Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
Second paper in series solves Einstein vacuum equations for three impulsive waves.
problem Solving local Cauchy problem for impulsive gravitational waves.
method Geometric commutators for energy estimates, fractional-derivative regularity, anisotropic Sobolev embedding.
result Scalar field becomes everywhere Lipschitz and C1,θ away from singular region. In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the a…
Study shows asymptotic solutions for scalar waves on long-range Minkowski-like spaces.
problem Analyzing scalar wave behavior on complex spacetimes.
method Proved existence of joint asymptotic expansions at null and timelike infinity.
result Logarithmic term in asymptotic expansions at null infinity is nonzero.
New method finds precise late-time behavior of wave equations.
problem Analyzing late-time behavior of wave equations with inverse-square potentials.
method Physical-space-based method for deriving late-time asymptotics.
result Sharp, uniform decay estimates in time for asymptotic late-time tails.
Constructs periodic solutions for wave equations, including Einstein's, with negative cosmological constant.
problem Finding periodic solutions for nonlinear wave equations, especially Einstein's equations with negative cosmological constant.
method Analytic continuation to construct periodic solutions.
result Infinite-dimensional families of time-periodic solutions for vacuum and Einstein-Maxwell-dilaton-scalar fields.
Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry an…
Proves global existence and uniqueness of solutions for Einstein-scalar-field equations.
problem Global existence and uniqueness of solutions for specific Einstein-scalar-field equations.
method Proves global existence and uniqueness of classical solutions with small initial data and wake-like decaying null infinity.
result Global existence and uniqueness of solutions for the equations with wake-like decaying null infinity.
Study the full bosonic string action in Minkowski space to Riemannian manifolds.
problem Modeling the full bosonic string action across different target manifolds.
method Investigate the action for Minkowski space as the domain and Riemannian manifolds as targets, coupling wave map equation to potentials.
result Establish existence results for the scalar and two-form potentials.
New solutions to scalar wave equation on Kerr black holes show infinite local energy near event horizon.
problem Blue-shift instabilities on Kerr black hole spacetimes.
method Unified treatment of blue-shift instabilities using scattering theory.
result Solutions with infinite local energy near event horizon and future null infinity.
We explore a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently been motivated by a new geometric perspective by Bray. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely …
Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.
problem Understanding the behavior and stability of charged scalar fields on near-extremal Reissner-Nordström spacetimes.
method Global integrated energy decay and boundedness estimates for solutions to the charged scalar field equation.
result Established global, weighted integrated energy decay and boundedness estimates for solutions on (near-)extremal Reissner-Nordström(--de Sitter) spacetimes.
This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgroun…
The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.
problem Analyzing boundedness and decay of Teukolsky equations on Kerr backgrounds.
method Adapting techniques from scalar waves, uniform-in-frequency estimates for Teukolsky PDEs were obtained.
result Solutions of Teukolsky equation on subextremal Kerr backgrounds remain bounded and decay in time.
Proves local existence and extension principle for Einstein Yang--Mills system with spherical symmetry.
problem Local existence and stability of the spherically symmetric Einstein Yang--Mills system.
method Employed an L2-based method to prove local existence and establish an extension principle. result Established local existence and extension principle for the SSEYM with H1 data. New approach simplifies proof of wave equations on black holes.
problem Global existence and decay for semilinear wave equations on extremal Reissner-Nordström black holes.
method Develops a new approach based on weaker estimates, avoiding near-horizon sharp estimates.
result Simpler and more streamlined proof without requiring near-horizon sharp estimates.
We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on …
Geometric wave propagator defined on Riemannian manifolds.
problem Wave equation on Riemannian manifolds.
method Geometric approach to constructing propagator as oscillatory integral.
result Explicit small time asymptotic expansion of subprincipal symbol.
The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa…
Work consists of introduction, two chapters, conclusion and four applications. In this work is examined the condition, with which the wave space metrics of Riemann- Cartan is the solution of Einstein equation in the void. Geometric structures were for this purpose studied on the differentiated variety: connectedness, c…
Study differential invariants for Kundt waves, solving equivalence problem.
problem Equivalence problem for Kundt waves metrics.
method Scalar differential invariants with respect to the equivalence pseudo-group.
result Finitely represented algebra of invariants on the generic stratum.
These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spa…
Solves the Cauchy problem for linearised Einstein equation on globally hyperbolic spacetimes.
problem Initial value problem for gravitational waves on globally hyperbolic vacuum spacetimes.
method Proves the solution map is an isomorphism of locally convex topological vector spaces and solves linearised constraint equations on closed manifolds.
result Well-posedness of the Cauchy problem for gravitational waves on globally hyperbolic spacetimes.
