Critical Thresholds and Conditional Stability for Euler Equations and Related Models

Abstract

We are concerned with time regularity of the velocity field for a class of so-called restricted flows, where the velocity field, u, is governed by the Newtonian law,

$$ {{\partial }_{t}}u + u \cdot \nabla u = F,\quad x \in I{{R}^{n}}, $$

((1))

which shows up in different contexts dictated by the different modeling of F’s, where F stands for a general forcing acting on the flow. The examples range from Euler/Navier-Stokes equations to Euler-Poisson equations. There has been an enormous amount of papers related to the study of global behavior of solutions to (1) with possible associated laws (say, conservation of mass, energy etc). For the question of global behavior of strong solutions the choice of the initial data and/or damping forces is decisive. The classical methods of analysis include energy method (for small initial perturbations due to the nonlinearity) and the singularity prediction ( the finite life span is often due to a global condition of large enough initial (generalized) energy, staying outside a critical threshold ball).