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. finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical th reshold. We investigate a class of Euler-Poisson equations, ranging from one-dimensi onal problems with or without various forcing mechanisms to multi-dimension al isotropic models with geometrical symmetry. These models are shown to ad mit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-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 in viscid Burgers' solution always forms a shock discontinuity, except for the non-generic case of increasing initial profile, u(0)(') greater than or eq ual to 0. In contrast, we show that the corresponding one-dimensional Euler -Poisson equation with zero background has global smooth solutions as long as its initial (rho (0), u(0))-configuration satisfies u(0)(') greater than or equal to -root 2k rho (0) - see (2.11) below, allowing a finite, critic al negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forc ing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena.