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On large‐time energy concentration in solutions to the Navier‐Stokes equations in general domains
Author(s) -
Skalák Z.
Publication year - 2011
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200900399
Subject(s) - mathematics , energy (signal processing) , domain (mathematical analysis) , operator (biology) , identity (music) , mathematical analysis , resolution (logic) , class (philosophy) , energy method , combinatorics , mathematical physics , pure mathematics , physics , statistics , computer science , biochemistry , chemistry , repressor , artificial intelligence , transcription factor , acoustics , gene
Let Ω ⊆ ℝ 3 be a uniformly regular domain of the class C 3 or Ω = ℝ 3 . Let A denote the Stokes operator and {E λ ; 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 E a‐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.