geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

stability properties of implicit runge kutta methods new stability concepts bs stability and bsi stability internal bs stability are introduced bs stability is a modification and extension of b stability and enables the derivation of order results for runge kutta methods applied to general nonlinear stiff initial value problems these order results b consistency b convergence are not affected by stiffness several classes of implicit runge kutta methods are shown to be bs stable gauss radau ia and radau iia schemes

cyclic length in the tame brauer group of the function field of a p adic curve let f be the function field of a smooth curve over the p adic number field qp we show that for each prime to p number n the n torsion subgroup h 2 f n nbr f is generated by z n cyclic classes in fact the z n length is equal to two it follows that the brauer dimension of f is two first proved in 20 and any f division algebra of period n and index n 2 is decomposable

global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

einstein s equations near spatial infinity a new class of space times is introduced which in a neighbourhood of spatial infinity allows an expansion in negative powers of a radial coordinate einstein s vacuum equations give rise to a hierarchy of linear equations for the coefficients in this expansion it is demonstrated that this hierarchy can be completely solved provided the initial data satisfy certain constraints

asymptotically simple solutions of the vacuum einstein equations in even dimensions we show that a set of conformally invariant equations derived from the fefferman graham tensor can be used to construct global solutions of vacuum einstein equations in all even dimensions this gives in particular a new simple proof of friedrich s result on the future hyperboloidal stability of minkowski space time and extends its validity to even dimensions

pointwise decay for the maxwell field on black hole space times abstract in this article we study the pointwise decay properties of solutions to the maxwell system on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish peeling estimates as well as a t 4 rate of decay on compact regions for all the components of the maxwell tensor

tensor products of division algebras and fields abstract this paper began as an investigation of the question of whether d 1 f d 2 is a domain where the d i are division algebras and f is an algebraically closed field contained in their centers we present an example where the answer is no and also study the picard group and brauer group properties of f 1 f f 2 where the f i are fields finally as part of our example we have results about division algebras and brauer groups over curves specifically we give a splitting criterion for certain brauer group elements on the product of two curves over f

decay of axisymmetric solutions of the wave equation on extreme kerr backgrounds abstract we study the cauchy problem for the wave equation g 0 on extreme kerr backgrounds specifically we consider regular axisymmetric initial data prescribed on a cauchy hypersurface 0 which connects the future event horizon with spacelike or null infinity and we solve the linear wave equation on the domain of dependence of 0 we show that the spacetime integral of an energy type density is bounded by the initial conserved flux corresponding to the stationary killing field t and we derive boundedness of the non degenerate energy flux corresponding to a globally timelike vector field n finally we prove uniform pointwise boundedness and power law decay for up to and including the event horizon h

brauer hilbertian fields let f be a field of characteristic p p 0 allowed and let f t be the rational function field in one variable over f we say f is brauer hilbertian if the following holds for every in the brauer group br f t of exponent prime to p there are infinitely many specializations t a f such that the specialization br f is defined and has exponent equal to that of we show every global field is brauer hilbertian and if k is hilbertian and f is finite separable over k t f is brauer hilbertian

extending convergence of bdf methods for a class of nonlinear strongly stiff problems abstract in 1995 a new approach was presented to overcome the drawbacks of existing convergence theory for stiff initial value problems by auzinger et al bit 36 4 1996 635 652 and illustrated by means of a convergence argument for the implicit euler scheme in the present paper our intention is to obtain extending quantitive convergence results of bdf methods for a class of nonlinear strongly stiff problems and to illustrate further this new approach some numerical examples confirm our results

