Asymptotic energy and enstrophy concentration in solutions to the Navier–Stokes equations in R 3

Abstract

Let A be the Stokes operator. We show as the main result of the paper that if w is a nonzero global weak solution to the Navier-Stokes equations in R 3 satisfying the strong energy inequality, then the energy of the solution w concentrates asymptotically in frequencies smaller than or equal to the finite number C(1/2) = lim sup t→∞ ||A 1/2 w(t)|| 2 /||w(t)|| 2 in the sense that lim t→∞ ||E λ w(t)||/||w(t)|| = 1 for every λ > C(1/2), where {E λ ; λ ≥ 0} is the resolution of identity of A. We also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of w defined as ||A 1/2 w||.

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

“…It then follows from (19) that there exists t 1 > T 0 dependent only on c 0 , c 1 , α, γ, and λ 0 such that ||E λ u(t)||/||u(t)|| ≥ 1/2 for every t ≥ t 1 and λ ≥ λ 0 . We can now proceed exactly in the same way as in the proof of Lemma 4.4 in [11] with only one exception: Since we know that t 1 depends only on c 0 , c 1 , α, γ, and λ 0 , we get (20), where c > 0 does not depend on u but only on c 0 , c 1 , α, γ, and λ 0 . The proof of Theorem 2.3 is complete.…”

“…It will lead (applying also the results from [9]) to a substantial extension of the class of the initial conditions described in [6] and also to the better estimate of the convergence rate in (32). In fact, we will present two different estimates of 1 − ||E λ u(t)||/||u(t)||, the first one based on the results from [10] and [11] and the second one obtained by the same method as the estimate (32).…”

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

“…The equality (9) follows immediately from the second equalities in (13) and (14). (10) is now a consequence of (15).…”

“…In this paper, we will focus on the large-time energy concentration in global solutions. This problem has already been studied in several recent papers, see [6,[11][12][13][14][15]. Suppose that w is a nonzero global weak solution of (1)-(4) satisfying the strong energy inequality (the definition of these concepts is presented further in this section).…”

“…Accordingly, the condition (5) plays an important role in the study of large-time energy concentration. Up to now it has been known to hold in several cases: [13].…”