Vacuum Initial Data Research Articles

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Viscosity solutions for the relativistic inhomogeneous Vlasov equation in Schwarzschild outer space-time in the presence of the Yang-Mills field

This paper provide evidence of the existence and uniqueness result for the viscosity solutions of inhomogeneous Vlasov equation. we consider the Cauchy-Dirichlet problem for the relativistic Vlasov equation with near vacuum initial data where the distribution function depends on the time, the position, the momentum and the non-Abelian charge of particles. We consider this equation on aSchwarzschild outer space-time with Yang-Mills field.

Pp-wave initial data

An initial data characterization for vacuum pp-wave spacetimes in dimension four is constructed. This is a vacuum initial data set plus some extra conditions guaranteeing that the data development is a subset of a vacuum pp-wave. Some of the extra conditions only depend on the same quantities used to construct the vacuum initial data, namely the first and the second fundamental forms while others are related to a conformal Killing initial data characterization (CKID).

Global Well-Posedness of Classical Solutions to the Cauchy Problem of Two-Dimensional Barotropic Compressible Navier--Stokes System with Vacuum and Large Initial Data

The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data.

Observations and predictions from past lightcones

In a general Lorentzian manifold M, the past lightcone of a point is a proper subset of M that does not carry enough information to determine the rest of M. In special circumstances however, say if M is a globally hyperbolic Cauchy development of vacuum initial data on a Cauchy surface S and there is a point whose past lightcone contains S, then the contents of such a lightcone does determine M up to isometry. We present results that describe what properties of M guarantee that past lightcones do indeed determine all or at least significant portions of M. Null lines and observer horizons, which are well known features of the de-Sitter spacetime, play a prominent role.

Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints

Systematic numerical investigations of the asymptotics of near Schwarzschild vacuum initial data sets is carried out by inspecting solutions to the parabolic–hyperbolic and to the algebraic–hyperbolic forms of the constraints, respectively. One of our most important findings is that the concept of near Schwarzschild configurations, applied previously in [, ], is far too restrictive. It is demonstrated that by relaxing the conditions on the freely specifiable part of the data a more appropriate notion of near Schwarzschild initial data configurations can be defined which allows us to generate asymptotically flat initial data configurations.

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

In this paper we continue earlier investigations of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by R\'{a}cz. Motivated by the strong evidence from these works that the resulting vacuum initial data sets are generically not asymptotically flat we analyse the asymptotics of the solutions of a modified formulation by a combination of analytical and numerical techniques. We conclude that the vacuum initial data sets generated with this new formulation are generically asymptotically flat.

Type D conformal initial data

For a vacuum initial data set of the Einstein field equations it is possible to carry out a conformal rescaling or conformal compactification of the data giving rise to an initial data set for the Friedrich vacuum conformal equations. When will the data development with respect to the conformal equations of this set be a conformal extension of a type D solution? In this work we provide a set of necessary and sufficient conditions on a set of initial data for the conformal equations that guarantees that the data development of the conformal equations has a subset that is conformal to a vacuum type D solution of the Einstein’s equations. In particular we find the conditions under which this vacuum solution corresponds to the Kerr solution. Using our results we are able to show that there are no obstructions to extend the Petrov type of the physical spacetime to the unphysical spacetime if the conformal data are hyperboloidal.

Bartnik mass via vacuum extensions

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

Vacuum initial data on from Killing vectors

We construct compact initial data of constant mean curvature for Einstein’s 4d vacuum equations with positive, where is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer (2004 Class. Quantum Grav. 21 73). We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are —symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential ‘symmetry breaking’ by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the ‘round donut’ of topology . In the absence of symmetry breaking, the Lichnerowicz equation for a symmetric momentum on reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

Causal geodesic incompleteness of spacetimes arising from IMP gluing

In 2002, Isenberg-Mazzeo-Pollack (IMP) constructed a series of vacuum initial data sets via a gluing construction. In this paper, we investigate some local geometry of these initial data sets as well as implications regarding their spacetime developments. In particular, we state conditions for the existence of outer trapped surfaces near the center of the IMP gluing neck and thence use a generalization of the Penrose incompleteness theorem to deduce null incompleteness of the resulting spacetimes.