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

factorization theory from commutative to noncommutative settings abstract we study the non uniqueness of factorizations of non zero divisors into atoms irreducibles in noncommutative rings to do so we extend concepts from the commutative theory of non unique factorizations to a noncommutative setting several notions of factorizations as well as distances between them are introduced in addition arithmetical invariants characterizing the non uniqueness of factorizations such as the catenary degree the invariant and the tame degree are extended from commutative to noncommutative settings we introduce the concept of a cancellative semigroup being permutably factorial and characterize this property by means of corresponding catenary and tame degrees also we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization applying the abstract machinery we develop we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero sum sequences over a ray class group we also determine catenary degrees and the permutable tame degree for the semigroup of non zero divisors of the ring of n n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

a proof of price s law on schwarzschild black hole manifolds for all angular momenta price s law states that linear perturbations of a schwarzschild black hole fall off as t 2 ell 3 for t to infty provided the initial data decay sufficiently fast at spatial infinity moreover if the perturbations are initially static i e their time derivative is zero then the decay is predicted to be t 2 ell 4 we give a proof of t 2 ell 2 decay for general data in the form of weighted l 1 to l infty bounds for solutions of the regge wheeler equation for initially static perturbations we obtain t 2 ell 3 the proof is based on an integral representation of the solution which follows from self adjoint spectral theory we apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates

b convergence properties of multistep runge kutta methods by using the theory of b convergence for general linear methods to the special case of multistep runge kutta methods a series of b convergence results for multistep runge kutta methods is obtained and it is proved that the family of algebraically stable r step s stage multistep runge kutta methods with parameters al a2 ar presented by burrage in 1987 is optimally bconvergent of order at least s and b convergent of order s 1 provided that r s and aj o j 1 2 r furthermore this family of methods is optimally b convergent of order s 1 if some other additional conditions are satisfied

a proof of the uniform boundedness of solutions to the wave equation on slowly rotating kerr backgrounds 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 mathcal m g which are sufficiently close to a schwarzschild metric with parameter m 0 and whose killing fields span the null generator of the event horizon we show uniform boundedness on the exterior for solutions to the wave equation g 0 the most fundamental statement is at the level of energy we show that given a suitable foliation then there exists a constant c depending only on the parameter m and the choice of the foliation such that for all solutions a suitable energy flux through is bounded by c times the initial energy flux through 0 this energy flux is positive definite and does not degenerate at the horizon i e it agrees with the energy as measured by a local observer it is shown that a similar boundedness statement holds for all higher order energies again without degeneration at the horizon this leads in particular to the pointwise uniform boundedness of in terms of a higher order initial energy on 0 note that in view of the very general assumptions the separability properties of the wave equation or geodesic flow on the kerr background are not used in fact the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high frequency geometric optics regime

microlocal analysis of asymptotically hyperbolic and kerr de sitter spaces with an appendix by semyon dyatlov in this paper we develop a general systematic microlocal framework for the fredholm analysis of non elliptic problems including high energy or semiclassical estimates which is stable under perturbations this framework described in sect 2 resides on a compact manifold without boundary hence in the standard setting of microlocal analysis many natural applications arise in the setting of non riemannian b metrics in the context of melrose s b structures these include asymptotically de sitter type metrics on a blow up of the natural compactification kerr de sitter type metrics as well as asymptotically minkowski metrics the simplest application is a new approach to analysis on riemannian or lorentzian or indeed possibly of other signature conformally compact spaces such as asymptotically hyperbolic or de sitter spaces including a new construction of the meromorphic extension of the resolvent of the laplacian in the riemannian case as well as high energy estimates for the spectral parameter in strips of the complex plane these results are also available in a follow up paper which is more expository in nature vasy in uhlmann g ed inverse problems and applications inside out ii 2012 the appendix written by dyatlov relates his analysis of resonances on exact kerr de sitter space which then was used to analyze the wave equation in that setting to the more general method described here

modern convergence theory for stiff initial value problems abstract in this paper we give a brief review of available theoretical results about convergence and error structures for discretizations of stiff initial value problems we point out limitations of the various approaches and discuss some recent developments

