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The local wellposedness (after imposing gauge conditions, for reasons already described in Rafe Mazzeo and Peter Michor's answers) of Einstein's equations is a result more than 60 years old! The original proof was published by Yvonne Choquet-Bruhat in 1952 (before she took her current last name), and yes, more modern presentations usually use heavily the notion of Sobolev spaces on manifolds.

In 1969 Choquet-Bruhat and Robert Geroch showed that, essentially due to the hyperbolic nature of the equations, one can define the maximal manifold $M$ into which all solutions embed. With this notion one can then formulate the question of "global Cauchy problem" as a question of studying the geometric properties of this maximal solution.

The local wellposedness (after imposing gauge conditions, for reasons already described in Rafe Mazzeo and Peter Michor's answers) of Einstein's equations is a result more than 60 years old! The original proof was published by Yvonne Choquet-Bruhat in 1952 (before she took her current last name), and yes, more modern presentations usually use heavily the notion of Sobolev spaces on manifolds.

In 1969 Choquet-Bruhat and Robert Geroch showed that, essentially due to the hyperbolic nature of the equations, one can define the maximal manifold $M$ into which all solutions embed. With this notion one can then formulate the question of "global Cauchy problem" as a question of studying the geometric properties of this maximal solution.

The local wellposedness (after imposing gauge conditions, for reasons already described in Rafe Mazzeo and Peter Michor's answers) of Einstein's equations is a result more than 60 years old! The original proof was published by Yvonne Choquet-Bruhat in 1952 (before she took her current last name). And yes, that proof already heavily used the notion of Sobolev spaces on manifolds.

The local wellposedness (after imposing gauge conditions, for reasons already described in Rafe Mazzeo and Peter Michor's answers) of Einstein's equations is a result more than 60 years old! The original proof was published by Yvonne Choquet-Bruhat in 1952 (before she took her current last name), and yes, more modern presentations usually use heavily the notion of Sobolev spaces on manifolds.

Some further reading:

Coming back to the local well-posedness problem in 3+1 dimensions: in the regime of "classical" solutions, existence, uniqueness, and regularity follows from a very similar analysis to that of Hughes, Kato, Marsden. To close the argument one needs to work in the Sobolev space $H^{5/2+}$ (or classically $H^3$ if you don't want to deal with fractional numbers of derivatives). One may ask about the optimal regularity for the local well-posedness statements. The scaling regularity should be $H^{3/2}$. However, the equation is quasilinear, and it is generally the case that compared to semilinear problems we cannot expect to go all the way down to scaling critical (see for example these two papers of Hans Lindblad 1 2). This problem motivated a body of literature studying the below-classical-regularity local existence problem for quasilinear wave equations. Between the groups of Klainerman-Rodnianski and that of Hart Smith and Daniel Tataru (esp. this paper and this other one), for general quasilinear wave equations the sharp exponent $H^{2+}$ was obtained. For the Einstein vacuum equations specifically, though, one may do better: a recently posted series of pre-prints by Klainerman, Rodnianski, and Jeremie Szeftel obtains a local existence theorem at the regularity level $H^2$ (these are arXiv items: http://arxiv.org/abs/1204.1772 http://arxiv.org/abs/1204.1767 http://arxiv.org/abs/1204.1768 http://arxiv.org/abs/1204.1769 http://arxiv.org/abs/1204.1770 http://arxiv.org/abs/1204.1771 http://arxiv.org/abs/1301.0112).

Some further reading: