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2.6 Free boundary problems

In applying general relativity one would like to have solutions of the Einstein-matter equations modelling material bodies. As will be discussed in Section 3.1 there are solutions available for describing equilibrium situations. However, dynamical situations require solving a free boundary problem if the body is to be made of fluid or an elastic solid. We will now discuss the few results which are known on this subject. For a spherically symmetric self-gravitating fluid body in general relativity, a local-in-time existence theorem was proved in [206]. This concerned the case in which the density of the fluid at the boundary is non-zero. In [285] a local existence theorem was proved for certain equations of state with vanishing boundary density. These solutions need not have any symmetry but they are very special in other ways. In particular, they do not include small perturbations of the stationary solutions discussed in Section 3.1. There is no general result on this problem up to now.

Remarkably, the free boundary problem for a fluid body is also poorly understood in classical physics. There is a result for a viscous fluid [319], but in the case of a perfect fluid the problem was wide open until recently. A major step forward was taken by Wu [352], who obtained a result for a fluid that 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 that grow with unbounded exponential rates (cf. the discussion in [41]). 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. For the generalization of the problem considered by Wu to the case of a fluid with rotation, Christodoulou and Lindblad [109Jump To The Next Citation Point] have obtained estimates that look as if they should be enough to obtain an existence theorem. Strangely, it proved very difficult to complete the argument. This point deserves some further comment. In many problems the heart of an existence proof is obtaining suitable estimates. Then more or less standard approximation techniques can be used to obtain the desired conclusion (for a discussion of this see [148], Section 3.1). In the problem studied in [109] it is an appropriate approximation method that is missing. More recently Lindblad was able to obtain an existence result using a different approach involving the Nash-Moser theorem and a detailed analysis of the linearized system about a given solution. He treated the incompressible case in [231] while in the case of a compressible fluid with non-vanishing boundary density the linearized analysis has been carried out [230] .

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 [145] that it is possible to treat the Cauchy problem for fluids in general relativity in Lagrangian coordinates.

In the case of a fluid with non-vanishing boundary density it is not only the evolution equations that cause problems. It is already difficult to construct suitable solutions of the constraints. A theorem on this has recently been obtained by Dain and Nagy [128]. There remains an undesirable technical restriction, but the theorem nevertheless provides a very general class of physically interesting initial data for a self-gravitating fluid body in general relativity.

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