]. Some of the results of [228] were recently generalized to the case of a relativistic
fluid [124]. The proofs of the development of singularities are by
contradiction and so do not give information about what happens
when the smooth solution breaks down. One of the things that can
happen is the formation of shock waves and it is known that, at
least in certain cases, solutions can be extended in a physically
meaningful way beyond the time of shock formation. The extended
solutions only satisfy the equations in the weak sense. For the
classical Euler equations there is a well-known theorem on global
existence of weak solutions in one space dimension which goes
back to [114]. This has been generalized to the relativistic case. Smoller
and Temple treated the case of an isentropic fluid with linear
equation of state [229] while Chen analysed the cases of polytropic equations of
state [59] and flows with variable entropy [60]. This means that there is now an understanding of this question
in the relativistic case similar to that available in the
classical case.
In space dimensions higher than one there are no general global existence theorems. For a long time there were also no uniqueness theorems for weak solutions even in one dimension. It should be emphasized that weak solutions can easily be shown to be non-unique unless they are required to satisfy additional restrictions such as entropy conditions. A reasonable aim is to find a class of weak solutions in which both existence and uniqueness hold. In the one-dimensional case this has recently been achieved by Bressan and collaborators (see [47, 49, 48] and references therein).
It would be desirable to know more about which quantities must blow up when a singularity forms in higher dimensions. A partial answer was obtained for classical hydrodynamics by Chemin [58]. The possibility of generalizing this to relativistic and self-gravitating fluids was studied by Brauer [45]. There is one situation in which a smooth solution of the classical Euler equations is known to exist for all time. This is when the initial data are small and the fluid initially is flowing uniformly outwards. A theorem of this type has been proved by Grassin [118]. There is also a global existence result due to Guo [121] for an irrotational charged fluid in Newtonian physics, where the repulsive effect of the charge can suppress the formation of singularities.
A question of great practical interest for physics is that of the stability of equilibrium stellar models. Since, as has already been pointed out, we know so little about the global time evolution for a self-gravitating fluid ball, even in the Newtonian case, it is not possible to say anything rigorous about nonlinear stability at the present time. We can, however, make some statements about linear stability. The linear stability of a large class of static spherically symmetric solutions of the Einstein-Euler equations within the class of spherically symmetric perturbations has been proved by Makino [172] (cf. also [165] for the Newtonian problem). The spectral properties of the linearized operator for general (i.e. non-spherically symmetric) perturbations in the Newtonian problem have been studied by Beyer [36]. This could perhaps provide a basis for a stability analysis, but this has not been done.
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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 |