the interior of charged black holes and the problem of uniqueness in general relativity we consider a spherically symmetric double characteristic initial value problem for the real einstein maxwell scalar field equations on the initial outgoing characteristic the data is assumed to satisfy the price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat cauchy surface we establish that the heuristic mass inflation scenario put forth by israel and poisson is mathematically correct in the context of this initial value problem in particular the maximal future development has a future boundary over which the space time is extendible as a c0 metric but along which the hawking mass blows up identically thus the space time is inextendible as a c1 metric in view of recent results of the author in collaboration with i rodnianski which rigorously establish the validity of price s law as an upper bound for the decay of scalar field hair the c0 extendibility result applies to the collapse of complete asymptotically flat spacelike initial data where the scalar field is compactly supported this shows that under christodoulou s c0 formulation the strong cosmic censorship conjecture is false for this system 2005 wiley periodicals inc

the instability of naked singularities in the gravitational collapse of a scalar field one of the fundamental unanswered questions in the general theory of relativity is whether naked singularities that is singular events which are visible from inflnity may form with positive probability in the process of gravitational collapse the conjecture that the answer to this question is in the negative has been called cosmic censorship the present paper which is a continuation of the work in 1 2 3 and 4 addresses this question in the context of the spherical gravitational collapse of a scalar fleld the problem of a spherically symmetric self gravitating scalar fleld is formulated in terms of a 2 dimensional quotient space time manifold q with boundary see 1 the boundary ofq corresponds to the set of flxed points of the group action the center of symmetry which is a timelike geodesic i the manifoldq is endowed with a lorentzian metric gab an area radius function i i r

factorizations in upper triangular matrices over information semialgebras abstract an integral domain or a commutative cancellative monoid is atomic if every nonzero nonunit element factors into atoms and it satisfies the accp if every ascending chain of principal ideals eventually stabilizes the interplay between these two properties has been investigated since the 1970s an atomic domain or monoid satisfies the finite factorization property ffp if every element has only finitely many factorizations and it satisfies the bounded factorization property bfp if for each element there is a bound for the number of atoms in each of its factorizations these two properties have been systematically studied since they were introduced by anderson anderson and zafrullah in 1990 noetherian domains satisfy the bfp while dedekind domains satisfy the ffp it is well known that for commutative cancellative monoids and in particular for multiplicative monoids of integral domains ffp bfp accp atomic for n 2 we show that each of these four properties transfers back and forth between an information semialgebra s certain commutative cancellative semiring and the multiplicative monoid t n s consisting of n n upper triangular matrices over s we also show that a similar transfer behavior takes place if one replaces t n s by its submonoid u n s consisting of upper triangular matrices with units along their main diagonals as a consequence we find that the atomic chain ffp bfp accp atomic also holds for the two classes comprising the noncommutative monoids t n s and u n s finally we construct various rational information semialgebras to verify that in general none of the established implications is reversible

martingale solutions for the stochastic nonlinear schrodinger equation in the energy space we consider a stochastic nonlinear schrodinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic nlse in h 1 on compact manifolds and bounded domains we construct a martingale solution using a modified faedo galerkin method based on the littlewood paley decomposition for the 2d manifolds with bounded geometry we use the strichartz estimates to show the pathwise uniqueness of solutions

stability and instability of the cauchy horizon for the spherically symmetric einstein maxwell scalar field equations this paper considers a trapped characteristic initial value problem for the spherically symmetric einstein maxwell scalar field equations for an open set of initial data whose closure contains in particular reissner nordstrdata the future boundary of the maximal domain of development is found to be a light like surface along which the curvature blows up and yet the metric can be continuously extended beyond it this result is related to the strong cosmic censorship conjecture of roger penrose the principle of determinism in classical physics is expressed mathemat ically by the uniqueness of solutions to the initial value problem for certain equations of evolution indeed in the context of the einstein equations of general relativity where the unknown is the very structure of space and time uniqueness is equivalent on a fundamental level to the validity of this principle the question of uniqueness may thus be termed the issue of the predictability of the equation the present paper explores the issue of predictability in general relativity since the work of leray it has been known that for the einstein equations contrary to common experience uniqueness for the cauchy problem in the large does not generally hold even within the class of smooth solutions in other words uniqueness may fail without any loss in regularity such failure is thus a global phenomenon the central question is whether this violation of predictability may occur in solutions representing actual physical processes physical phenomena and concepts related to the general theory of relativity namely gravitational collapse black holes angular momentum etc must cer tainly come into play in the study of this problem unfortunately the math ematical analysis of this exciting problem is very difficult at present beyond reach for the vacuum einstein equations in the physical dimension conse

