Bielefeld Analysis Seminar

This talk will focus on the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on new Strichartz estimates for stochastic kinetic transport. The derivation of these estimates involves a local in time dispersion analysis using properties of stochastic flows, and a time-splitting argument to extend the local in time results to arbitrary time intervals. This is based on joint work with Benjamin Gess and Sebastian Herr.

Join Zoom Meeting
Meeting-ID: [622 2841 8411]
Passcode: [058508]

This talk will focus on well-known examples of wave equations, primarily, the classical nonlinear wave equation with a focusing power nonlinearity and the wave maps equation. Both equations exhibit rich dynamics, in particular self-similar blow up solutions. I will report on results concerning the stability of such solutions in the energy supercritical regime and highlight the underlying spectral structures which one needs to understand in order to prove such results. The main focus will lie on results in optimal topologies in terms of L²-based Sobolev spaces, which in turn require the derivation of Strichartz estimates for wave equations with self-similar potentials. This is based on joint work with Roland Donninger.

Join Zoom Meeting
Meeting-ID: [622 2841 8411]
Passcode: [058508]

The controllability problem for a given partial differential equation (PDE) consists in sending any initial condition to zero with a right-hand-side active only in a given subregion ω. It is tightly connected to non-concentration properties for solutions of the said PDE: if no solution can concentrate outside of ω, then controllability from ω holds, and vice versa. The talk will cover two examples of such phenomena. First, I will explain how controllability of the heat equation is implied by so-called spectral estimates for frequency-localized functions. These spectral estimates are themselves equivalent to an equidistribution property of ω. I will give many examples, including some original results on non-compact manifolds, and a brief idea of the proof. The tools involved draw from spectral theory, harmonic analysis, spectral theory and geometric analysis. I will then talk about a similar problem for the damped wave equation. In that case, concentration of waves along geodesics of the manifold must be avoided to achieve controllability. When the damping is continuous, the Geometric Control Condition (GCC) gives a sharp condition on the control set: the damping must capture every geodesic in some finite time. I will present some generalizations of the GCC by Burq-Gérard and myself for discontinuous dampings on tori.

This talk concerns supercritical wave maps from Minkowski space into slightly perturbed versions of the perfectly round d-sphere. In a work by Shatah and Tahvildar-Zadeh the existence of self-similar blowup solutions to the wave maps equation has already been established in this setting. However, since these solutions were obtained via variational methods proving any form of stability for these solutions has remained out of reach. Based on a joint work with R. Donninger and B. Schörkhuber I will present a construction of self-similar blowup solutions in the above described setting which are somehow close to the ground-state self-similar solution for the unperturbed sphere. This proximity allowed us to rigorously prove, using perturbative methods, their asymptotic nonlinear stability.

We consider particles interacting with a smooth mean-field potential and attempt to quantify the speed of convergence (log-Sobolev constant) of the associated Langevin dynamics in terms of the number N of particles and the strength of the interaction. Our main interest is in relating the scaling of the log-Sobolev constant as a function of N, to properties of the free energy of the model. We show that a certain notion of convexity of the free energy implies uniform-in-N bounds on the log-Sobolev constant. In some cases, this convexity criterion is sharp — for instance, in the Curie–Weiss model where we prove uniform bounds on the log-Sobolev constant up to the critical temperature, which is optimal. Our proof does not involve the dynamics. Instead, we decompose the measure describing interactions between particles with inspiration from renormalisation group arguments for lattice models, adapted here to a lattice-free setting in the simplest case of mean-field interactions. Our results apply more generally to non mean-field, possibly random settings, provided each particle interacts with sufficiently many others. Based on joint work with Roland Bauerschmidt and Thierry Bodineau.

We review a method to obtain optimal Poincaré-Hardy inequalities on the hyperbolic spaces. Then we show how to transfer the basic idea to the discrete setting. This yields optimal Poincaré-Hardy-type inequalities on weakly spherically symmetric graphs which include fast enough growing trees and anti-trees. Moreover, this method yields optimal weights which are larger at infinity than the optimal weights constructed via the Fitzsimmons ratio of the square root of the minimal positive Green's function. Joint work with Christian Rose.

Exciting topics among the dynamics of nonlinear evolution equations are the occurrence and stability of finite time blowup solutions. In fact, the wave maps equation, Yang–Mills equation, and focusing semilinear wave equation all admit self-similar blowup solutions in closed form. In this talk, I will present a comprehensive stability theory for self-similar blowup solutions of these nonlinear wave equations, which was obtained recently in my PhD thesis. The underlying analysis is based on coordinate systems that are adapted to self-similarity and compatible with the wave evolution. This allows the study of the wave flow near self-similar blowup solutions in spacetime regions that reach from the backward light cone towards the future light cone of the respective singularity.

A seminal result of Jordan, Kinderlehrer and Otto states that Fokker–Planck equations on Euclidean spaces can be interpreted as a gradient flow of the relative entropy through optimal transport. In the first part of the talk, I will briefly recall this theory in the usual setting and highlight some iconic generalisations and their consequences. In the second part, based on a joint work with Thomas Leblé, I will present some of the ideas we used to generalise this result to the setting of interacting spin systems, where the underlying space is a countable product of compact Riemannian manifolds, and the range of the interaction is potentially infinite. In particular, I will define the free energy, and the infinite-volume diffusion, partial differential equation and gradient flow that model this situation and show they are related through an Evolution Variational Inequality. Our approach establishes in particular that at high enough temperature the free energy decays exponentially fast along the dynamic, which is new.