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Solutions to the Navier‐Stokes equations with the large time energy concentration in the low frequencies
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.201000014
Subject(s) - lambda , norm (philosophy) , energy (signal processing) , identity (music) , operator (biology) , mathematics , focus (optics) , navier–stokes equations , physics , mathematical analysis , mathematical physics , combinatorics , thermodynamics , quantum mechanics , statistics , compressibility , chemistry , optics , biochemistry , repressor , political science , acoustics , transcription factor , law , gene
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 L σ 2 such that if u 0 belongs to such a class and u is a global weak solution with u(0) = u 0 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 {E λ λ > 0} is the resolution of the identity of the Stokes operator and ||·|| denotes the standard L 2 ‐norm. We present some estimates of this concentration rate.