- Product | Existence Theorems in Partial Differential Equations
- Existence and comparison theorems for partial differential equations of Riccati type
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Moreover, the solutions appear to still be discrete, so that a simple perturbation argument starting from the spherical case does not seem feasible. In another development it was shown that a linearized analysis indicates the existence of stationary non-static solutions [ 10 ]. It would be desirable to study the question of linearization stability in this case, which, if the answer were favourable, would give an existence proof for solutions of this kind.

A solution of the Einstein equations is called spatially homogeneous if there exists a group of symmetries with three-dimensional spacelike orbits. In this case there are at least three linearly independent spacelike Killing vector fields. For most matter models the field equations reduce to ordinary differential equations. Kinetic matter leads to an integro-differential equation. The most important results in this area have been reviewed in a recent book edited by Wainwright and Ellis[ 81 ].

See, in particular, part two of the book. There remain a host of interesting and accessible open questions. The spatially homogeneous solutions have the advantage that it is not necessary to stop at just existence theorems; information on the global qualitative behaviour of solutions can also be obtained. An important open question concerns the mixmaster solution, as discussed in [ 73 ]. The most detailed results on global inhomogeneous solutions of the Einstein equations obtained up to now concern spherically symmetric solutions of the Einstein equations coupled to a massless scalar field with asymptotically flat initial data.

In a series of papers Christodoulou [ 17 , 16 , 19 , 18 , 20 , 21 , 22 , 26 ] has proved a variety of deep results on the global structure of these solutions. Particularly notable are his proofs that naked singularities can develop from regular initial data [ 22 ] and that this phenomenon is unstable with respect to perturbations of the data [ 26 ]. In related work Christodoulou [ 23 , 24 , 25 ] has studied global spherically symmetric solutions of the Einstein equations coupled to a fluid with a special equation of state the so-called two-phase model.

Solutions of the Einstein equations with cylindrical symmetry which are asymptotically flat in all directions allowed by the symmetry represent an interesting variation on asymptotic flatness. Since black holes are incompatible with this symmetry, one may hope to prove geodesic completeness of solutions under appropriate assumptions.

This has been accomplished for the Einstein vacuum equations and for the source-free Einstein-Maxwell equations in[ 7 ], building on global existence theorems for wave maps[ 31 , 30 ]. For a quite different point of view on this question see[ 83 ]. In the context of spatially compact spacetimes it is first necessary to ask what kind of global statements are to be expected. In a situation where the model expands indefinitely it is natural to pose the question whether the spacetime is causally geodesically complete towards the future. In a situation where the model develops a singularity either in the past or in the future one can ask what the qualitative nature of the singularity is.

It is very difficult to prove results of this kind. As a first step one may prove a global existence theorem in a well-chosen time coordinate. The aim is then to show that, in the maximal Cauchy development of data belonging to a certain class, a time coordinate of the given type exists and exhausts the expected interval.

## Product | Existence Theorems in Partial Differential Equations

The first result of this kind for inhomogeneous spacetimes was proved by Moncrief in [ 58 ]. This result concerned Gowdy spacetimes.

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These are vacuum spacetimes with two commuting Killing vectors acting on compact orbits. The area of the orbits defines a natural time coordinate. This result was extended to more general vacuum spacetimes with two Killing vectors in [ 6 ].

Another attractive time coordinate is constant mean curvature CMC time. For a general discussion of this see [ 70 ]. A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models collisionless matter, wave maps was proved in [ 74 ]. That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in [ 71 ]. Related results have been obtained for spherical and hyperbolic symmetry [ 69 , 11 ].

There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [ 32 , 50 , 35 ] and solutions of the Einstein-Vlasov system with spherical and plane symmetry[ 63 ]. This leads to some simplification in comparison to the general problem, but with present techniques it is still very hard to obtain results of this kind. In [ 39 ] Friedrich proved a result on the stability of de Sitter space.

This concerns the Einstein vacuum equations with positive cosmological constant. His result is as follows. Consider initial data induced by de Sitter space on a regular Cauchy hypersurface. Then all initial data vacuum with positive cosmological constant near enough to these data in a suitable Sobolev topology have maximal Cauchy developments which are geodesically complete.

## Existence and comparison theorems for partial differential equations of Riccati type

In fact the result gives much more detail on the asymptotic behaviour than just this and may be thought of as proving a form of the cosmic no hair conjecture in the vacuum case. This conjecture says roughly that the de Sitter solution is an attractor for expanding cosmological models with positive cosmological constant. This result is proved using conformal techniques and, in particular, the regular conformal field equations developed by Friedrich. There are results obtained using the regular conformal field equations for negative or vanishing cosmological constant [ 41 , 43 ] but a detailed discussion of their nature would be out of place here.

