Local and global existence theorems for the Einstein equations

DOI10.12942/lrr-2000-1zbMath0944.83002OpenAlexW2949469144WikidataQ39120933 ScholiaQ39120933MaRDI QIDQ5915355

Alan D. Rendall

Publication date: 25 July 2000

Published in: Living Reviews in Relativity (Search for Journal in Brave)

Full work available at URL: http://www.livingreviews.org/lrr-2000-1

zbMATH Keywords

Cauchy problemEinstein equationspecial relativityexistence theoremsingularity structureNewtonian theory

Mathematics Subject Classification ID

Einstein's equations (general structure, canonical formalism, Cauchy problems) (83C05) Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory (83-02) Space-time singularities, cosmic censorship, etc. (83C75) Special relativity (83A05) Exact solutions to problems in general relativity and gravitational theory (83C15)

Related Items

Local smooth solutions to the 3-dimensional isentropic relativistic Euler equations, Characteristic evolution and matching, Characteristic evolution and matching, Critical phenomena in gravitational collapse, The rest-frame instant form of metric gravity., Characteristic evolution and matching, A proof of Price's law for the collapse of a self-gravitating scalar field, Generalized pseudo-Riemannian geometry

Cites Work

• Quasi-spherical metrics and prescribed scalar curvature
• Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics
• Asymptotic behaviour of the gravitational field and the nature of singularities in Gowdy spacetimes
• Compressible fluid flow and systems of conservation laws in several space variables
• Neighborhoods of Cauchy horizons in cosmological spacetimes with one Killing field
• On space-times with U(1)\(\times U(1)\) symmetric compact Cauchy surfaces
• Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation
• On the Cauchy problem for Boltzmann equations: Global existence and weak stability
• Remarques sur l'apparition de singularités dans les écoulements euleriens compressibles. (Remarks on the appearance of singularities in compressible Euler flows)
• Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant
• Stationary spherically symmetric models in stellar dynamics
• Formation of singularities in three-dimensional compressible fluids
• The problem of a self-gravitating scalar field
• Remarks on cosmological spacetimes and constant mean curvature surfaces
• Partial differential equations. 4th ed
• The boost problem in general relativity
• The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties
• Propagation of moments and regularity for the 3-dimensional Vlasov- Poisson system
• Smooth static solutions of the Einstein/Yang-Mills equations
• Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data
• Pseudodifferential operators and nonlinear PDE
• On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations
• Global existence proof for relativistic Boltzmann equation
• Global existence of solutions of the spherically symmetric Vlasov- Einstein system with small initial data
• The Boltzmann equation and its applications
• The instability of naked singularities in the gravitational collapse of a scalar field.
• Global foliations of vacuum spacetimes with \(T^2\) isometry
• Smooth irrotational flows in the large to the Euler-Poisson system in \(\mathbb{R}^{3+1}\)
• On spherically symmetric stellar models in general relativity
• Global existence for the Einstein-Boltzmann equation in the flat Robertson-Walker spacetime
• The initial boundary value problem for Einstein's vacuum field equation
• Stable steady states in stellar dynamics
• Global solutions of the relativistic Euler equations
• On the asymptotic behavior of spherically symmetric wave maps
• On the spherical symmetry of static stellar models
• The Newtonian limit for asymptotically flat solutions of the Vlasov- Einstein system
• Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting
• The mathematical theory of dilute gases
• Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II
• On ``asymptotically flat space-times with \(G_2\)-invariant Cauchy surfaces
• On the existence of rotating stars in general relativity
• On the Newtonian limit of general relativity
• The ``Two and one-half dimensional relativistic Vlasov Maxwell system
• Conservation laws for relativistic fluid dynamics
• Gravitational fields near space-like and null infinity
• Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry
• Nonlinear stability of homogeneous models in Newtonian cosmology
• The relativistic Vlasov-Maxwell system in two space dimensions: I. II
• Stationary black holes: Uniqueness and beyond
• Stationary and static stellar dynamic models with axial symmetry
• Isotropic cosmological singularities. III: The Cauchy problem for the inhomogeneous conformal Einstein-Vlasov equations
• Isotropic cosmological singularities. II: The Einstein-Vlasov system
• Isotropic cosmological singularities. I: Polytropic perfect fluid spacetimes
