BREAKDOWN OF SMOOTH SOLUTIONS OF THE 3-DIMENSIONAL EULER-POISSON SYSTEM

Results concerning the occurrence of (kinematical) singularities obtai ned by Majda et al. [Commun, Math. Phys. 94, 61-66 (1984)] for the inc ompressible Euler equations and of Chemin [Commun. Math. Phys. 133, 32 3-329 (1990)] fur the compressible Euler equations are generalized for the compressible Euler-Poisson system. This generalization is applied to two situations of physical interest, namely, either the evolution of a compact body with a freely falling boundary or a cosmological sol ution with finite, spatially periodic, deviations of a Newtonian, Frie dman-like cosmological model. Both situations are briefly reviewed, Fa r the compact body the solutions belong to a special class, introduced by Makino [Patterns and Waves (North-Holland, Amsterdam, 1986), pp. 4 59-479], In Sec. III, uniqueness is shown for these and therewith one of the severe disadvantages of these solution is eliminated. In both s ituations the qualitative behavior is similar to the gravitation free case in the sense that only some of the kinematical quantities of the fluid and the gradient of the matter variable diverge; in other words, no specific ''gravitation singularity'' appears. The differences betw een the two situations considered here is that, for technical reasons, a nonlinear function w = M (rho) has to be introduced as a new matter variable for the compact body, Because rho has compact support the bl ow-up of grad w in the L-infinity-norm implies two possibilities, one being that the singularity is in the interior of the body. In that cas e the blow-up of grad w implies the blow-up of grad rho. If, on the ot her hand, the singularity is near the boundary of the body, then no pr ecise information is available. (C) 1998 American Institute of Physics . [S0022-2488(98)03401-X].