cosmological post newtonian expansions to arbitrary order we prove the existence of a large class of one parameter families of solutions to the einstein euler equations that depend on the singular parameter epsilon v t c 0 epsilon epsilon 0 where c is the speed of light and v t is a typical speed of the gravitating fluid these solutions are shown to exist on a common spacetime slab m cong 0 t times mathbb t 3 and converge as epsilon searrow 0 to a solution of the cosmological poisson euler equations of newtonian gravity moreover we establish that these solutions can be expanded in the parameter epsilon to any specified order with expansion coefficients that satisfy epsilon independent nonlocal symmetric hyperbolic equations

optimal gradient estimates and asymptotic behaviour for the vlasov poisson system with small initial data the vlasov poisson system describes interacting systems of collisionless particles for solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t 3 at late times in this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay so that in n dimensions for n geqq 3 the derivative of the density of order k decays as t n k an asymptotic formula for the solution at late times is also obtained

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in euclidean and minkowski spaces abstract in this paper we investigate the existence of rotationally symmetric entire graphs resp entire spacelike graphs with prescribed k th mean curvature function in euclidean space r n 1 resp minkowski spacetime l n 1 as a previous step we analyze the associated homogeneous dirichlet problem on a ball which is not elliptic for k 1 and then we prove that it is possible to extend the solutions moreover a sufficient condition for uniqueness is given in both cases

the formation of black holes in spherically symmetric gravitational collapse we consider the spherically symmetric asymptotically flat einstein vlasov system we find explicit conditions on the initial data with adm mass m such that the resulting spacetime has the following properties there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2m the timelike lines r c 0 2m are incomplete and for r 2m the metric converges asymptotically to the schwarzschild metric with mass m the initial data that we construct guarantee the formation of a black hole in the evolution we give examples of such initial data with the additional property that the solutions exist for all r 0 and all schwarzschild time i e we obtain global existence in schwarzschild coordinates in situations where the initial data are not small some of our results are also established for the einstein equations coupled to a general matter model characterized by conditions on the matter quantities

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

the classical limit of the relativistic vlasov maxwell system solutions of the relativistic vlasov maxwell system of partial differential equations are considered in three space dimensions the speed of light c appears as a parameter in this system for smooth cauchy data classical solutions are shown to exist on a time interval that is independent ofc then using an integral representation for the electric and magnetic fields due to glassey and strauss 6 conditions are given under which solutions of the relativistic vlasov maxwell system converge in pointwise sense to solutions of the non relativistic vlasov poisson system at the asymptotic rate of 1 c asc tends to infinity

stable self similar blow up dynamics for the three dimensional relativistic gravitational vlasov poisson system r f x v dv describes the mechanical state of a stellar system subject to its own gravity smooth initial data yield unique global in time solutions from a celebrated result by pfaffelmoser 37 there exists a hierarchy of physical models which aim at taking into account further relativistic effects the simplest one is the three dimensional relativistic gravitational vlasov poisson system tf v 1 v xf ef vf 0 which we study here a striking feature as observed by glassey schaeffer 18 is that this system now admits finite blow up solutions nevertheless the existence argument is purely obstructive and provides no insight into the description of the singularity formation we exhibit in this paper a family of finite time blow up self similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations our analysis applies to the four dimensional gravitational vlasov poisson system as well whose study began in 26

higher regularity of the tangential fields in the relativistic vlasov maxwell system abstract it is shown that the tangential electric and magnetic fields in the glassey strauss representation formulas are in fact bounded in l loc t l x 2 for some 0

remarks on cosmological spacetimes and constant mean curvature surfaces we give a general condition which ensures the existence of constant mean curvature cmc cauchy surfaces in cosmological spacetimes however there is an example of a spacetime which does not satisfy this condition and does not admit any cmc cauchy surfaces we discuss conditions under which cmc surfaces may exist

classical solutions and the glassey strauss theorem for the 3d vlasov maxwell system r glassey and w strauss have proved in arch rational mech anal 92 1986 59 90 that c1 solutions to the relativistic vlasov maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded the present paper simplifies their proof

on the newtonian limit of general relativity we establish rigorous results about the newtonian limit of general relativity by applying to it the theory of different time scales for non linear partial differential equations as developed in 4 1 8 roughly speaking we obtain a priori estimates for solutions to the einstein s equations an intermediate but fundamental step to show that given a newtonian solution there exist continuous one parameter families of solutions to the full einstein s equations the parameter being the inverse of the speed of light which for a finite amount of time are close to the newtonian solution these one parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions this procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the newtonian solution in any a priori given sobolev norm in some intuitive sense these relativistic solutions by being close to the newtonian one have little extra radiation content although actually this should be so only in the case of the characteristic initial data formulation along future directed light cones our results are local in the sense that they do not include the treatment of asymptotic regions global results are admittedly very important in particular they would say how differentiable the solutions are with respect to the parameter but their treatment would involve the handling of tools even more technical than the ones used here on the other hand this local theory is all that is needed for most problems of practical numerical computation

