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Todd A. Oliynyk

Publications and source records attributed to Todd A. Oliynyk.

At least 37 records · Page 2Linked to original sources
arXiv2020

A priori estimates for relativistic liquid bodies

We demonstrate that a sufficiently smooth solution of the relativistic Euler equations that represents a dynamical compact liquid body, when expressed in Lagrangian coordinates, determines a solution to a system of non-linear wave equations with acoustic boundary conditions. Using this wave formulation, we prove that these solutions satisfy energy estimates without loss of derivatives. Importantly, our wave formulation does not require the liquid to be irrotational, and the energy estimates do not rely on divergence and curl type estimates employed in previous works.

math.AP↗
arXiv2019

Dynamical relativistic liquid bodies I: constraint propagation

We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary conditions. We show that solutions of our new wave formulation determine solutions of the relativistic Euler equations that satisfy the vacuum boundary conditions provided the initial data is chosen to make a specific set of constraints vanish on the initial hypersurface. Moreover, we prove that these constraints propagate. This article is the first step of a two step strategy to establish the local-in-time existence and uniqueness of solutions to the relativistic Euler equations representing dynamical liquid bodies in vacuum.

gr-qc↗
arXiv2018

Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant

We introduce a new method for establishing the future non-linear stability of perturbations of FLRW solutions to the Einstein-Euler equations with a positive cosmological constant and a linear equation of state of the form $p = K ρ$. The method is based on a conformal transformation of the Einstein-Euler equations that compactifies the time domain and can handle the equation of state parameter values $0<K\leq 1/3$ in a uniform manner. It also determines the asymptotic behavior of the perturbed solutions in the far future.

gr-qc↗
arXiv2018

Cosmological Newtonian limits on large spacetime scales

We establish the existence of $1$-parameter families of $ε$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $Λ>0$ and a linear equation of state $p=ε^2 K ρ$, $0<K\leq 1/3$, for the parameter values $0<ε< ε_0$. These solutions exist globally on the manifold $M=(0,1]\times \mathbb{R}^3$, are future complete, and converge as $ε\searrow 0$ to solutions of the cosmological Poisson-Euler equations. They represent inhomogeneous, nonlinear perturbations of a FLRW fluid solution where the inhomogeneities are driven by localized matter fluctuations that evolve to good approximation according to Newtonian gravity.

gr-qc↗
arXiv2018

Newtonian Limits of Isolated Cosmological Systems on Long Time Scales

We establish the existence of $1$-parameter families of $ε$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $Λ>0$ and a linear equation of state $p=ε^2 K ρ$, $0<K\leq 1/3$, for the parameter values $0<ε< ε_0$. These solutions exist globally to the future, converge as $ε\searrow 0$ to solutions of the cosmological Poison-Euler equations of Newtonian gravity, and are inhomogeneous non-linear perturbations of FLRW fluid solutions.

gr-qc↗
arXiv2016

Quantum Joint Distributions

In this paper we provide a method for constructing joint distributions for an arbitrary set of observables on finite dimensional Hilbert spaces irrespective of whether the observables commute or not. These distributions have a number of desirable properties: they agree with the standard quantum mechanical ones if the observables commute, they also depend continuously on the observables, and under unitary transformations they behave in a reasonable manner.

quant-ph↗
arXiv2016

Classical-quantum limits

We introduce a new approach to analyzing the interaction between classical and quantum systems that is based on a limiting procedure applied to multi-particle Schrödinger equations. The limit equations obtained by this procedure, which we refer to as the classical-quantum limit, govern the interaction between classical and quantum systems, and they possess many desirable properties that are inherited in the limit from the multi-particle quantum system. As an application, we use the classical-quantum limit equations to identify the source of the non-local signalling that is known to occur in the classical-quantum hybrid scheme of Hall and Reginatto. We also derive the first order correction to the classical-quantum limit equation to obtain a fully consistent first order approximation to the Schrödinger equation that should be accurate for modeling the interaction between particles of disparate mass in the regime where the particles with the larger masses are effectively classical.

quant-ph↗
arXiv2015

The Newtonian limit on cosmological scales

We establish the existence of a wide class of inhomogeneous relativistic solutions to the Einstein-Euler equations that are well approximated on cosmological scales by solutions of Newtonian gravity. Error estimates measuring the difference between the Newtonian and relativistic solutions are provided.

gr-qc↗
arXiv2014

Cosmological Newtonian limit

We describe a wide class of inhomogeneous relativistic solutions that are well approximated on cosmological scales by solutions of Newtonian gravity. Error estimates measuring the difference between the Newtonian and relativistic solutions are given. The solutions presented here unambiguously demonstrate that it is possible for Newtonian gravity to provide a good approximation to General Relativity on cosmological scales.

gr-qc↗
arXiv2013

Lagrange coordinates for the Einstein-Euler equations

We derive a new symmetric hyperbolic formulation of the Einstein-Euler equations in Lagrange coordinates that are adapted to the Frauendiener-Walton formulation of the Euler equations. As an application, we use this system to show that the densitized lapse and zero shift coordinate systems for the vacuum Einstein equations are equivalent to Lagrange coordinates for a fictitious fluid with a specific equation of state.

gr-qc↗
arXiv2011

On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary

We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=Kρ^2$ that have a fluid vacuum boundary. Near the fluid vacuum boundary, the sound speed for these solutions are monotonically decreasing, approaching zero where the density vanishes. Moreover, the fluid acceleration is finite and bounded away from zero as the fluid vacuum boundary is approached. The existence results of this article also generalize in a straightforward manner to equations of state of the form $p=Kρ^\frac{γ+1}γ$ with $γ> 0$.

gr-qc↗
arXiv2011

Dynamical elastic bodies in Newtonian gravity

Well-posedness for the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed.

gr-qc↗
arXiv2011

There are no magnetically charged particle-like solutions of the Einstein Yang-Mills equations for Abelian models

We prove that there are no magnetically charged particle-like solutions for Abelian models in Einstein Yang-Mills, but for non-Abelian models the possibility remains open. An analysis of the Lie algebraic structure of the Yang-Mills fields is essential to our results. In one key step of our analysis we use invariant polynomials to determine which orbits of the gauge group contain the possible asymptotic Yang-Mills field configurations. Together with a new horizontal/vertical space decomposition of the Yang-Mills fields this enables us to overcome some obstacles and complete a dynamical system existence theorem for asymptotic solutions with nonzero total magnetic charge. We then prove that these solutions cannot be extended globally for Abelian models and begin an investigation of the details for non-Abelian models.

gr-qc↗
arXiv2010

A rigorous formulation of the cosmological Newtonian limit without averaging

We prove the existence of a large class of one-parameter families of cosmological solutions to the Einstein-Euler equations that have a Newtonian limit. This class includes solutions that represent a finite, but otherwise arbitrary, number of compact fluid bodies. These solutions provide exact cosmological models that admit Newtonian limits but, are not, either implicitly or explicitly, averaged.

gr-qc↗
arXiv2010

The fast Newtonian limit for perfect fluids

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson-Euler equations of Newtonian gravity. The results presented here generalize those of \cite{Oli06} to allow for a larger class of initial data. As in \cite{Oli06}, the proof is based on a non-local symmetric hyperbolic formulation of the Einstein-Euler equations which contain a singular parameter $\ep=v_T/c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\ep\searrow 0$.

gr-qc↗