Go to previous page Go up Go to next page

2.3 Questions of differentiability

Solving the Cauchy problem for a system of partial differential equations involves specifying a set of initial data to be considered, and determining the differentiability properties of solutions. Thus, two regularity properties are involved - the differentiability of the allowed data, and that of the corresponding solutions. Normally, it is stated that for all data with a given regularity, solutions with a certain type of regularity are obtained. For instance, in Section 2.2 we chose both types of regularity to be “infinitely differentiable”. The correspondence between the regularity of data and that of solutions is not a matter of free choice. It is determined by the equations themselves, and in general the possibilities are severely limited. A similar issue arises in the context of the Einstein constraints, where there is a correspondence between the regularity of free data and full data.

The kinds of regularity properties that can be dealt with in the Cauchy problem depend, of course, on the mathematical techniques available. When solving the Cauchy problem for the Einstein equations, it is necessary to deal at least with nonlinear systems of hyperbolic equations. (There may be other types of equations involved, but they will be ignored here.) For general nonlinear systems of hyperbolic equations the standard technique is the method of energy estimates. This method is closely connected with Sobolev spaces, which will now be discussed briefly.

Let u be a real-valued function on Rn. Let

( sum s integral )1/2 ||u || = |Diu |2(x)dx . (2) s i=0
The space of functions for which this quantity is finite is the Sobolev space Hs(Rn). Here, |Diu |2 denotes the sum of the squares of all partial derivatives of u of order i. Thus, the Sobolev space Hs is the space of functions, all of whose partial derivatives up to order s are square integrable. Similar spaces can be defined for vector valued functions by taking a sum of contributions from the separate components in the integral. It is also possible to define Sobolev spaces on any Riemannian manifold, using covariant derivatives. General information on this can be found in [30]. Consider now a solution u of the wave equation in Minkowski space. Let u(t) be the restriction of this function to a time slice. Then it is easy to compute that, provided u is smooth and u(t) has compact support for each t, the quantity ||Du(t)|| 2s + ||@tu(t)||2s is time independent for each s. For s = 0 this is just the energy of a solution of the wave equation. For a general nonlinear hyperbolic system, the Sobolev norms are no longer time-independent. The constancy in time is replaced by certain inequalities. Due to the similarity to the energy for the wave equation, these are called energy estimates. They constitute the foundation of the theory of hyperbolic equations. It is because of these estimates that Sobolev spaces are natural spaces of initial data in the Cauchy problem for hyperbolic equations. The energy estimates ensure that a solution evolving from data belonging to a given Sobolev space on one spacelike hypersurface will induce data belonging to the same Sobolev space on later spacelike hypersurfaces. In other words, the property of belonging to a Sobolev space is propagated by the equations. Due to the locality properties of hyperbolic equations (existence of a finite domain of dependence), it is useful to introduce the spaces Hs loc, which are defined by the condition that whenever the domain of integration is restricted to a compact set, the integral defining the space s H is finite.

In the end, the solution of the Cauchy problem should be a function that is differentiable enough so that all derivatives that occur in the equation exist in the usual (pointwise) sense. A square integrable function is in general defined only almost everywhere and the derivatives in the above formula must be interpreted as distributional derivatives. For this reason, a connection between Sobolev spaces and functions whose derivatives exist pointwise is required. This is provided by the Sobolev embedding theorem. This says that if a function u on n R belongs to the Sobolev space s H loc and if k < s- n/2, then there is a k times continuously differentiable function that agrees with u except on a set of measure zero.

