Critical thresholds in Euler-Poisson equations

. We present a preliminary study of a new phenomena associated with the Euler-Poisson equations | the so called critical threshold phenomena, where the answer to questions of global smoothness vs. nite time breakdown depends on whether the initial conguration crosses an intrinsic, O(1) critical threshold. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problem with or without various forcing mechanisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience nite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always f...