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…
Proves global existence for quasilinear wave equations with weak null condition.
problem Global existence for quasilinear wave equations with weak null condition.
method p-weighted energy method, hierarchical structure in semilinear terms, robust methods.
result Proves global existence for a larger class of quasilinear wave equations.
This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_gψ = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general ψ or (II2) |a| < M, ψ axisymmetric. In either case, we prove a version of 'integrated local energy decay', specificall…
Establishes existence of maximal globally hyperbolic development for Einstein equations.
problem Initial value problem for generalised Einstein equations.
method Generalised Lorentz gauge, adapted from Ringström's approach.
result Existence and geometric uniqueness of maximal globally hyperbolic development.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
Study of Bondi-Sachs formalism for massless scalar field with zero cosmological constant.
problem Analyzing the Bondi-Sachs formalism for Einstein's massless scalar field equations.
method Asymptotic expansions and peeling property for Bondi-Sachs metrics and scalar fields.
result Positivity of Bondi energy-momentum under specific conditions.
This is the second in a series of papers in which we take a systematic study of gauge field theories such as the Maxwell equations and the Yang-Mills equations, on curved space-times. In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We show that if we a…
This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial da…
We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface …
Classifies solutions to vacuum weighted Einstein equations on pr-waves.
problem Classifying solutions to vacuum weighted Einstein field equations on pr-waves.
method Classifying solutions using smooth metric measure spacetimes of dimension 4.
result Provides examples of solutions with special geometric properties.
Abstract reviews geometric wave and Dirac equations on manifolds.
problem Analyzing geometric equations on manifolds.
method Well-posedness and stability for initial value problems, structure of equations on black-hole spacetimes, index theorem for hyperbolic Dirac operators, properties of Green-hyperbolic operators.
result Results on the structure of wave and Dirac equations on black-hole spacetimes, including the Kerr solution.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…
This paper solves a complex equation using Lie symmetry approach to find solitary wave and multiple soliton solutions.
problem Solving a (3 + 1)-dimensional KdV type equation.
method Lie group of transformation method to find infinitesimal generators and commutator table.
result Exact solutions of KdV type equation in explicit form.
Geometric flow on curves in S^3 generates YO equations solutions.
problem Modeling short wave-long wave interaction.
method Simple geometric flow on curves in S3. result Constructs transverse curves for YO equations periodic solutions.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
Proves Strong Cosmic Censorship conjecture for specific spacetimes.
problem Proving the Strong Cosmic Censorship conjecture for certain spacetimes.
method Analyzing curvature invariants and expansion-normalised variables.
result Proven for orthogonal Bianchi class B perfect fluids and vacuum spacetimes.
Researchers analyze wave equations for spacetime perturbations with electromagnetic and gravitational effects.
problem Analyzing perturbations in Reissner-Nordström spacetime.
method Deriving and analyzing a system of coupled wave equations for the Weyl and Ricci curvatures.
result Combined energy-Morawetz and rp-estimates for the system of wave equations in the case of small charge. We show that axisymmetric extremal horizons are unstable under linear scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This resu…
Study proves well-posedness and scattering for wave equations on hyperbolic spaces with singular data.
problem Proving well-posedness and scattering for wave equations on hyperbolic spaces with singular initial data.
method Using weak-Lp spaces and dispersive estimates on Lorentz spaces, the study establishes global well-posedness and exponential asymptotic stability. result Developed a scattering theory and constructed wave operators in a singular framework.
Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.
problem Interaction of three impulsive gravitational waves in Einstein vacuum equations.
method Geometric estimates and wave estimates to prove local solution and continuity.
result Local solution to Einstein vacuum equations with three impulsive gravitational waves, Lipschitz continuity.
Study connects Riemann-Finsler geometry to Lorentz-violating scalar fields.
problem Exploring the connection between Riemann-Finsler geometries and Lorentz-violating scalar fields.
method Deriving quadratic actions and classical relativistic point-particle lagrangians in various spacetime dimensions.
result Support for open conjectures about Riemann-Finsler geometries in Lorentz-violating field theories.
Global solution found for biharmonic wave maps into spheres.
problem Solving biharmonic wave maps into spherical targets.
method Reformulated as a conservation law, solved with Ginzburg-Landau approximation.
result Global weak solution constructed in the energy space.
Proves wave equation solutions in Kerr-de Sitter spacetime have specific asymptotic expansions.
problem Analyzing solutions to wave equations in Kerr-de Sitter spacetime.
method Developed a Fredholm setup for quasinormal modes and analyzed trapping of lightlike geodesics.
result Proves asymptotic expansions of wave equation solutions up to a decay order.
The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…