black holes without spacelike singularities it is shown that for small spherically symmetric perturbations of asymptotically flat two ended reissner nordstrom data for the einstein maxwell real scalar field system the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously under additional assumptions it is shown that the hawking mass blows up identically along this bifurcate null hypersurface and thus the metric cannot be extended twice differentiably in fact it cannot be extended in a weaker sense characterized at the level of the christoffel symbols the proof combines estimates obtained in previous work with an elementary cauchy stability argument there are no restrictions on the size of the support of the scalar field and the result applies to both the future and past boundary of spacetime in particular it follows that for an open set in the moduli space of solutions around reissner nordstrom there is no spacelike component of either the future or the past singularity

asymptotically flat initial data with prescribed regularity at infinity we prove the existence of a large class of asymptotically flat initial data with non vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space like infinity in terms of powers of a radial coordinate

hidden symmetries and decay for the wave equation on the kerr spacetime energy and decay estimates for the wave equation on the exterior region of slowly rotating kerr spacetimes are proved the method used is a generalisation of the vector field method that allows the use of higher order symmetry operators in particular our method makes use of the second order carter operator which is a hidden symmetry in the sense that it does not correspond to a killing symmetry of the spacetime

hyperboloidal layers for hyperbolic equations on unbounded domains we show how to solve hyperbolic equations numerically on unbounded domains by compactification thereby avoiding the introduction of an artificial outer boundary the essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification we construct a new layer method based on this idea called the hyperboloidal layer the method is demonstrated on numerical tests including the one dimensional maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques

decay and non decay of the local energy for the wave equation on the de sitter schwarzschild metric we describe an expansion of the solution of the wave equation on the de sitter schwarzschild metric in terms of resonances the principal term in the expansion is due to a resonance at 0 the error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an varepsilon derivative loss in the angular directions

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

phase space analysis on some black hole manifolds abstract the schwarzschild and reissner nordstrom solutions to einstein s equations describe space times which contain spherically symmetric black holes we consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type we show that for solutions with initial data which decay at infinity and at the bifurcation sphere a weighted l 6 norm in space decays like t 1 3 this weight vanishes at the event horizon but not at infinity to obtain this control we require only an loss of angular derivatives

division algebras over surfaces abstract we consider here a number of topics concerning the theory of division algebras over the function field of a surface one result relates the obstruction for ramification data to be from a division algebra and third etale cohomology another result shows this obstruction is always zero when the surface is spec of a regular local ring with some mild assumptions at the same time we study the brauer group of this function field as it relates to the brauer group of the function field of the henselization finally we prove a result which says that brauer group elements which look like they are of prime index q unequal to any characteristic must have all their ramification split by a cyclic galois extension of the same degree this last result requires a primitive q root of one

uniqueness of smooth stationary black holes in vacuum small perturbations of the kerr spaces the goal of the paper is to prove a perturbative result concerning the uniqueness of kerr solutions a result which we believe will be useful in the proof of their nonlinear stability following the program started in ionescu and klainerman invent math 175 35 102 2009 we attempt to remove the analyticity assumption in the the well known hawking carter robinson uniqueness result for regular stationary vacuum black holes unlike ionescu and klainerman in invent math 175 35 102 2009 which was based on a tensorial characterization of the kerr solutions due to mars class quant grav 16 2507 2523 1999 we rely here on hawking s original strategy which is to reduce the case of general stationary space times to that of stationary and axi symmetric spacetimes for which the carter robinson uniqueness result holds in this reduction hawking had to appeal to analyticity using a variant of the geometric carleman estimates developed in ionescu and klainerman invent math 175 35 102 2009 in this paper we show how to bypass analyticity in the case when the stationary vacuum space time is a small perturbation of a given kerr solution our perturbation assumption is expressed as a uniform smallness condition on the mars simon tensor the starting point of our proof is the new local rigidity theorem established in alexakis et al hawking s local rigidity theorem without analyticity http arxiv org abs 0902 1173v1 gr qc 2009

