Abstract: 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 governed by the Ricatti-like equation $\lambda_t+u\cdot \nabla\lambda+\lambda^2= \langle l, \nabla F r\rangle$.
We focus our investigation on four prototype models associated with different forcing $F$, ranging from simple linear damping and a viscous dusty medium models to the main thrust of the paper --- the restricted models of Euler/Navier-Stokes equations and Euler-Poisson equations.
In particular, we address the question of the time regularity for these models, that is, whether they admit a finite time breakdown, a global smooth solution, or an intermediate scenario of critical threshold phenomena where global regularity depends on initial configurations.
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.
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