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

spatially homogeneous solutions of the vlasov nordstrom fokker planck system the vlasov nordstrom fokker planck system describes the evolution of self gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation we study the spatially homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions additionally we study the long time asymptotic behavior of the system and prove that even in the absence of friction solutions possess a non trivial asymptotic profile an exact formula for the long time limit of the particle density is derived in the ultra relativistic case

analytical and numerical solution to the two potential zakharov shabat inverse scattering problem an analytical and a numerical method are presented in order to solve the inverse scattering problem associated with the two potential zakharov shabat coupled mode equations the numerical solution which uses leapfrogging in space and time represents a direct numerical solution to the coupled gel fand levitan marchenko glm integral equations as an extension of the authors previous work on glm equations of simpler form the analytical method which is applied for one pole reflection coefficients consists in constructing appropriate differential operators which transform the coupled glm equations to ordinary linear differential equations an application of these methods for nonuniform transmission line synthesis is presented

initial data sets of minimal energy the total mass of a gravitational system the adm mass is considered as a functional of the initial data sets for sources and gravitation from the condition that such a functional has a minimum under a restricted class of variations tensorial equations for the gravitational initial data are obtained the solutions to these equations whose existence for a large class of source fields is asserted represent initial data sets for gravitation satisfying the constraint equations and may be considered as having no gravitational radiation

hypersurfaces of prescribed mean curvature in lorentzian manifolds we give a new existence proof for closed hypersurfaces of prescribed mean curvature in lorentzian manifolds

asymptotic behavior and orbital stability of galactic dynamics in relativistic scalar gravity the nordstrom vlasov system is a relativistic lorentz invariant generalization of the vlasov poisson system in the gravitational case the asymptotic behavior of solutions and the non linear stability of steady states are investigated it is shown that solutions of the nordstrom vlasov system with energy greater or equal to the mass satisfy a dispersion estimate in terms of the conformal energy when the energy is smaller than the mass we prove the existence and non linear orbital stability of a class of static solutions isotropic polytropes against general perturbations the proof of orbital stability is based on a variational problem associated to the minimization of the energy functional under suitable constraints

a new continuation criterion for the relativistic vlasov maxwell system the global existence of solutions to the relativistic vlasov maxwell system given sufficiently regular finite energy initial data is a longstanding open problem the main result of glassey and strauss arch ration mech anal 92 59 90 1986 shows that a solution f e b remains c 1 as long as the momentum support of f remains bounded alternate proofs were later given by bouchut et al arch ration mech anal 170 1 15 2003 and klainerman and staffilani commun pure appl anal 1 103 125 2002 we show that only the boundedness of the momentum support of f after projecting to any two dimensional plane is needed for f e b to remain c 1

positive rotationally symmetric solutions for a dirichlet problem involving the higher mean curvature operator in minkowski space abstract in this paper we investigate the global behavior of positive rotationally symmetric solutions for dirichlet problem involving the k th mean curvature operator in minkowski space l n 1 which is not elliptic for k 1 the proofs of our main results are based upon bifurcation techniques

problems with different time scales for nonlinear partial differential equations consider a system of partial differential equations u t varepsilon 1 p 0 p 1 u here 0 varepsilon ll 1 is a small constant p 0 p 0 partial partial x is a differential operator with constant coefficients and p 1 p 1 x t u partial partial x is a quasilinear operator the coefficients of p 0 p 1 are of order o 1 it is shown how to prepare the initial data such that the solutions u x t and their derivatives can be bounded independently of varepsilon applications are discussed

the newtonian limit for asymptotically flat solutions of the vlasov einstein system it is shown that there exist families of asymptotically flat solutions of the einstein equations coupled to the vlasov equation describing a collisionless gas which have a newtonian limit these are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition a central role in the proof is played by energy estimates in unweighted sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation

evolution of spacelike surfaces in mathrm ads _3 by their lagrangian angle we study spacelike hypersurfaces m in an anti de sitter spacetime n of constant sectional curvature kappa kappa 0 that evolve by the lagrangian angle of their gaus maps in the two dimensional case we prove a convergence result to a maximal spacelike surface if the gaus curvature k of the initial surface m subset n and the sectional curvature of n satisfy k kappa

the newtonian limit for perfect fluids we prove that there exists a class of non stationary solutions to the einstein euler equations which have a newtonian limit the proof of this result is based on a symmetric hyperbolic formulation of the einstein euler equations which contains a singular parameter vt c where vt is a characteristic velocity scale associated with the fluid and c is the speed of light the symmetric hyperbolic formulation allows us to derive e independent energy estimates on weighted sobolev spaces these estimates are the main tool used to analyze the behavior of solutions in the limit 0

