Remarkably, the free boundary problem for a fluid body is also
poorly understood in classical physics. There is a result for a
viscous fluid [226], 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 [244], 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 [26]). 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 [80] have obtained estimates that look as if they should be enough
to obtain an existence theorem. It has, however, not yet been
possible 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 [108], Section 3.1). In the problem studied in [80] it is an appropriate approximation method that is missing.
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 [105] 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 [91]. 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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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 |