In the existence and uniqueness theorems stated in Section 2.2, the assumptions on the initial data for the vacuum Einstein equations can be weakened to say that hab should belong to s H loc and kab to Hsl-oc1. Then, provided s is large enough, a solution is obtained that belongs to Hsloc. In fact, its restriction to any spacelike hypersurface also belongs to Hs loc, a property that is a priori stronger. The details of how large s must be would be out of place here, since they involve examining the detailed structure of the energy estimates. However, there is a simple rule for computing the required value of s. The value of s needed to obtain an existence theorem for the Einstein equations using energy estimates is that for which the Sobolev embedding theorem, applied to spatial slices, just ensures that the metric is continuously differentiable. Thus the requirement is that s > n/2 + 1 = 5/2, since n = 3. It follows that the smallest possible integer s is three. Strangely enough, the standard methods only give uniqueness up to diffeomorphisms for s > 4. The reason is that in proving the uniqueness theorem a diffeomorphism must be carried out, which need not be smooth. This apparently leads to a loss of one derivative. In [10Jump To The Next Citation Point] local existence and uniqueness for the vacuum Einstein equations was proved using a gauge condition defined by elliptic equations for which this loss does not occur. In that case the gap of one derivative is eliminated. On the other hand, the occurrence of elliptic equations as part of the reduced Einstein equations with this gauge makes the result intrinsically global, and it is not clear whether it can be localized in space. Another interesting aspect of the main theorem of [10] is that it includes a continuation criterion for solutions. There exists a definition of Sobolev spaces for an arbitrary real number s, and hyperbolic equations can also be solved in the spaces with s not an integer [334]. Presumably these techniques could be applied to prove local existence for the Einstein equations with s any real number greater than 5/2. In any case, the condition for local existence has been weakened to s > 2 using other techniques, as discussed in Section 2.4.

Consider now C oo initial data. Corresponding to these data there is a development of class Hs for each s. It could conceivably be the case that the size of these developments shrinks with increasing s. In that case, their intersection might contain no open neighbourhood of the initial hypersurface, and no smooth development would be obtained. Fortunately, it is known that the Hs developments cannot shrink with increasing s [87], and so the existence of a Co o solution is obtained for Co o data. It appears that the s H spaces with s sufficiently large are the only spaces containing the space of smooth functions for which it has been proved that the Einstein equations are locally solvable.

What is the motivation for considering regularity conditions other than the apparently very natural o o C condition? One motivation concerns matter fields and will be discussed in Section 2.5. Another is the idea that assuming the existence of many derivatives that have no direct physical significance seems like an admission that the problem has not been fully understood. A further reason for considering low regularity solutions is connected to the possibility of extending a local existence result to a global one. If the proof of a local existence theorem is examined closely it is generally possible to give a continuation criterion. This is a statement that if a solution on a finite time interval is such that a certain quantity constructed from the solution is bounded on that interval, then the solution can be extended to a longer time interval. (In applying this to the Einstein equations we need to worry about introducing an appropriate time coordinate.) If it can be shown that the relevant quantity is bounded on any finite time interval where a solution exists, then global existence follows. It suffices to consider the maximal interval on which a solution is defined, and obtain a contradiction if that interval is finite. This description is a little vague, but contains the essence of a type of argument that is often used in global existence proofs. The problem in putting it into practice is that often the quantity whose boundedness has to be checked contains many derivatives, and is therefore difficult to control. If the continuation criterion can be improved by reducing the number of derivatives required, then this can be a significant step toward a global result. Reducing the number of derivatives in the continuation criterion is closely related to reducing the number of derivatives of the data required for a local existence proof.

A striking example is provided by the work of Klainerman and Machedon [210Jump To The Next Citation Point] on the Yang-Mills equations in Minkowski space. Global existence in this case was first proved by Eardley and Moncrief [135Jump To The Next Citation Point], assuming initial data of sufficiently high differentiability. Klainerman and Machedon gave a new proof of this, which, though technically complicated, is based on a conceptually simple idea. They prove a local existence theorem for data of finite energy. Since energy is conserved this immediately proves global existence. In this case finite energy corresponds to the Sobolev space H1 for the gauge potential. Of course, a result of this kind cannot be expected for the Einstein equations, since spacetime singularities do sometimes develop from regular initial data. However, some weaker analogue of the result could exist.

  Go to previous page Go up Go to next page