  We study the velocity gradients of the fundamental Eulerian equation,
$\partial_t u +u\cdot \nabla u=F$, which shows up in different contexts
dictated by the different modeling of $F$'s. To this end we utilize a basic
description for the spectral dynamics of $\nabla u$, expressed in terms of the
(possibly complex) eigenvalues, $\lambda=\lambda(\nabla u)$, which are shown to
be governed by the Ricatti-like equation $\lambda_t+u\cdot
\nabla\lambda+\lambda^2= < l, \nabla F r>$.
  We address the question of the time regularity of four prototype models
associated with different forcing $F$. Using the spectral dynamics as our
essential tool in these investigations, we obtain a simple form of a critical
threshold for the linear damping model and we identify the 2D vanishing
viscosity limit for the viscous irrotational dusty medium model. Moreover, for
the $n$-dimensional restricted Euler equations we obtain $[n/2]+1$ global
invariants, interesting for their own sake, which enable us to precisely
characterize the local topology at breakdown time, extending previous studies
in the $n=3$-dimensional case. Finally, as a forth model we introduce the
$n$-dimensional restricted Euler-Poisson (REP)system, identifying a set of
$[n/2]$ global invariants, which in turn yield (i) sufficient conditions for
finite time breakdown, and (ii) characterization of a large class of
2-dimensional initial configurations leading to global smooth solutions.
Consequently, the 2D restricted Euler-Poisson equations are shown to admit a
critical threshold.