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

immersed hypersurfaces with constant weingarten curvature in a recent paper korevaar 1 5 used the alexandrov reflection principle to show that closed embedded hypersurfaces in n 1 lhn 1 or the upper hemisphere of s 1 are umbilic spheres provided a certain function f of the principal curvatures t t 22 is constant he only had to assume that f is positive on the positive cone 212i 0 vi and that f is elliptic on the component f of 21 f 0 which contains here f is said to be elliptic if 0f d21 0 for all i 1 inn this generalization of earlier sphere theorems see 9 and 10 for references cannot be extended to hypersurface immersions in view of recent counterexamples 1 11 however assuming additional curvature conditions walter derived in 10 global results for hypersurface immersions in a space n l c of constant curvature c which have a constant higher mean curvature function h here h is the r th symmetric function of the principal curvatures it was shown that such hypersurfaces are of constant mean curvature h1 provided they have nonnegative sectional curvature and non negative principal curvatures as a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified here we show that it is not necessary to assume all principal curvatures to be non negative moreover we extend waiter s result to general symmetric functions f f 2 let m be a smooth connected and compact manifold without boundary then we have the following result

a numerical solution to the zakharov shabat inverse scattering problem a novel numerical method to solve the inverse scattering problem associated with the zakharov shabat coupled mode equations is developed in this problem the coupling coefficient for two propagating modes is determined from reflection data both codirectional and contradirectional coupling can be accommodated as well as the presence of bound states analytical results based on an independent analytical method are derived in order to demonstrate the accuracy of the proposed method which is applicable to a broad class of reflection data

entire spacelike hypersurfaces of constant gaus curvature in minkowski space we prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed gaus curvature in minkowski space the proof is based on barrier constructions and local a priori estimates

eigenvalues and eigenfunctions associated with the gel fand levitan equation it is shown here that the solutions of the gel fand levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class the details are sketched for the case of rational reflection coefficients and carried out for the simplest class of examples

a vector field method for massless relativistic transport equations and applications abstract in this article we present a vector field method for the study of solutions to massless relativistic transport equations compared to 10 we remove the lorentz boosts of the commutation vector fields and we prove a new functional inequality in order to derive pointwise decay estimates on the solutions it makes our method more suitable for the study of systems coupling a massless vlasov equation with an equation which has a different speed of propagation we also believe that our approach can be adapted more easily to the context of curved backgrounds in the second part of this paper we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless vlasov poisson system no compact support assumption is required on the data and in particular the initial decay in the velocity variable is optimal in order to compensate the slow decay rate of the particle density near the light cone we take advantage of the null structure of the equations

post newtonian expansions for perfect fluids we prove the existence of a large class of dynamical solutions to the einstein euler equations that have a first post newtonian expansion the results here are based on the elliptic hyperbolic formulation of the einstein euler equations used in 15 which contains a singular parameter epsilon v t c where vt is a characteristic velocity associated with the fluid and c is the speed of light as in 15 energy estimates on weighted sobolev spaces are used to analyze the behavior of solutions to the einstein euler equations in the limit epsilon searrow 0 and to demonstrate the validity of the first post newtonian expansion as an approximation

on linear landau damping for relativistic plasmas via gevrey regularity abstract we examine the phenomenon of landau damping in relativistic plasmas via a study of the relativistic vlasov poisson system both on the torus and on r 3 linearized around a sufficiently nice spatially uniform kinetic equilibrium we find that exponential decay of spatial fourier modes is impossible under modest symmetry assumptions however by assuming the equilibrium and initial data are sufficiently regular functions of velocity for a given wavevector in particular that they exhibit a kind of gevrey regularity we show that it is possible for the mode associated to this wavevector to decay like exp t with 0 1 if the magnitude of the wavevector exceeds a certain critical size which depends on the character of the interaction we also give a heuristic argument why one should not expect such rapid decay for modes with wavevectors below this threshold