on the regular and the asymptotic characteristic initial value problem for einstein s vacuum field equations the regular characteristic initial value problem for einstein s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations this is achieved by making use of the spin frame formalism instead of the harmonic gauge condition the method is applied to the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on part of past null infinity and on an incoming null hypersurface a uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity

on hyperboloidal cauchy data for vacuum einstein equations and obstructions to smoothness of scri the relationship between the geometric properties of hyperboloidal cauchy data for vacuum einstein equations at the conformal boundary of the initial data surface and between the space time geometry is analyzed in detail we prove that a necessary condition for existence of a smooth or a polyhomogeneous scri i e a scri around which the metric is expandable in terms ofr j log i r rather than in terms ofr j is the vanishing of the shear of the conformal boundary of the initial data surface we derive the boundary constraints which have to be satisfied by an initial data set for compatibility with friedrich s conformal framework we show that a sufficient condition for existence of a smooth scri not necessarily complete is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data we also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non vanishing of the weyl tensor at the conformal boundary

weak null singularities in general relativity we construct a class of spacetimes without symmetry assumptions satisfying the vacuum einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2 sphere the metric of these spacetimes extends continuously beyond the singularities while the christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries the construction shows moreover that the singularities are stable in a suitable sense these singularities are stronger than the impulsive gravitational spacetimes considered by luk rodnianski and are conjecturally the generic type of singularity in the interior of black holes arising from gravitational collapse

indecomposable and noncrossed product division algebras over function fields of smooth p adic curves we construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves x over zp this is done by defining an index preserving morphism s br k x br k x which splits res br k x br k x where k x is the completion of k x at the special fiber and using it to lift indecomposable and noncrossed product division algebras over k x

boundedness of the total energy of relativistic membranes evolving in a curved spacetime abstract we establish a global existence theory for the equation governing the evolution of a relativistic membrane in a curved background spacetime when the spacetime metric is a perturbation of the minkowski metric relying on a hyperboloidal foliation and generalizing an observation made by lefloch and ma we revisit a theorem established by lindblad who treated membranes in the flat minkowski spacetime and provide a simpler proof of existence our result is valid in a curved spacetime and most importantly establishes the important property that the total energy of the membrane is globally bounded in time

geometrical classification of killing tensors on bidimensional flat manifolds valence two killing tensors in the euclidean and minkowski planes are classified under the action of the group which preserves the type of the corresponding killing web the classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic the approach allows one to classify both characteristic and noncharacteristic killing tensors

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

cauchy problems for the conformal vacuum field equations in general relativity cauchy problems for einstein s conformal vacuum field equations are reduced to cauchy problems for first order quasilinear symmetric hyperbolic systems the hyperboloidal initial value problem where cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two surface is discussed and translated into the conformally related picture it is shown that for conformal hyperboloidal initial data of classhs s 4 there is a unique up to questions of extensibility development which is a solution of the conformal vacuum field equations of classhs it provides a solution of einstein s vacuum field equations which has a smooth structure at past null infinity

decay for solutions of the wave equation on kerr exterior spacetimes iii the full subextremal case a m 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 jaj m or axisymmetry arxiv 1010 5132 providing the complete proof of denitive boundedness and decay results for the scalar wave equation on kerr backgrounds in the general subextremal jaj m case without symmetry assumptions the essential ideas of the proof together with explicit constructions of the most dicult mul

on the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes we analyze the appearance of logarithmic terms at the critical sets of friedrich s cylinder representation of spatial infinity it is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions conversely it is proved that under the additional hypothesis that the spacetime has constant adm mass aspect and vanishing dual adm mass aspect by which we mean the limit of the bondi dual mass aspect on to the critical set this condition is also necessary for a spacetime to admit a smooth conformal representation at the critical sets

