A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

We show that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.Research supported in part by NSF DMS 9101517     

The structure of the space of solutions of Einstein's equations II: several Killing fields and the Einstein-Yang-Mills equations

The space of solutions of Einstein's vacuum equations is shown to have conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a nontrivial set of Killing fields. Similar results are shown for the coupled Einstein-Yang-Mills system. Combined with an appropriate slice theorem, the results show that the space of geometrically equivalent solutions is a stratified manifold with each stratum being a symplectic manifold characterized by the symmetry type of its members.     

A class of plane symmetric dust solutions

A class of plane symmetric solutions containing dust is considered. It is argued that, however inhomogeneous the mass distribution, matter on each plane of symmetry has no net attraction to matter on other planes. It is shown that the geodesic distance between two thin sheets of dust, separated by a particular Kasner region, is zero a finite time after the initial singularity, quickly reaches a maximum, and then decreases ast ^–1/3     

High-codimensional static bifurcations of strongly nonlinear oscillator

The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.     

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.This research was supported through a fellowship from the Consejo Nacional de Investigaciones Cientificas y Technicas de la Republica Argentina.     

Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat. Received: 15 July 1996 / Accepted: 12 March 1997     

The Initial Singularity of Ultrastiff Perfect Fluid Spacetimes Without Symmetries

We consider the Einstein equations coupled to an ultrastiff perfect fluid and prove the existence of a family of solutions with an initial singularity whose structure is that of explicit isotropic models. This family of solutions is ‘generic’ in the sense that it depends on as many free functions as a general solution, i.e., without imposing any symmetry assumptions, of the Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.     

On Nodal Sets for Dirac and Laplace Operators

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a Δ-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal. Received: 28 October 1996 / Accepted: 3 March 1997     

A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction

In this paper, new class of solutions of the Tzitzéica equation are derived by using the classical Lie symmetry analysis. The important aspect of this paper however is the fact that the analysis results in a new class of solitary-like solutions, the so-called cusp-solitary solutions.

This special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold. The studied equation was originally found in the field of geometry, otherwise the Tzitzéica equation takes place in many branches of non-linear sciences. Therefore, explicit class of solutions connected by a physical meaning are of great importance. The analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation. A complete characterization of the group properties is given. The classical Lie point symmetries are derived and algebraic properties are determined. Similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of Jacobian elliptic functions. It is worth to mention that the application of known powerful algebraic methods (e.g. special function transform methods) are not appropriate to study the solution manifold. Hence, the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture.     

Blowup of Smooth Solutions for Relativistic Euler Equations

We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial ``generalized' momentum is sufficiently large.     

A solution to the Navier-Stokes inequality with an internal singularity

We consider weak solutions to the time dependent Navier-Stokes equations of incompressible fluid flow in three dimensional space with an external force that always acts against the direction of the flow. We show that there exists a solution with an internal singularity. The speed of the flow reaches infinity at this singular point. In addition, the solution has finite kinetic energy.The author was supported in part by a Sloan Foundation Fellowship     

Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation

It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.     

On Lipschitz solutions of the constant astigmatism equation

We show that the solutions of the constant astigmatism equation that correspond to a class of surfaces found by Lipschitz in 1887, exactly match the Lie symmetry invariant solutions and constitute a four-dimensional manifold. The two-dimensional orbit space with respect to the Lie symmetry group is described. Our approach relies on the link between constant astigmatism surfaces and orthogonal equiareal patterns. The counterpart sine–Gordon solutions are shown to be Lie symmetry invariant as well.     

Traveling periodic waves of chemical activity

It is shown that within the manifold of exact solutions a system of reaction-diffusion equations admits only travelling waves with planar symmetry. A derivation of the generic form of approximate (asymptotic) cylindrical and spiral travelling periodic wave solutions is given. If an exact solution homogeneous in space and periodic in time is admitted by the system of reaction-diffusion equations, then travelling periodic spiral waves are admissble as approximate solutions. This is the theoretical explanation for the travelling periodic waves of chemical activity observed in recent experiments.     

A new similarity solution of the Burgers equation with linear damping

The direct method of Clarkson and Kruskal is used to study the symmetry reductions of the Burgers equation with linear damping. The classical similarity reductions reported previously is recovered. The new, nonclassical similarity reduction is obtained. The new similarity solution is given, and it is also obtained by means of the singular manifold method. The nonclassical method is used to demonstrate that the new exact solution is indeed a nonclassical similarity solution. This work has been supported by the Postdoctoral Science Foundation of China.     

Electrostatic and optical resonances of arrays of cylinders

The solutions of the electrostatic potential problem for the square and hexagonal arrays of circular cylinders with zero applied field (homogeneous or resonant solutions) are studied. We show that for non-touching cylinders, the set of resonances is discrete except in the neighbourhood of one point, at which the dielectric constant of the array has an essential singularity. For arrays of touching cylinders, the set is well represented by a continuous distribution. This representation enables the derivation of the asymptotic form of the expansion for the dielectric constant of the array when the dielectric constant of the cylinders is large. The known value of the first term in the expansion enables us to derive the second term. The physical characteristics of the resonant solutions are studied. Metals achieve values of dielectric constant which are close to the resonant values (real and negative) for certain wavelengths. Curves are given which enable the prediction of those wavelengths at which the optical resonances of both arrays occur, for any area fraction and composition of a columnar cerment film.     

Spatial Kasner Solution and an Infinite Slab with a Constant Energy Density

We study the solutions of the Einstein equations in the presence of a thick infinite slab with constant energy density. When there is an isotropy in the plane of the slab, we find an explicit exact solution that matches with the Rindler and Weyl-Levi-Civita spacetimes outside the slab. We also show that there are solutions that can be matched with general anisotropic Kasner spacetime outside the slab. In any case, it is impossible to avoid the presence of the Kasner type singularities in contrast to the well-known case of spherical symmetry, where by matching the internal Schwarzschild solution with the external one, the singularity in the center of coordinates can be eliminated.

     

Gravitational Collapse of Self-Similar Perfect Fluid in 2 + 1 Gravity

Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. These include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth and second kinds. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either a black hole or a null singularity. It is also shown that one solution can have two different kinds of kinematic self-similarity.     

The fermion determinant in two dimensional Minkowski space: Zeros and related properties

We present a detailed analysis of the non-abelian determinant of massless fermions in two dimensional Minkowski space. In the framework of the external field problem, the determinant vanishes if the out- and ingoing vacua are orthogonal; the gauge potentials for which this happens are identified. Causality implies that the effective action obtained from the sum of fermion loops has the right singularity at a zero of the determinant. Such a zero can be reached by a continuous deformation of a potential with non-vanishing determinant. The set of zeros exhibits a rich structure.Work partially supported by the Swiss National Science Foundation