Abstracts

Christopher Alexander

Abstract In this short talk I will show that the expanding flat (k=0) pressureless Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime is unstable to spherical perturbations in the absence of a cosmological constant. We will see that under and overdense perturbations are k<0 and k>0 FLRW spacetimes to leading order respectively, with higher order perturbations corresponding to accelerations away from these spacetimes. We conjecture that such accelerations may account for the anomalous acceleration observed in our universe today, without the requirement for the existence of dark energy. This research builds upon the work of Smoller, Temple and Vogler (2018) and is an ongoing collaboration with Blake Temple.

Muhammed Ali Mehmood

Abstract The hard-congestion model is an example of a two-phase free-congested system that finds many applications in the study of congestion phenomena, such as traffic flow, crowd dynamics and granular flows. Very little is known about the existence/uniqueness of weak or strong solutions, even in one dimension. In this talk I will discuss a recent work which proves the existence of weak and measure-valued solutions to the hard-congestion model on the real line, by studying the 'hard-congestion limit' of the dissipative Aw-Rascle system. We will introduce the theory of the so-called 'duality solutions', which is a class of measure-valued solutions for conservation laws. In particular, I will provide a definition of duality solutions for our hard-congestion model and give an existence result for both weak and duality solutions.

Jonathan Ben-Artzi

Abstract The Vlasov-Maxwell system describes the behavior of a plasma subject to electromagnetic forces. The hyperbolic nature of Maxwell’s equations (finite speed of propagation of the electromagnetic fields) makes this system particularly difficult to analyze. Indeed, the global existence of solutions with large initial data is still an open problem, even after decades of research. Glassey and Strauss (1987) proved the regularity of solutions with small initial data. In this talk I will present a recent result on the long-time behavior of these solutions: scattering toward a self-similar state with explicit convergence rates. Consequently, one obtains the expected decay rate of $t^{-3}$ for the charge density. For plasmas that are globally neutral a better rate is obtained, suggesting that there is some damping mechanism. This is joint work with Stephen Pankavich.

Immanuel Ben Porat

Abstract The quasi-neutral limit is a hydrodynamical limit, i.e. a derivation of a fluid equation from a kinetic equation with a quasi-neutral scaling. In this talk, we explain how to derive the two dimensional incompressible Euler equation as a quasi-neutral limit of the Vlasov-Poisson equation using a modulated energy approach. We propose a strategy which enables to treat solutions where the gradient of the velocity is merely $\mathrm{BMO}$, in accordance to the celebrated Yudovich theorem. Joint work with Mikaela Iacobelli and Alexandre Rege.

Esther Bou Dagher

Abstract We study phase transitions associated to a nonlinear interpolation inequality on the sphere. Using carré du champ methods, we distinguish symmetry and symmetry breaking regimes, with a transition that can be of first or second order. An entropy approach coupled with a nonlinear flow allows us to establish new estimates and results, including optimality cases in a large family of Gagliardo-Nirenberg-Sobolev interpolation inequalities. This is a joint work with Jean Dolbeault.

Manuel Del Pino

Abstract In the classical Water Wave Problem, we construct new overhanging solitary waves by a procedure resembling desingularization of the gluing of constant mean curvature surfaces by tiny catenoid necks. The solutions here predicted have long been numerically detected. This is joint work with Juan Davila, Monica Musso, and Miles Wheeler.

Nicolas Dirr

Abstract We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing.

Richard Medina Rodriguez

Abstract In this talk we will talk about the hard spheres Boltzmann equation near its hydrodynamic limit in a cylinder and we will study its longtime behavior near its steady state, the standard Maxwellian. First, we will generalize hypocoercivity techniques in smooth domains by Bernou, Carrapatoso, Mischler and Tristani to cylindrical ones, meaning we will prove constructive gains of regularity to elliptical problems on cylinders, which will be used to construct a norm on which the linearized problem will be coercive. Afterwards we will use the fact that we are near the hydrodynamic limit to employ a trajectory method to obtain $L^\infty$ estimates on the solutions of the perturbed linearized problem. Finally, with this a priori estimates we will be able to construct solutions to the nonlinear Boltzmann equation by employing an $L^2-L^\infty$ method.

