Conditions for asymptotic energy and enstrophy concentration in solutions to the Navier–Stokes equations

2009

Ann. Univ. Ferrara

Self Cite

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||.

Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 89%

“…Proof In [7] this lemma was proved for the case of uniformly regular domains of the class C 3 . For the whole space R 3 the proof can be repeated exactly in the same way.…”

Section: Lemma 41 Let W Be a Nonzero Global Strong Solution Of (21)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It is due to the fact that for R 3 the inequality (3.3) holds for every δ ∈ (0, 1/2] in contrast to [7], where δ ∈ [1/4, 1/2].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2009

Ann. Univ. Ferrara

Self Cite

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

2011

Z Angew Math Mech

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.

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

2011

Z Angew Math Mech

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.