The Problem of a Self-Gravitating Scalar Field with Positive Cosmological Constant

We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.     

Non Linear Perturbations of Kerr Spacetime in External Regions and the Peeling Decay

We prove, outside the influence region of a ball of radius R ₀ centred in the origin of the initial data hypersurface, Σ₀, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr. This external region is the “far” part of the outer region of the perturbed Kerr spacetime. Moreover, if we assume that the corrections to the Kerr metric decay sufficiently fast, o(r ⁻³), we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling theorem”.     

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the “Double Null Foliation Gauge”

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region much larger than the one provided by the Cauchy–Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the a priori estimates for the Weyl equations, associated with the “Bel-Robinson norms”. In particular, if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a “geometric” way the Einstein equations, then we show how the Cauchy–Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally, we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.     

Projecting Massive Scalar Fields to Null Infinity

It is known that, in an asymptotically flat spacetime, null infinity cannot act as an initial-value surface for massive scalar fields. Exploiting tools proper of harmonic analysis on hyperboloids and global norm estimates for the wave operator, we show that it is possible to circumvent such obstruction at least in Minkowski spacetime. Hence we project norm-finite solutions of the Klein–Gordon equation of motion in data on null infinity and, eventually, we interpret them in terms of boundary free field theory. Submitted: May 7, 2007. Accepted: July 16, 2007.     

Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces \({{\bf \mathcal{S}}}\) in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang A (Gibbons–Penrose inequality for surfaces in Schwarzschild Spacetime. arXiv:1303.1863, 2013) find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of \({{\bf \mathcal{S}}}\) onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional Euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang’s condition explicitly in terms of the time height function of \({{\bf \mathcal{S}}}\) over a hyperplane and the geometry of the projection of \({{\bf \mathcal{S}}}\) along its past null cone onto this hyperplane. We also include, in Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.     

On the Cartan Curvatures of a Null Curve in Minkowski Spacetime

We reconstruct the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and we characterize pseudo-spherical null curves and Bertrand null curves.Mathematics Subject Classification(2000). 53B30, 53A04.     

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.     

Global Existence of Solutions to Nonlinear Wave Equations

T. Global in time solutions of the equations Ou = H(u, u') on Minkowski space-time are considered. Results available so far involve complicated decay and energy estimates and also careful choice of Banach spaces and associated ordinary differential inequalities. This work tries to simplify some of the existing arguments and to develop a new technique for other nonlinear evolution equations. The method is motivated by the work of Christodoulou and Baez, Segal, and Zhou, on nonlinear wave equations. The key idea is to use the Penrose conformal compactification that transforms the equations from Minkowski space to the Einstein universe in order to change the global existence question to the local one.     

TIME-ASYMPTOTIC STABILITY OF BOUNDARY-LAYERS FOR A HYPERBOLIC RELAXATION SYSTEM

This work is concerned with time-asymptotic stability of boundary-layers for a typical hyperbolic relaxation system. Under a nonclassical requirement characterizing a class of boundary conditions for the typical system, we prove the global (in time) existence and asymptotic decay of solutions with initial data close to the steady solutions or relaxation boundary-layers.     

Micropolar fluid system in a space of distributions and large time behavior

We analyze the well-posedness of the initial value problem for the generalized micropolar fluid system in a space of tempered distributions and also prove the existence of the stationary solutions. The asymptotic stability of solutions is showed in this space, and as a consequence, a criterium for vanishing small perturbations of initial data (stationary solution) at large time is obtained. A fast decay of the solutions is obtained when we assume more regularity on the initial data.     

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.     

On strong solutions to the compressible Hall-magnetohydrodynamic system

In this paper, we consider strong/classical solutions to the 3D compressible Hall-magnetohydrodynamic system. First, we prove the existence of local strong solutions with positive density. Then the existence of global small solutions with small initial data is proved. Optimal time decay rate is also established.     

Accretive growth kinematics via null evolution curve

It is known that the kinematics on the Lorentzian surfaces changes according to the casual characters of the vector fields. Suspicions, the character of the generator curve affects the surface growth. Therefore, we determine the model of the growth function in the three-dimensional Minkowski spacetime with a null generating curve. Moreover, the proposed method is illustrated with various examples.     

Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

This paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove similar results for the wave equation on the exterior of a very slowly rotating Kerr spacetime with recent work applying the vector-field method to the relativistic Vlasov equation.     

Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

In this paper, we study time‐asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space‐time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time‐asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.     

Causal cells: spacetime polytopes with null hyperfaces

We consider polyhedra and 4-polytopes in Minkowski spacetime—in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, “tetrahedral diamonds” and 4-parallelotopes. We propose a “most regular” representative of each class. The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four lightrays in a tetrahedral pattern. We speculate on the relevance of this construct for discretizations of curved spacetime and for quantum gravity.     

Solution globale des equations de Maxwell-Dirac-Klein-Gordon

We prove the global existence on Minkowski space time of a solution of the Cauchy problem for the non linear system of coupled Maxwell, Dirac and Klein-Gordon equations, for small data with appropriate decay at space-like infinity. The method uses the conformal mapping of Minkowski space time onto a bounded open set of the Einstein cylinder.     

Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.     

Local Propagation of Impulsive GravitationalWaves

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data. © 2015 Wiley Periodicals, Inc.     

Global existence for three‐dimensional incompressible isotropic elastodynamics

The existence of global‐in‐time classical solutions to the Cauchy problem for incompressible, nonlinear, isotropic elastodynamics for small initial displacements is proved. The generalized energy method is used to obtain strong dispersive estimates that are needed for long‐time stability. This requires the use of weighted local decay estimates for the linearized equations, which are obtained as a special case of a new general result for certain isotropic symmetric hyperbolic systems. In addition, the pressure that arises as a Lagrange multiplier to enforce the incompressibility constraint is estimated as a nonlinear term. The incompressible elasticity equations are inherently linearly degenerate in the isotropic case; i.e., the equations satisfy a null condition necessary for global existence in three dimensions. © 2007 Wiley Periodicals, Inc.