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 [68, 70, 69] 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 . The possibility of generalizing this to relativistic and self-gravitating fluids was studied by Brauer . 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 . There is also a global existence result due to Guo  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. 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  (cf. also  for the Newtonian problem). A nonlinear stability result for solutions of the Euler-Poisson system was proved in  under the assumption of global existence. The spectral properties of the linearized operator for general (i.e. non-spherically symmetric) perturbations in the Newtonian problem have been studied by Beyer . This could perhaps provide a basis for a stability analysis, but this has not been done.
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