Breakdown of smooth solutions of the three-dimensional Euler–Poisson system
1998; American Institute of Physics; Volume: 39; Issue: 2 Linguagem: Inglês
10.1063/1.532370
ISSN1527-2427
Autores Tópico(s)Cosmology and Gravitation Theories
ResumoResults concerning the occurrence of (kinematical) singularities obtained by Majda et al. [Commun. Math. Phys. 94, 61–66 (1984)] for the incompressible Euler equations and of Chemin [Commun. Math. Phys. 133, 323–329 (1990)] for 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 solution with finite, spatially periodic, deviations of a Newtonian, Friedman-like cosmological model. Both situations are briefly reviewed. For the compact body the solutions belong to a special class, introduced by Makino [Patterns and Waves (North-Holland, Amsterdam, 1986), pp. 459–479]. In Sec. III, uniqueness is shown for these and therewith one of the severe disadvantages of these solution is eliminated. In both situations 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 between the two situations considered here is that, for technical reasons, a nonlinear function w=M(ϱ) has to be introduced as a new matter variable for the compact body. Because ϱ has compact support the blow-up of grad w in the L∞-norm implies two possibilities, one being that the singularity is in the interior of the body. In that case the blow-up of grad w implies the blow-up of grad ϱ. If, on the other hand, the singularity is near the boundary of the body, then no precise information is available.
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