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

colliot thelene s conjecture and finiteness of u invariants let k be a number field and f a function field in one variable over k we prove that the ramification of a p torsion element in br f on a regular proper model over the ring of integers in k can be split in an extension of degree p 3 using this result we show that colliot thelene s conjecture on 0 cycles of degree 1 implies finiteness for the u invariant of the function field of a curve over a totally imaginary number field and period index bounds for the brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

on the existence of analytic null asymptotically flat solutions of einstein s vacuum field equations this paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for einstein s field equations for analytic data on past null infinity and on an incoming null hypersurface

conformal killing initial data we find necessary and sufficient conditions ensuring that the vacuum development of an initial dataset of einstein s field equations admits a conformal killing vector we refer to these conditions as conformal killing initial data ckid and they extend the well known killing initial data kid that have been known for a long time the procedure used to find the ckid is a classical argument which is reviewed and presented in a form that may have an independent interest based on identifying a suitable propagation identity and checking the well posedness of the corresponding initial value problem as example applications we review the derivation of the kid conditions as well as give a more thorough treatment of the homothetic kid conditions than was previously available in the literature we find necessary and sufficient conditions ensuring that the vacuum development of an initial dataset of einstein s field equations admits a conformal killing vector we refer to these conditions as conformal killing initial data ckid and they extend the well known killing initial data kid that have been known for a long time the procedure used to find the ckid is a classical argument which is reviewed and presented in a form that may have an independent interest based on identifying a suitable propagation identity and checking the well posedness of the corresponding initial value problem as example applications we review the derivation of the kid conditions as well as give a more thorough treatment of the homothetic kid conditions than was previously available in the literature

on the global uniqueness for the einstein maxwell scalar field system with a cosmological constant part 2 structure of the solutions and stability of the cauchy horizon this paper is the second part of a trilogy dedicated to the following problem given spherically symmetric characteristic initial data for the einstein maxwell scalar field system with a cosmological constant lambda with the data on the outgoing initial null hypersurface given by a subextremal reissner nordstrom black hole event horizon study the future extendibility of the corresponding maximal globally hyperbolic development as a suitably regular lorentzian manifold in the first paper of this sequence costa et al class quantum gravity 32 015017 2015 we established well posedness of the characteristic problem with general initial data in this second paper we generalize the results of dafermos ann math 158 875 928 2003 on the stability of the radius function at the cauchy horizon by including a cosmological constant this requires a considerable deviation from the strategy followed in dafermos ann math 158 875 928 2003 focusing on the level sets of the radius function instead of the red shift and blue shift regions we also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin in the third and final paper costa et al on the global uniqueness for the einstein maxwell scalar field system with a cosmological constant part 3 mass inflation and extendibility of the solutions arxiv 1406 7261 2015 we will consider the issue of mass inflation and extendibility of solutions beyond the cauchy horizon

uniform decay of local energy and the semi linear wave equation on schwarzschild space we provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a schwarzschild background our estimate implies that such solutions have asymptotic behavior phi o left r 1 big t r big frac 1 2 right as long as the source term is bounded in the norm 1 frac 2m r 1 cdot 1 t r 1 l 1 big h 3 omega r 2dr d omega big in particular this gives scattering at small amplitudes for non linear scalar fields of the form square g varphi lambda varphi p varphi for all 2 p

