The "Topics in General Relativity" seminar is the seminar of the Holzegel group at Mathematics Münster. It takes place every Tuesday at 12:00 at the Westfälische Wilhelms-Universität. For further details/to receive e-mails concerning this seminar, feel free to contact Allen Juntao Fang, the current organiser of the seminar. The page for previous semesters may be found here: Summer 2022, Winter 2022-2023, Summer 2023, Winter 2023-2024, and Summer 2024 .
The "PDE colloquium" takes places Tuesdays at 14:15 at the Westfälische Wilhelms-Universität, in Room SRZ 203 (unless otherwise announced) of the Seminarraumzentrum building (Orléans-Ring 12). If you want to be on the mailing list, please send an email to pde.colloquium@uni-muenster.de. The seminar's webpage for further details may be found here.
Léo Bigorgne (8th October 2024, room SRZ 216/217)
PDE ColloquiumTitle: Decay for massless Vlasov fields on Schwarzschild spacetime
Abstract: Our goal consists in developing a commutation vector field approach for the solutions to the massless Vlasov equation on the exterior of a Schwarzschild black hole. We would like an approach that verifies two properties. First, it has to provide decay estimates for the energy flux induced by the energy-momentum tensor of Vlasov fields and their derivatives. Secondly, we would like an approach that is compatible with the ones used to study wave equations on black hole spacetimes. For this, we make use of a weight function that captures the concentration phenomenon of future-trapped null geodesics in phase space. Moreover, we consider the associated symplectic gradient $V_-$, which turns out to enjoy good commutation properties with the massless Vlasov operator. By using a well-chosen modification of $V_-$, we construct a $W_{x,p}^{1,1}$ norm for which any smooth solution to the massless Vlasov equation verifies an integrated energy decay estimate without relative degeneration. This is joint work with Renato Velozo. The lecture will take place as part of the Imperial Day in Muenster.(https://www.uni-muenster.de/MathematicsMuenster/events/2024/imperialday.shtml).
Dimitri Cobb (22nd October 2024, room SRZ 203)
PDE ColloquiumTitle: Global Existence and Uniqueness of Unbounded Solutions in the 2D Euler Equations
Abstract: In this talk, we will study unbounded solutions of the incompressible Euler equations in two dimensions of space. The main interest of these solutions is that the usual function spaces in which solutions are defined (for example based on finite energy conditions like $L^2$ or $H^s$) are not compatible with the symmetries of the problem, namely Galileo invariance and scaling transformation. In addition, many real world problems naturally involve infinite energy solutions, typically in geophysics. After presenting the problem and giving an overview of previous results, we will state our result: existence and uniqueness of global Yudovich solutions under a certain sublinear growth assumption of the initial data. The proof is based on an integral decomposition of the pressure and local energy balance, leading to global estimates in local Morrey spaces. This work was done in collaboration with Herbert Koch (Universität Bonn).
Carla Cederbaum (29th October 2024, room SRZ 203)
PDE ColloquiumTitle: Coordinates are messy in General (Relativity)
Abstract: In General Relativity, one is interested in ?asymptotically Euclidean? Riemannian manifolds, that is, manifolds that look almost like Euclidean space outside some compact set. For such manifolds ? typically accompanied by additional structure such as a (0,2)-tensor field and called ?initial data sets? ?, one is interested in understanding asymptotic geometric invariants such as their ?mass?, ?angular momentum?, and ?center of mass?. To study the latter two, one usually assumes the existence of so-called ?Regge--Teitelboim coordinates? on the asymptotic ?end? of the manifold. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge--Teitelboim coordinates. We will also show that (asymptotic) harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge--Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.
Benedikt Miethke (5th November 2024, room MA 503)
Topics in General RelativityTitle: C^0-inextendibility of the Kasner spacetime
Abstract: The Kasner spacetime is a cosmological model of an anisotropic expanding universe without matter and an exact solution to the Einstein vacuum equations. Due to its curvature singularity (at the Big Bang), it is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this talk we will review some obstructions to continuous extendibility and prove that the Kasner spacetime is even inextendible as a Lorentzian manifold with a continuous metric. We do so by adapting the proof of the C^0-inextendibility of the maximal analytically extended Schwarzschild spacetime established by Jan Sbierski.
