Title: Critical thresholds in Euler-Poisson equations

Authors: Shlomo Engelberg, Hailiang Liu and Eitan Tadmor

Issue: Volume 50 (2001), Issue 1, 109-158

Abstract: 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 threshold.\par We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems 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 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.\par A case in point is the simple one-dimensional problem where the unforced inviscid Burgers solution always forms a shock discontinuity, except for the non-generic case of increasing initial profile, $u_0' \geq 0$. 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 $u_0' \geq -\sqrt{2k\rho_0}$ -- see (1.1) below, allowing a finite, critical negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena.