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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 [209214Jump To The Next Citation Point]. A central aspect is that of Strichartz inequalities. These allow one to go beyond the theory based on L2 spaces and use Sobolev spaces based on the Lebesgue Lp spaces for p /= 2. 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 Rn. Differentiability properties of u correspond, roughly speaking, to fall-off properties of its Fourier transform ^u. This is because the Fourier transform converts differentiation into multiplication. The Fourier transform is decomposed as sum ^u = fi^u, where fi is a dyadic partition of unity. The statement that it is dyadic means that all the fi 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 sum u = ui, where ui is the inverse Fourier transform of fi^u. The component ui of u contains only frequencies of the order i 2. 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 c 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 c 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 equations 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 [214] demonstrate that the time of existence of solutions of the reduced Einstein equations in harmonic coordinates depends only on the H2+e norm of the initial data for any e > 0. Combining this with the results of [243] gives an existence result in the same space. It is of interest to try to push the existence theorem to the limiting case e = 0 or even to the slightly weaker assumption on the data that the curvature is square integrable. This 2 L curvature conjecture (local existence in this setting) is extremely difficult but interesting progress has been made by Klainerman and Rodnianski in [215] where the structure of null hypersurfaces was analysed under very weak hypotheses. For this purpose the authors developed an invariant form of Littlewood-Paley theory [216]. This uses the asymptotics of solutions of the heat equation on a manifold and is coordinate-independent.

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.

It may be that the technique of using parabolic equations as a tool to better understand hyperbolic equations can be carried much further. In [333] Tao presents ideas how the harmonic map heat flow could be used to define a high quality gauge for the study of wave maps.

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