Solutions to the Navier‐Stokes equations with the large time energy concentration in the low frequencies

Abstract

In this paper we study the large time behavior of the solutions to the Navier‐Stokes equations. We focus on the solutions exhibiting the large time energy concentration in the low frequencies and describe some classes of the initial conditions from Lσ2 such that if u0 belongs to such a class and u is a global weak solution with u(0) = u0 satisfying the strong energy inequality, then \begin{eqnarray*}1 ‐ \frac {||E_\lambda u(t)||} {||u(t)||} \rightarrow 0, \text{ if } t \rightarrow \infty \end{eqnarray*} for every λ > 0, where {Eλ λ > 0} is the resolution of the identity of the Stokes operator and ||·|| denotes the standard L2‐norm. We present some estimates of this concentration rate.

The Asymptotic Properties of Turbulent Solutions to the Navier-Stokes Equations

“…The diameter of the annulus determines the rate of the exponential decay of the solution, and it can be arbitrarily narrow. The ball is centered in the origin of the coordinates, and can have an arbitrarily small diameter (for the results presented in this section, see [15], [16], [17] and [18]). The following theorem and the ensuing remarks provide precise information.…”

“…In Theorem 4, we will present an example of a concrete class of initial conditions such that if u 0 belongs to this class and u is a turbulent solution with u(0) = u 0 , then the energy of the solution concentrates asymptotically in frequencies from an arbitrarily small ball in the frequency space centered in the origin of the coordinates. We also present two estimates of the rate of energy concentration (see [18]).…”

On large‐time energy concentration in solutions to the Navier‐Stokes equations in general domains

“…In [16] we studied in detail some other classes of initial conditions yielding solutions with large-time energy concentration in small frequencies and proved also some estimates of this concentration rate. We present here one of the results from [16]:…”