The other result on global existence for small data is that of Christodoulou and Klainerman on the stability of Minkowski space[ 28 ]. The formulation of the result is close to that given in Section 4.

Suppose then that initial data are given which are asymptotically flat and sufficiently close to those induced by Minkowski space on a hyperplane. Then Christodoulou and Klainerman prove that the maximal Cauchy development of these data is geodesically complete. They also provide a wealth of detail on the asymptotic behaviour of the solutions. The proof is very long and technical.

The central tool is the Bel-Robinson tensor which plays an analogous role for the gravitational field to that played by the energy-momentum tensor for matter fields.

Apart from the book of Christodoulou and Klainerman itself some introductory material on geometric and analytic aspects of the proof can be found in [ 8 ] and [ 27 ] respectively. Penrose suggests that the initial singularity in a cosmological model should be such that the Weyl tensor tends to zero or at least remains bounded. There is some difficulty in capturing this by a geometric condition and it was suggested by [ 79 ] that a clearly formulated geometric condition which, on an intuitive level, is closely related to the original condition, is that the conformal structure should remain regular at the singularity.

Singularities of this type are known as conformal or isotropic singularities. Then it has been shown [ 61 , 36 ] that solutions with an isotropic singularity are determined uniquely by certain free data given at the singularity. The data which can be given is, roughly speaking, half as large as in the case of a regular Cauchy hypersurface. The method of proof is to derive an existence and uniqueness theorem for a suitable class of singular hyperbolic equations.

Generalizations of this by Anguige and Tod have been discussed in [ 80 ]. Related work was done earlier in a somewhat simpler context by Moncrief[ 59 ] who showed the existence of a large class of spacetimes with Cauchy horizons. In Section 2. They can be thought of as generalizations of the data induced by Minkowski space on a hyperboloid.

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In the case of Minkowski space the solution admits a conformal compactification where a conformal boundary, null infinity, can be added to the spacetime. It can be shown that in the case of the maximal development of hyperboloidal data a piece of null infinity can be attached to the spacetime. For small data, i. See [ 40 ] and references therein for more details.

Making contact between hyperboloidal data and asymptotically flat initial data is much more difficult and there is as yet no complete picture. An account of the results obtained up to now is given in [ 43 ]. If the relation between hyperboloidal and asymptotically flat initial data could be understood it would give a very different approach to the problem treated by Christodoulou and Klainerman Section 4.

Most textbooks on general relativity discuss the fact that Newtonian gravitational theory is the limit of general relativity as the speed of light tends to infinity.

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It is a non-trivial task to give a precise mathematical definition of this statement. Once a definition has been given the question remains whether this definition is compatible with the Einstein equations in the sense that there are general families of solutions of the Einstein equations which have a Newtonian limit in the sense of the chosen definition. A theorem of this kind was proved in [ 68 ], where the matter content of spacetime was assumed to be a collisionless gas described by the Vlasov equation.

For another suggestion as to how this problem could be approached see [ 44 ]. That there is an abundant supply of families of this kind is the result of [ 68 ]. Note however that for k too large serious restrictions arise [ 67 ]. The latter fact corresponds to the well-known divergent behaviour of higher order post-Newtonian approximations. Apart from the interest of the Newtonian limit, Newtonian gravitational theory itself may provide interesting lessons for general relativity. This is no less true for existence theorems than for other issues.

In this context it is also interesting to consider a slight generalization of Newtonian theory, the Newton-Cartan theory.

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This allows a nice treatment of cosmological models, which are in conflict with the sometimes implicit assumption in Newtonian gravitational theory that only isolated systems are considered. It is also unproblematic to introduce a cosmological constant into the Newton-Cartan theory. Three global existence theorems have been proved in Newtonian cosmology. The first[ 9 ] is an analogue of the cosmic no hair theorem cf. Section 4. It asserts that homogeneous and isotropic models are nonlinearly stable if the matter is described by dust or a polytropic fluid with pressure.

Thus it gives information about global existence and asymptotic behaviour for models arising from small but finite perturbations of homogeneous and isotropic data. The second and third results concern collisionless matter and the case of vanishing cosmological constant. The second[ 65 ] says that data which constitute a periodic but not necessarily small perturbation of a homogeneous and isotropic model which expands indefinitely give rise to solutions which exist globally in the future.

In the standard Cauchy problem, which has been the basic set-up for all the previous sections, initial data are given on a spacelike hypersurface.