arXiv subjects

Matthias Ostermann

Publications and source records attributed to Matthias Ostermann.

6 recordsLinked to original sources

arXiv2024

On stable self-similar blowup for corotational wave maps and equivariant Yang-Mills connections

We consider corotational wave maps from Minkowski spacetime into the sphere and the equivariant Yang-Mills equation for all energy-supercritical dimensions. Both models have explicit self-similar finite time blowup solutions, which continue to exist even past the singularity. We prove the nonlinear asymptotic stability of these solutions in spacetime regions that approach the future light cone of the singularity. For this, we develop a general functional analytic framework in adapted similarity coordinates that allows to evolve the stable wave flow near a self-similar blowup solution in such spacetime regions.

math.AP↗

arXiv2024

Stable blowup for focusing semilinear wave equations in all dimensions

We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial symmetry in all space dimensions and for all superlinear powers. This result covers for the first time the whole energy-supercritical range without symmetry restrictions.

math.AP↗

arXiv2024

A globally stable self-similar blowup profile in energy supercritical Yang-Mills theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

math.AP↗

arXiv2024

A characterization of the subspace of radially symmetric functions in Sobolev spaces

In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.

math.FA↗

arXiv2014

Direct Minkowski Functional analysis of large redshift surveys: a new high--speed code tested on the luminous red galaxy Sloan Digital Sky Survey-DR7 catalogue

As deeper galaxy catalogues are soon to come, it becomes even more importantto measure large-scale fluctuations in the catalogues with robust statistics that cover all moments of the galaxy distribution.In this paper we reinforce a direct analysis of galaxy data by employing the Germ-Grain method to calculate thefamily of Minkowski Functionals. We introduce a new code, suitable for the analysis of large data sets without smoothingand without the construction of excursion sets. We provide new tools to measure correlation properties, putting emphasis onexplicitly isolating non-Gaussian correlations with the help of integral-geometric relations. As a first application we present the analysis of large-scale fluctuations in the luminous red galaxy sample of Sloan Digital Sky Survey data release 7 data. We findsignificant deviations from the $Λ$ cold dark matter mock catalogues on samples as large as $500h^{-1}$Mpc (more than $3σ$)and slight deviations of around $2σ$ on $700h^{-1}$Mpc, and we investigate possible sources of these deviations.

astro-ph.CO↗

arXiv2012

Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation

In this first paper we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativistic analogue of Zel'dovich's approximation for the description of structure formation in cosmology, and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zel'dovich's approximation and investigate its implications by functionally evaluating relevant variables, and we address the singularity problem. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to put into practice nonperturbatively, e.g. problems of structure formation, backreaction problems, nonlinear properties of gravitational radiation, and light-propagation in realistic inhomogeneous universe models. With this model we also provide the key-building blocks for initializing a fully relativistic numerical simulation.

gr-qc↗