Remarkably, the free boundary problem for a fluid body is also poorly understood in classical physics. There is a result for a viscous fluid  but in the case of a perfect fluid the problem was wide open until very recently. Now a major step forward has been taken by Wu , who obtained a result for a fluid which is incompressible and irrotational. There is a good physical reason why local existence for a fluid with a free boundary might fail. This is the Rayleigh-Taylor instability which involves perturbations of fluid interfaces which grow with unbounded exponential rates. (Cf. the discussion in .) It turns out that in the case considered by Wu this instability does not cause problems and there is no reason to expect that a self-gravitating compressible fluid with rotation in general relativity with a free boundary cannot also be described by a well-posed free boundary value problem.
One of the problems in tackling the initial value problem for a dynamical fluid body is that the boundary is moving. It would be very convenient to use Lagrangian coordinates, since in those coordinates the boundary is fixed. Unfortunately, it is not at all obvious that the Euler equations in Lagrangian coordinates have a well-posed initial value problem, even in the absence of a boundary. It was, however, recently shown by Friedrich  that it is possible to treat the Cauchy problem for fluids in general relativity in Lagrangian coordinates.
|Local and Global Existence Theorems for the Einstein
Alan D. Rendall
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