Spectral Dynamics of the Velocity Gradient Field in Restricted Flows

الأصل · EN

We study the velocity gradients of the fundamental Eulerian equation, ∂ₜ u +u· ∇ 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 ∇ u, expressed in terms of the (possibly complex) eigenvalues, λ=λ(∇ u), which are shown to be governed by the Ricatti-like equation λₜ+u· ∇λ+λ²= < l, ∇ 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.