• A class of counterexamples to Jeans' theorem for the Vlasov-Einstein system
• Self-gravitating relativistic fluids: The formation of a free phase boundary in the phase transition from soft to hard
• Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large
• The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations
• A mathematical theory of gravitational collapse
• Examples of naked singularity formation in the gravitational collapse of a scalar field
• The unique limit of the Glimm scheme
• A regularity theorem for solutions of the spherically symmetric Vlasov- Einstein system
• Finite energy solutions of the Yang-Mills equations in \(\mathbb{R}^{3+1}\)
• Self-gravitating relativistic fluids: A two-phase model
• Einstein equations and conformal structure: Existence of anti-de Sitter-type space-times
• Fuchsian equations in Sobolev spaces and blow-up
• The semigroup generated by \(2 \times 2\) conservation laws
• Self-gravitating relativistic fluids:the continuation and termination of a free phase boundary.
• Global foliations of matter spacetimes with Gowdy symmetry
• Asymptotic stability of the relativistic Maxwellian
• Global aspects of the Cauchy problem in general relativity
• Asymptotic properties of linear field equations in minkowski space
• Existence and properties of spherically symmetric static fluid bodies with a given equation of state
• The formation of black holes and singularities in spherically symmetric gravitational collapse
• On the regularity of spherically symmetric wave maps
• Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations
• Strong cosmic censorship in polarised Gowdy spacetimes
• A necessary and sufficient condition for York data to specify an asymptotically flat spacetime
• The initial value problem for a class of general relativistic fluid bodies
• On the definition of post-newtonian approximations
• On the StrongL¹Trend to Equilibrium for the Boltzmann Equation
• Remarks on the abstract form of nonlinear cauchy-kovalevsky theorems
• Blow-up Surfaces for Nonlinear Wave Equations, I
• The Cauchy problem for quasi–linear hyperbolic evolution problems with a singularity in the time
• Well-posedness in Sobolev spaces of the full water wave problem in 3-D
• Black Hole Uniqueness Theorems
• Global smooth solutions to Euler equations for a perfect gas
• Blow-up surfaces for nonlinear wave equations, II
• Initial-boundary value problem for the spherically symmetric Einstein equations for a perfect fluid
• Growth rates for the linearized motion of fluid interfaces away from equilibrium
• Bounded variation solutions of the spherically symmetric einstein‐scalar field equations
• On the structure of conformal singularities in classical general relativity
• Static solutions of the spherically symmetric Vlasov–Einstein system
• The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models
• Stability of Gaseous Stars in Spherically Symmetric Motions
• Global dynamics of the mixmaster model
• Existence and non-existence results for global constant mean curvature foliations
• Spatially self-similar spherically symmetric perfect-fluid models
• Static Axially Symmetric Solutions of Einstein–Yang-Mills-Dilaton Theory
• Conformal “Thin-Sandwich” Data for the Initial-Value Problem of General Relativity
• Curvature blow up in Bianchi VIII and IX vacuum spacetimes
• A class of plane symmetric perfect-fluid cosmologies with a Kasner-like singularity
• Timelike self-similar spherically symmetric perfect-fluid models
• Cosmological spacetimes not covered by a constant mean curvature slicing
• Dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations which are locally rotationally symmetric
• Asymptotic self-similarity breaking at late times in cosmology
• On local and global aspects of the Cauchy problem in general relativity
• Asymptotic behavior in polarized T2-symmetric vacuum space–times
• Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry
• Conservation laws for the relativistic P-system
• Constant mean curvature solutions of the Einstein constraint equations on closed manifolds
• Existence of maximal hypersurfaces in some spherically symmetric spacetimes
• The blow-up problem for exponential nonlinearities
• The spectrum of adiabatic stellar oscillations
• Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry
• Hyperbolic reductions for Einstein's equations
• A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds
• Regularity of the Gain Term and Strong $L^1 $ Convergence to Equilibrium for the Relativistic Boltzmann Equation
• Analytic description of singularities in Gowdy spacetimes
• Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics
• Dynamical Systems in Cosmology
• Solutions in the large for nonlinear hyperbolic systems of equations
• Critical phenomena in gravitational collapse
• Critical phenomena in gravitational collapse
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item