Construction of vacuum initial data by the conformally covariant split system

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-constant mean curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split system defined on compact manifolds with or without boundaries. In the former case, the boundary corresponds to an apparent horizon in the constructed initial data. The case with a cosmological constant is then considered separately. Finally, to demonstrate the applicability of the conformal covariant split system in numerical studies, we provide numerical examples of solutions on manifolds and .

Global existence of the classical solutions for the inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time

In this paper, we consider the Cauchy problem for the spatially inhomogeneous relativistic Boltzmann equation with near vacuum initial data. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson–Walker space-time. Unique global (with respect to the direction of time corresponding to the expansion of the universe) classical solution is obtained. This is done in a suitable weighted space.

The relativistic Boltzmann equation on a spherically symmetric gravitational field

In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. We consider this equation on a spherically symmetric gravitational field spacetime. The collision kernel considered here is for the hard potentials case. We prove the existence of a unique global (in time) mild solution in a suitable weighted space.

Smoothly Compactifiable Shear-Free Hyperboloidal Data is Dense in the Physical Topology

We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Holder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.

Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time

In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson-Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space.

Vacuum type D initial data

A vacuum type D initial data set is a vacuum initial data set of the Einstein field equations whose data development contains a region where the space–time is of Petrov type D. In this paper we give a systematic characterisation of a vacuum type D initial data set. By systematic we mean that the only quantities involved are those appearing in the vacuum constraints, namely the first fundamental form (Riemannian metric) and the second fundamental form. Our characterisation is a set of conditions consisting of the vacuum constraints and some additional differential equations for the first and second fundamental forms These conditions can be regarded as a system of partial differential equations on a Riemannian manifold and the solutions of the system contain all possible regular vacuum type D initial data sets. As an application we particularise our conditions for the case of vacuum data whose data development is a subset of the Kerr solution. This has applications in the formulation of the nonlinear stability problem of the Kerr black hole.

Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier–Stokes system with vacuum and large initial data

For periodic initial data with density allowed to vanish initially, we establish the global existence of strong and weak solutions to the two-dimensional barotropic compressible Navier–Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is λ=ρβ with β>4/3. These results generalize and improve the previous ones due to Vaigant–Kazhikhov [Sib. Math. J. 36 (1995) 1283–1316] who required β>3. Moreover, we also prove that the densities for both the strong and weak solutions remain bounded from above independently of time. As a consequence, it is shown that both the strong and weak solutions converge to the equilibrium state as time tends to infinity.

Local non-negative initial data scalar characterization of the Kerr solution

For any vacuum initial data set, we define a local, non-negative scalar quantity which vanishes at every point of the data hypersurface if and only if the data are {\em Kerr initial} data. Our scalar quantity only depends on the quantities used to construct the vacuum initial data set which are the Riemannian metric defined on the initial data hypersurface and a symmetric tensor which plays the role of the second fundamental form of the embedded initial data hypersurface. The dependency is {\em algorithmic} in the sense that given the initial data one can compute the scalar quantity by algebraic and differential manipulations, being thus suitable for an implementation in a numerical code. The scalar could also be useful in studies of the non-linear stability of the Kerr solution because it serves to measure the deviation of a vacuum initial data set from the Kerr initial data in a local and algorithmic way.

On complete stationary vacuum initial data

We describe a proof of M.T. Anderson’s result (Anderson, 2000) on the rigidity of complete stationary initial data for the Einstein vacuum equations in spacetime dimension 3 + 1 , under an extra assumption on the norm of the stationary Killing vector field. The argument only involves basic comparison geometry along with some Bochner–Weitzenböck formula techniques. We also discuss on the possibility to extend these techniques to higher dimensions.