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

the initial boundary value problem for einstein s vacuum field equation we study the initial boundary value problem for einstein s vacuum field equation we prescribe initial data on an orientable compact 3 dimensional manifold s with boundary and boundary conditions on the manifold t re 0 we assume the boundaries and 0 of s and t to be identified in the natural way furthermore we prescribe certain gauge source functions which determine the evolution of the fields provided that all data are smooth and certain consistency conditions are met on we show that there exists a smooth solution to einstein s equation ric g 0 on a manifold which has after an identification a neighbourhood of s in t s as a boundary the solution is such that s is space like the initial data are induced by the solution on s and in the region of t where the solution is defined t is time like and the boundary conditions are satisfied

on the hyperbolicity of einstein s and other gauge field equations it is shown that einstein s vacuum field equations respectively the conformal vacuum field equations in a frame formalism imply a symmetric hyperbolic system of reduced propagation equations for any choice of coordinate system and frame field and conformal factor certain freely specifiable gauge source functions occurring in the reduced equations reflect the choice of gauge together with the initial data they determine the gauge uniquely their choice does not affect the isometry class conformal class of a solution of an initial value problem by the same method symmetric hyperbolic propagation equations are obtained from other gauge field equations irrespective of the gauge using the concept of source functions one finds that einstein s field equation considered as second order equations for the metric coefficients are of wave equation type in any coordinate system

reliability of perturbation theory in general relativity the relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one parameter families of solutions on the one hand the coefficients of the taylor expansion in the parameter of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations on the other hand it is the converse question does any solution of the perturbation equations come from taylor expanding some family of exact solutions which is of importance for the mathematical justification of the use of perturbation theory this converse question is called the one of the reliability of perturbation theory using and completing recent results on the characteristic initial value problem the local reliability of perturbation theory for general relativity in vacuum is proven very generally this result is then generalized to the einstein yang mills equations and therefore in particular to the einstein maxwell ones

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

a new class of obstructions to the smoothness of null infinity expansions of the gravitational field arising from the development of asymptotically euclidean time symmetric conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6 to this end a certain representation of spatial infinity as a cylinder is used this setup is based on the properties of conformal geodesics it is found that these expansions suggest that null infinity has to be non smooth unless the newman penrose constants of the spacetime and some other higher order quantities of the spacetime vanish as a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders furthermore the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity then the initial data must in fact be schwarzschildean around spatial infinity

fast and slow solutions in general relativity the initialization procedure we apply recent results in the theory of pde specifically in problems with two different time scales to einstein s equations near their newtonian limit the results imply a justification to post newtonian approximations when initialization procedures to different orders are made on the initial data we determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations using the results mentioned the first post newtonian equations and their solutions in terms of green s functions are presented in order to indicate how to proceed in calculations with this approach

axisymmetric fully spectral code for hyperbolic equations abstract we present a fully pseudo spectral scheme to solve axisymmetric hyperbolic equations of second order with the chebyshev polynomials as basis functions the numerical grid is based on the lobbato for two spatial directions and radau for the time direction collocation points the method solves two issues of previous algorithms which were restricted to one spatial dimension namely i the inversion of a dense matrix and ii the acquisition of a sufficiently good initial guess for non linear systems of equations for the first issue we use the iterative bi conjugate gradient stabilized method which we equip with a pre conditioner based on a singly diagonally implicit runge kutta sdirk method in this paper the sdirk method is also used to solve issue ii the numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution as an application we solve general relativistic wave equations on a black hole space time in so called hyperboloidal slices and reproduce some recent results available in the literature

stability and b convergence of general linear methods abstract this paper continues earlier work by the same author concerning the stability and b convergence of general linear methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space in a previous paper we have proved that bh consistency resp bh consistency together with bh stability implies optimal b convergence resp b convergence in this paper by means of bs and bsi stability properties the sufficient conditions for a method to be b bh bh or bh consistent bh or weakly bh stable b or optimally b convergent respectively are further established

calculating initial data for the conformal einstein equations by pseudo spectral methods abstract we present a numerical scheme for determining hyperboloidal initial data sets for the conformal field equations by using pseudo spectral methods this problem is to split into two parts the first step is the determination of a suitable conformal factor which transforms from an initial data set in physical space time to a hyperboloidal hypersurface in the ambient conformal manifold this is achieved by solving the yamabe equation a nonlinear second order equation the second step is a division by the conformal factor of certain fields which vanish on i the zero set of the conformal factor the challenge there is to numerically obtain a smooth quotient both parts are treated by pseudo spectral methods the nonlinear equation is solved iteratively while the division problem is treated by transforming the problem to the coefficient space solving it there by the qr factorisation of a suitable matrix and then transforming back these hyperboloidal initial data can be used to generate general relativistic space times by evolution with the conformal field equations