Alpár Mészáros

Abstract In this talk we will revisit the classical problem on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. We will demonstrate that the theory of optimal transport via gradient flows can bring new perspectives, when it comes to consider confining potentials or nonlocal drift terms within the problem. In particular, we provide quantitative convergence rates in the 2-Wasserstein distance for the singular limit, which are global in time thanks to the contractive property arising from the external potentials. The talk will be based on a recent joint work with Noemi David and Filippo Santambrogio.

Søren Mikkelsen

Abstract It is a fundamental problem in mathematical physics to derive macroscopic transport equation from the underlying microscopic transport equations. In this talk, we will consider problems of this kind. To be precise we will consider solutions to a time-dependent Schrödinger equation for a potential localised at the points of a Poisson point process. For these solutions we will present a result stating that the phase-space distribution converges in the annealed Boltzmann-Grad limit to a semiclassical Wigner measure which solves the linear Boltzmann equation..

Shrish Parmeshwar

Abstract A long-standing topic of interest is to understand solutions of the incompressible 2D Euler equations where the vorticity of the solution stays highly concentrated around a finite number of points on some interval of time, in some sense approximating the behaviour of point vortices. There are a large class of steady states that satisfy this behaviour, and also solutions that exhibit this behaviour dynamically on finite time intervals. We exhibit solutions of 2D Euler that are genuinely dynamic, and also retain this concentration of vorticity around points for all time: a configuration approximating two vortex pairs separating at linear speed, and a configuration approximating three vortices separating like a self-similar spiral at sublinear speed. Joint work with Juan Davila, Manuel Del Pino, and Monica Musso.

Matthew Schrecker

Abstract The steady states of many problems in the kinetic theory of galaxies and plasmas, as well as in 2D fluids, give rise to a Hamiltonian dynamical system. To understand the asymptotic stability properties of such steady states, it often important first to understand the decay rate of solutions to the Hamiltonian flow transport equation. In many important applications, these dynamical systems contain a non-degenerate elliptic point at which particles remain trapped by the Hamiltonian flow, obstructing the usual decay via phase mixing mechanism. In this talk, I will discuss recent results on decay via phase mixing for solutions of the pure transport equations driven by such Hamiltonians. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

Ali Taheri

Abstract Evolution equations rooted in mathematical physics (kinetic theory) and geometry are considered and gradient estimates of local and global types (including Hamilton-Souplet-Zhang and Li-Yau estimates) are established. The setting is that of a smooth metric measure space with evolving metric and potential (a geometric flow) and the estimates are established under different curvature conditions and lower bounds on the Bakry-Emery Ricci tensor. Applications to Harnack inequalities, spectral bounds, Logarithmic Sobolev inequalities and Liouville results are given. Further applications to the (super) Perelman-Ricci flow, entropy dissipation and classification of ancient/eternal solutions will also be given if time allows.

Milos Tasic

Abstract A longstanding problem of mathematical physics is to derive kinetic equations from microscopic evolution's of particles. An obstacle in such problems is to control correlations induced by unlikely events (typically called recollisions) In this talk we will introduce a diagrammatic expression for the aforementioned events in order to derive a corrector equation for ballistic annihilation.

Florian Theil

Abstract The justification of kinetic equations like the Boltzmann equation as a scaling limit of particle dynamics for large times is a longstanding problem; the main challenge is to control the size of the relevant density functions. We propose a regularisation strategy where the number of interactions per particles is bounded and derive uniform-in-time bounds for the discrepancy between the solution of the kinetic equation and the particle dynamics. This result is a consequence of a careful study of cancellations associated with recollisions.