# Mathematics Colloquia and Seminars

We discuss the global regularity for a range of different models governed by the Euler dynamics. We first discuss 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. We investigate various one-dimensional problems with or without forcing mechanisms as well as 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. The critical threshold phenomena hinges on a delicate balance in the equations between the Euler dynamics and the {\it global} forcing governed by the Poisson equation. In this context we also study a related range of Euler dynamics, which is driven by {\it localized} forcing -- the so called Restricted Euler dynamics. We present new results on the global regularity and finite time breakdown of restricted Euler-Poisson equations, the restricted Euler/Navier-Stokes equations, and other similar models. The novelty of our approach lies in analyzing the spectral dynamics associated with the velocity gradient fields of these restricted models.