instability results for the wave equation in the interior of kerr black holes abstract we prove that a large class of smooth solutions to the linear wave equation g 0 on subextremal rotating kerr spacetimes which are regular and decaying along the event horizon become singular at the cauchy horizon more precisely we show that assuming appropriate upper and lower bounds on the energy along the event horizon the solution has infinite non degenerate energy on any spacelike hypersurfaces intersecting the cauchy horizon transversally extrapolating from known results in the reissner nordstrom case the assumed upper and lower bounds required for our theorem are conjectured to hold for solutions arising from generic smooth and compactly supported initial data on a cauchy hypersurface this result is motivated by the strong cosmic censorship conjecture in general relativity

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

the cauchy problem for the wave equation in the schwarzschild geometry the cauchy problem is considered for the scalar wave equation in the schwarzschild geometry we derive an integral spectral representation for the solution and prove pointwise decay in time

the asymptotic characteristic initial value problem for einstein s vacuum field equations as an initial value problem for a first order quasilinear symmetric hyperbolic system the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown it is delineated how the same method can be applied to the standard cauchy problems for einstein s vacuum and conformal vacuum equations

convergence of parallel multistep hybrid methods for singular perturbation problems parallel multistep hybrid methods phms can be implemented in parallel with two processors accordingly have almost the same computational speed per integration step as bdf methods of the same order with the same stepsize but phms have better stability properties than bdf methods of the same order for stiff differential equations in the present paper we give some results on error analysis of a a stable phms for the initial value problems of ordinary differential equations in singular perturbation form our convergence results are similar to those of linear multistep methods such as bdf methods i e the convergence orders are equal to their classical convergence orders and no order reduction occurs some numerical examples also confirm our results

late time asymptotics for the wave equation on extremal reissner nordstrom backgrounds abstract we derive the precise late time asymptotics for solutions to the wave equation on extremal reissner nordstrom black holes and explicitly express the leading order coefficients in terms of the initial data our method is based on purely physical space techniques we derive novel weighted energy hierarchies and develop a singular time inversion theory which allow us to uncover the subtle contribution of both the near horizon and near infinity regions to the precise asymptotics we introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes our work confirms and in some cases extends the numerical and heuristic analysis of lucietti murata reall tanahashi ori sela and blaksley burko

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc

superradiant instabilities for short range non negative potentials on kerr spacetimes and applications abstract the wave equation g m a 0 on subextremal kerr spacetimes m m a g m a 0 a m does not admit real mode solutions as was established by shlapentokh rothman in this paper we show that the absence of real modes does not persist under the addition of an arbitrary short range non negative potential v to the wave equation or under changes of the metric g m a in the far away region of m m a retaining the causality of the stationary killing field t there in particular we first establish for any 0 a m the existence of real mode solutions to equation g m a v 0 for a suitably chosen time independent real potential v with compact support in space satisfying v 0 exponentially growing modes are also obtained after perturbing the potential v then as an application of the above result we construct a family of spacetimes m m a g m a d e f which are compact in space perturbations of m m a g m a have the same symmetries as m m a g m a and moreover admit real and exponentially growing modes these spacetimes contain stably trapped null geodesics but we also construct a more complicated family of spacetimes with normally hyperbolic trapped set admitting real and exponentially growing modes at the expense of having conic asymptotics the aforementioned results are in contrast with the case of stationary asymptotically flat or conic spacetimes m g with a globally timelike killing field t where real modes for equation g v 0 are always absent giving a useful zero frequency continuity criterion for showing stability for a smooth family of equations g v 0 with 0 1 and v 0 0 we show explicitly that this criterion fails on kerr spacetime