arXiv subjects

Uwe Brauer

Publications and source records attributed to Uwe Brauer.

9 recordsLinked to original sources

arXiv2025

Global existence of the irrotational Euler-Nordstr\"om equations with a positive cosmological constant: The gravitational field equation

Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordstr\"om system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$. We begin by analysing the Nordstr\"om equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order. To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^\mu$, with $\mu\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant.

math.AP↗

arXiv2025

The non-isentropic Einstein-Euler system written in a symmetric hyperbolic form

We cast the non--isentropic relativistic Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure $p$ and not the density $ρ$ as a variable. However, the system becomes degenerate when the pressure $p$ approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.

math-ph↗

arXiv2022

Global existence of a nonlinear wave equation arising from Nordström's theory of gravitation

We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus $\setT^3$ in the Sobolev spaces $H^m(\setT^3)$. We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up result if the conditions of our global existence theorems are not met.

math.AP↗

arXiv2020

Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler--Poisson system

We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right--hand sides in different functional setting, including weighted Sobolev spaces $H_{s,δ}$. An essential tool to achieve the continuity of the flow map is a low regularity energy estimate. We then apply these results to the Euler--Poisson system which describes various systems of physical interests.

math.AP↗

arXiv2017

Local existence of solutions to the Euler-Poisson system, including densities without compact support

Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density $ρ$ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for the adiabatic constant $γ=\frac{6}{5}$. The result is achieved by using weighted Sobolev spaces of fractional order and a new non linear estimate which allows to estimate the physical density by the regularised non linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtionan limit.

math.AP↗

arXiv2014

Elliptic Equations in Weighted Besov Spaces on Asymptotically Flat Riemannian Manifolds

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian an momentum constraints with constant mean curvature and with a background metric which satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space $B_{p,p}^s$, to asymptotically flat manifolds. We also consider the Brill--Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds Choquet--Bruhat, Isenberg and Pollack, and Maxwell, as well as they enable us to construct the initial data for the Einstein--Euler system.

math.AP↗

arXiv2013

Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein--Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove a local in time existence, uniqueness and well-posedness of classical solutions. The zero order term of our system contains an expression which might not be a $C^\infty$ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. Previously the authors constructed an initial data set for these of systems in the same type of weighted Sobolev spaces. We obtain the same lower bound for the regularity as the one of the classical result of Hughes, Kato and Marsden for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.

math.AP↗

arXiv2009

Well-Posedness of the Einstein-Euler System in Asymptotically Flat Spacetimes

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein--Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for elliptic and hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall for the Euler-Einstein system under the restriction of time symmetry and of Gamblin for the simpler Euler--Poisson system.

math.AP↗

arXiv1993

Trapped surfaces in spherical expanding open universes

Consider spherically symmetric initial data for a cosmology which, in the large, approximates an open $k = -1 ,Λ= 0$ Friedmann-Lema{\^ı}tre universe. Further assume that the data is chosen so that the trace of the extrinsic curvature is a constant and that the matter field is at rest at this instant of time. One expects that no trapped surfaces appear in the data if no significant clump of excess matter is to be found. This letter confirms this belief by displaying a necessary condition for the existence of trapped surfaces.This necessary condition, simply stated, says that a relatively large amount of excess matter must be concentrated in a small volume for trapped surfaces to appear.

gr-qc↗