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

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

permanent regimes for the vlasov maxwell equations with specular boundary conditions the subject matter of this paper concerns the existence of permanent regimes i e stationary or time periodic solutions for the vlasov maxwell system in a bounded domain we are looking for equilibrium configurations by imposing specular boundary conditions the main difficulty is the treatment of such boundary conditions our analysis relies on perturbative techniques based on uniform a priori estimates

asymptotic behavior of weak solutions for the relativistic vlasov maxwell equations with large light speed abstract we study here the behavior of time periodic weak solutions for the relativistic vlasov maxwell boundary value problem in a three dimensional bounded domain with strictly star shaped boundary when the light speed becomes infinite we prove the convergence toward a time periodic weak solution for the classical vlasov poisson equations

on the definition of post newtonian approximations a definition of post newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms this establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post newtonian equations which are used in practical calculations the breakdown of higher post newtonian approximations is examined within this framework it is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post newtonian level if one starts with a generic newtonian solution the most simple choice of gauge gives equations which are solvable at the 2pn level but which in general have no solutions in the case of the third post newtonian approximation

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

global weak solutions to the nordstrom vlasov system abstract the nordstrom vlasov system is a lorentz invariant model for a self gravitating collisionless gas we establish suitable a priori bounds on the solutions of this system which together with energy estimates and the smoothing effect of momentum averaging yield the existence of global weak solutions to the corresponding initial value problem in the process we improve the continuation criterion for classical solutions which was derived recently the weak solutions are shown to preserve mass

einstein s equations with asymptotically stable constraint propagation we introduce a proposal to modify einstein s equations by embedding them in a larger symmetric hyperbolic system the additional dynamical variables of the modified system are essentially first integrals of the original constraints the extended system of equations reproduces the usual dynamics on the constraint surface of general relativity and therefore naturally includes the solutions to einstein gravity the main feature of this extended system is that at least for a linearized version of it the constraint surface is an attractor of the time evolution this feature suggests that this system may be a useful alternative to einstein s equations when obtaining numerical solutions to full nonlinear gravity

well posed forms of the 3 1 conformally decomposed einstein equations we show that well posed conformally decomposed formulations of the 3 1 einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations we compute the characteristics structure and verify the constraint propagation of these new well posed formulations in these formulations the trace of the extrinsic curvature and the determinant of the 3 metric are singled out from the rest of the dynamical variables but are evolved as part of the well posed evolution system the only free functions are the lapse density and the shift vector we find that there is a 3 parameter freedom in formulating these equations in a well posed manner and that part of the parameter space found consists of formulations with causal characteristics namely characteristics that lie only within the lightcone in particular there is a 1 parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric

finite speed propagation of the solutions for the relativistic vlasov maxwell system in this paper we investigate the continuous dependence with respect to the initial data of the solutions for the 1d and 1 5d relativistic vlasov maxwell system more precisely we prove that these solutions propagate with finite speed we formulate our results in the framework of mild solutions i e the particle densities are solutions by characteristics and the electro magnetic fields are lipschitz continuous functions

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

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

global classical solutions to the 3d nordstrom vlasov system the nordstrom vlasov system describes the kinetic evolution of self gravitating collisionless matter in the framework of a relativistic scalar theory of gravitation we prove global existence and uniqueness of classical solutions for the corresponding initial value problem in three dimensions when the initial data for the scalar field are smooth and the initial particle density is smooth with compact support

yet another criterion for global existence in the 3d relativistic vlasov maxwell system abstract we prove that solutions of the 3d relativistic vlasov maxwell system can be extended as long as the quantity 1 t x max 1 r 3 d p 1 p 2 1 1 v f t x p is bounded in l x 2

numerical relativity as a tool for computational astrophysics abstract the astrophysics of compact objects which requires einstein s theory of general relativity for understanding phenomena such as black holes and neutron stars is attracting increasing attention in general relativity gravity is governed by an extremely complex set of coupled nonlinear hyperbolic elliptic partial differential equations the largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of einstein s equations for the first time since they were written over 80 years ago allowing one to attempt full 3d simulations of such exciting events as colliding black holes and neutron stars in this paper we review the computational effort in this direction and discuss a new 3d multi purpose parallel code called cactus for general relativistic astrophysics directions for further work are indicated where appropriate

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

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

exact solutions of the one dimensional acoustic wave equations for several new velocity profiles transmission and reflection coefficients after reviewing the relationship of the acoustic equation to the schrodinger equation solutions of the latter are used to construct new indices of refraction or equivalently sound velocity profiles svp for which the solutions of the acoustic equation are expressed in terms of elementary functions the new indices of refraction can be used to form layers to model more general inhomogeneous media just as piecewise constant slabs or other indices have been used in the past four of the new indices of refraction also provide especially simple closed form examples which can be used to describe how waves are scattered in media with such indices more general scattering problems are discussed in particular the notion of outgoing wave is gone into the five indices given in the present paper are the first of several which have been obtained from schrodinger equations with potentials for which the eigenfunctions can be found in closed form others will be given in later papers