Cohomological uniqueness of the Cauchy problem solutions for the Einstein–Maxwell equation

In this paper we analyze the Cauchy problem for the Einstein–Maxwell equation in the case of non-characteristic initial hypersurface. To find the correct notions of characteristic and Cauchy data we introduce a complex, which we call the Einstein–Maxwell complex. Then the Cauchy problem acquires correctness in terms of an associated spectral sequence. We define a Cauchy data in such way that they allow us to reconstruct a cohomologously unique formal solution.     

Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation

We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions underlying self-dual vacuum solutions of the Einstein equations, which arises from the commutation of multidimensional Hamiltonian vector fields.     

A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations

We study the conformal and conformal thin sandwich (CTS) methods as candidates for parameterizing the set vacuum initial data for the Cauchy problem of general relativity. To this end we consider a small family of symmetric conformal data. Within this family we obtain an existence result so long as the mean curvature has constant sign. When the mean curvature changes sign we find that solutions either do not exist, or they are not unique. In some cases solutions are shown to be non-unique. Moreover, the theory for mean curvatures with changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.     

Coordinate conditions in the theory of relativity

Different coordinate conditions are investigated for the Einstein equations; depending on them the field equations can belong to different types. The Cauchy problem for some is incorrect. The domain of influence of the initial data can be extended to the whole initial surface.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 46–49, February, 1982.     

Cauchy's problem and Huygens' principle for the linearized Einstein field equations

The linearized Einstein field equations with Einstein space-times as background are studied by use of the harmonic gauge. By means of Riesz' integration method a representation theorem for the solution of Cauchy's problem, using the constraints of the Cauchy data and the calculus of symmetric differential forms, is proved. We introduce some linear differential operators, which map the set of symmetric differential forms into the subset with vanishing divergence and trace and use these operators to derive necessary conditions for the validity of Huygens' principle from which it follows that the linearized field equations satisfy Huygens' principle only in flat space-times.Dedicated to Gudrun Schmidt on the occasion of her 50th birthday.     

Blow-up of test fields near Cauchy horizons

The behaviour of test fields near a compact Cauchy horizon is investigated. It is shown that solutions of nonlinear wave equations on Taub spacetime with generic initial data cannot be continued smoothly to both extensions of the spacetime through the Cauchy horizon. This is proved using an energy method. Similar results are obtained for the spacetimes of Moncrief containing a compact Cauchy horizon and for more general matter models.Partially supported by NSF grant DMS 9003256 and Office of Naval Research grant ONR NO 014 92 J 1245.     

Rough Solutions of Einstein Vacuum Equations in CMCSH Gauge

In this paper, we consider very rough solutions to the Cauchy problem for the Einstein vacuum equations in CMC spatial harmonic gauge, and obtain the local well-posedness result in H ^s , s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric g, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation ${\square_{\bf g} \phi=0}$ directly.     

Long-Time Asymptotics for the Nonlocal MKdV Equation

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation(nonlocal mKdV) qt(x, t)+qxxx(x, t)-6 q(x, t)q(-x,-t)qx(x, t) = 0, which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal m KdV equation. In contrast with the classical mKdV equation,we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.     

Existence theorems for π/n vortex scattering

The analysis of 90° vortex-vortex scattering is extended to /n scattering in all head-on collisions ofn vortices in the Abelian Higgs model. A Cauchy problem with initial data that describe the scattering ofn vortices is formulated. It is shown that this Cauchy problem has a unique global finite-energy solution. The symmetry of the solution and the form of the local analytic solution then show that /n scattering is realized.     

An imbedding of spacetimes

It is shown that any two-dimensional spacetimes with compact Cauchy surfaces can be causally isomorphically imbedded into the two-dimensional Einstein’s static universe. Also, it is shown that any two-dimensional globally hyperbolic spacetimes are conformally equivalent to a subset of the two-dimensional Einstein’s static universe.     

The Area of Horizons and the Trapped Region

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers, curvature estimates, together with a novel surgery construction for marginal surfaces. These results are applied to characterize the boundary of the trapped region. Supported in part by the NSF, under contract no. DMS 0407732 with the University of Miami. Supported in part by a Feodor-Lynen Fellowship of the Humboldt Foundation.     

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.     

On Darboux transformations for soliton equations in high-dimensional spacetime

The Darboux transformations for a class of completely integrable systems in the spacetimeR ^{n + 1}, which are much more general than the systems inLett. Math. Phys. 26, 199–209 (1989), are considered. The structure of the nonlinear evolution equations with space constraints is elucidated. It is pointed out that the inverse scattering method can be used to solve the Cauchy problem with initial data given on a noncharacteristic line.Supported by National Basic Research Project Nonlinear Science, NNSFC of China, FEYUT-SEDC-CHINA and Fok Ying-Tung Education Foundation of China.     

Sasaki–Einstein Manifolds and Volume Minimisation

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kähler–Einstein metric.     

Nonexistence of conformally flat slices in Kerr and other stationary spacetimes

It is proved that stationary solutions to the vacuum Einstein field equations with a nonvanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and nonboosted. The proof is based on results coming from a certain type of asymptotic expansion near null and spatial infinity--which also show that the development of Bowen-York-type data cannot have a development admitting a smooth null infinity--and from the fact that stationary solutions do admit a smooth null infinity.     

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein–Dirac–Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.     

Blow-up of Smooth Solutions to the Cauchy Problem for the Viscous Two-Phase Model

In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the viscous two-phase model in arbitrary dimensions under some restrictions on the initial data. The key of the proof is finding the relations between some physical quantities and establishing some inequalities.     

Numerical solution of the problem of computational time reversal in a quadrant

The problem of computational time reversal is posed as the inverse problem of the determination of an unknown initial condition with a finite support in a hyperbolic equation, given the Cauchy data at the lateral surface. Two such two-dimensional inverse problems are solved numerically in the case when the domain is a quadrant and the Cauchy data are given at finite parts of the coordinate axes. The previously obtained Lipschitz stability estimate implies refocusing of the time-reversed wave field in the case of a small amount of noise in the data. It also indicates the possibility of good performance of a proper numerical method. Such performance is demonstrated in this paper for a particular problem and a particular numerical method.