2.4 New techniques for rough solutions

Recently, new mathematical techniques have been developed to lower the threshold of differentiability required to obtain local existence for quasilinear wave equations in general and the Einstein equations in particular. Some aspects of this development will now be discussed following [154, 157]. A central aspect is that of Strichartz inequalities. These allow one to go beyond the theory based on spaces and use Sobolev spaces based on the Lebesgue spaces for . The classical approach to deriving Strichartz estimates is based on the Fourier transform and applies to flat space. The new ideas allow the use of the Fourier transform to be limited to that of Littlewood-Paley theory and facilitate generalizations to curved space.

The idea of Littlewood-Paley theory is as follows (see [1] for a good exposition of this). Suppose that we want to describe the regularity of a function (or, more generally, a tempered distribution) u on . Differentiability properties of u correspond, roughly speaking, to fall-off properties of its Fourier transform . This is because the Fourier transform converts differentiation into multiplication. The Fourier transform is decomposed as , where is a dyadic partition of unity. The statement that it is dyadic means that all the except one are obtained from each other by scaling the argument by a factor which is a power of two. Transforming back we get the decomposition , where is the inverse Fourier transform of . The component of u contains only frequencies of the order . In studying rough solutions of the Einstein equations, the Littlewood-Paley decomposition is applied to the metric itself. The high frequencies are discarded to obtain a smoothed metric which plays an important role in the arguments.

Another important element of the proofs is to rescale the solution by a factor depending on the cut-off applied in the Littlewood-Paley decomposition. Proving the desired estimates then comes down to proving the existence of the rescaled solutions on a time interval depending on in a particular way. The rescaled data are small in some sense and so a connection is established to the question of long-time existence of solutions of the Einstein equation for small initial data. In this way, techniques from the work of Christodoulou and Klainerman on the stability of Minkowski space (see Section  5.2) are brought in.

What is finally proved? In general, there is a close connection between proving local existence for data in a certain space and showing that the time of existence of smooth solutions depends only on the norm of the data in the given space. Klainerman and Rodnianski [157] demonstrate that the time of existence of solutions of the reduced Einstein equations in harmonic coordinates depends only on the norm of the initial data for any . The reason that this does not allow them to assert an existence result in the same space is that the constraints are needed in their proof and that an understanding of solving the constraints at this low level of differentiability is lacking.

The techniques discussed in this section, which have been stimulated by the desire to understand the Einstein equations, are also helpful in understanding other nonlinear wave equations. Thus, this is an example where information can flow from general relativity to the theory of partial differential equations.

 Theorems on Existence and Global Dynamics for the Einstein Equations Alan D. Rendall http://www.livingreviews.org/lrr-2002-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de