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

Abstract

Let Ω ⊆ ℝ3 be a uniformly regular domain of the class C3 or Ω = ℝ3. Let A denote the Stokes operator and {Eλ; λ > 0} be the resolution of identity of A. We show as the main result of the paper that if w is a nonzero global weak solution to the Navier‐Stokes equations in Ω satisfying the strong energy inequality, then there exists a nonnegative finite number a = a(w) such that for every ε > 0 \[lim_{t \rightarrow \infty} \frac {||(E_{a+\varepsilon}‐E_{a‐\varepsilon}) w(t)||} {||w(t)||} = 1, \] where we put Ea‐ε = 0 if a‐ε < 0. Thus, every nonzero global weak solution satisfying the strong energy inequality exhibits large‐time energy concentration in a particular frequency. Moreover, the solutions with the exponentially decreasing energy are characterized by the positivity of a. In Appendix, we present some further results describing in detail the large‐time behavior of w.