Leonhard Kehrberger (12th November 2024, room MA 503)
Topics in General RelativityTitle: Scattering, polyhomogeneity and explicit asymptotics for nonlinear waves near spacelike infinity
Abstract: I will present joint work with Istvan Kadar (to appear soon) that achieves the following: For a large class of quasilinear perturbations to the Minkowskian linear wave equation, we show that semi-global scattering solutions arising from data posed on an ingoing null cone and on past null infinity exist and obey sharp energy estimates. We then show that if the data admit a polyhomogeneous expansion up until some order, then so does the solution. Finally, we present an algorithm suitable for then computing the precise coefficients in the expansions towards future null infinity. This work is motivated by the desire to understand the asymptotic properties of gravitational radiation near future null infinity, and, in particular, solves the issue of summing the fixed-angular-mode estimates obtained in previous works from the series "The Case Against Smooth Null Infinity".
Thomas Stucker (19th November 2024, room MA 503)
Topics in General RelativityTitle: Quasinormal modes for the Kerr black hole
Abstract: The late-time behavior of solutions to the wave equation on Kerr spacetime is governed by inverse polynomial decay. However, at earlier time-scales, numerical simulations are found to be dominated by quasinormal modes (QNMs). These are exponentially damped oscillatory solutions with complex frequencies characteristic of the system. In this talk, I will present a rigorous characterization of QNMs for the scalar wave equation on Kerr. They are obtained as the discrete set of poles of the meromorphically continued cutoff resolvent. The construction combines the method of complex scaling near asymptotically flat infinity with microlocal methods near the black hole horizon. I will also discuss the distribution of QNMs in both the high and low energy regimes. In particular, I will present uniform low energy resolvent estimates, which exclude the accumulation of QNMs at zero energy.
Grigalius Taujanskas (3rd December 2024, room MA 503)
Topics in General RelativityTitle: On the scattering of finite energy Maxwell—Klein—Gordon fields
Abstract: In 1989 John Baez attempted to construct a scattering theory for Yang—Mills fields using conformal techniques, however ran into the problem that at the time the global well-posedness for merely finite energy data had not been proven. He therefore treated data with more derivatives, and provided a construction of a scattering operator corresponding to a distinguished element of the conformal group, but did not show invertibility of the wave operators. In the mid 90s Klainerman and Machedon were the first to obtain a finite energy well-posedness result in Minkowski space in the Coulomb gauge for, in the first instance, Maxwell—Klein—Gordon fields, and then for the Yang—Mills equations. Their construction relied crucially on the celebrated null structure in the nonlinearities. however even this theorem is not enough to obtain what Baez initially set out for. In an upcoming paper with J.-P. Nicolas we prove the finite energy global well-posedness of Maxwell—Klein—Gordon fields on the Einstein cylinder. I will discuss how this leads to the existence of finite energy scattering states on Minkowski space, completing one half of Baez’s goal, and mention the backward problem, which remains open.
Olivier Graf (17th December 2024, room MA 503)
Topics in General RelativityTitle: The linear stability of Schwarzschild-anti-de Sitter spacetimes
Abstract: Schwarzschild-adS spacetimes are stationary and spherically symmetric solutions to the Einstein equations with negative cosmological constant. They contain a black hole region and a conformal anti-de Sitter timelike boundary at infinity. In this talk I will present a result of linear stability for these spacetimes under gravitational perturbations preserving the anti-de Sitter boundary condition. As in the Schwarzschild case, the linearisation of the Einstein equations is governed by two Regge-Wheeler wave equations. I will show that the boundary conditions inherited by the Regge-Wheeler quantities can be decoupled into two boundary conditions: a Dirichlet boundary condition and a higher order ``Robin''-type boundary condition. I will show that these boundary conditions are conservative and yield to the decay of a coercive energy quantity. By red-shift and Carleman estimates for each spherical mode, one can obtain a 1/log(t) decay for the Regge-Wheeler quantities, which can further be infered for the full system of gravitational perturbations. I will also show how to construct quasimode solutions for the system of gravitational perturbations which prove that these bounds are optimal. This is joint work with Gustav Holzegel.