ \hbox{and}\ s\to\pm \infty. \] We show existence and non-existence results.

This is a joint work with Silvia Cingolani, Marco Gallo and Norihisa Ikoma.

We will discuss the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, I will explain how to get a uniform description of the spectrum located between the Landau levels. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, I will explain how to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and also how to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.

Joint work with R. Fahs, L. Le Treust and S. Vu Ngoc.

In this talk I consider steady gravity-capillary surface waves ‘riding’ a perfect fluid in Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). I will first demonstrate how the hydrodynamic problem can be formulated as two equations for two scalar functions of the horizontal spatial coordinates, namely the elevation of the free surface and the potential defining the gradient part (in the sense of the Hodge–Weyl decomposition) of the horizontal component of the tangential fluid velocity there. The formulation is nonlocal, has a variational structure and generalises the Zakharov–Craig–Sulem formulation for the classical water-wave problem, reducing to it in the irrotational limit.

Starting from the above formulation, one can derive the Kadomtsev–Petviashvili-I (KP-I) equation (strong surface tension) or the Davey–Stewartson (DS) system (weak surface tension) for such Beltrami flows using formal weakly nonlinear theory. These equations have ‘lump’ solutions and thus predict the existence of fully localised solitary water waves for the full problem. I will show how to rigorously reduce the full problem to a perturbation of the KP-I or DS equations and thus construct an existence proof for fully localised solitary waves ‘riding’ Beltrami flows.

In this talk, we present some reflections and recent developments on solving several longstanding open problems involving the singularities of entropy solutions for nonlinear conservation laws and related nonlinear partial differential equations through entropy analysis and associated methods. These problems are concerned with the minimal entropy conditions for well-posedness, cavitation/decavitation, concentration/deconcentration, and rigorous analysis for entropy solutions involving the singularities via the theory of divergence-measure fields, among others. Further related topics, perspectives, and open problems will also be addressed.

To each orthogonal projector of finite rank N on $L^2(R^d)$ is associated a point process on $R^d$ with N points, which gives the joint probability density of fermions that fill the image of the projector.

The study of the statistical properties of these fermions, in the large N limit, is linked to semiclassical spectral theory problems, some of them well studied (the Weyl law gives a law of large numbers), some of them new. In particular, the behaviour of the variance is linked with the properties of commutators involving spectral projectors, which are not so well understood.

In this talk, I will present my work in collaboration with Gaultier Lambert (KTH) on this topic.

In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov–Poisson equation, described a phenomenon he called “violent relaxation,” a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov–Poisson equation. In particular, I will discuss the critical role played by the interaction of the vacuum boundary for the kinetic Vlasov equation with the elliptic regularisation of the Poisson equation in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

We investigate certain questions arising in two-dimensional statistical hydrodynamics, by relying on principles of entropy maximization for the vorticity of a two-dimensional perfect fluid in a disc. In analogy with the entropy functions used in statistical mechanics and thermodynamics, we show that similar concavity properties hold for the 2d Euler equations when maximizing entropies at fixed energy levels. The proofs rely on rearrangement inequalities, a modification of the classical min-max principle, and the properties of the Euler–Lagrange equations for the corresponding constrained optimization.

On a background Minkowski spacetime, the Euler equations (both relativistic and not) are known to admit unstable homogeneous solutions with finite-time shock formation. Such shock formation can be suppressed on cosmological spacetimes whose spatial slices expand at an accelerated rate. However situations with decelerated expansion, which are relevant in our early universe, are not as well understood. I will present some recent joint work in this direction, based on collaborations with David Fajman, Maciej Maliborski, Todd Oliynyk and Max Ofner.