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On backward-time behavior of the solutions to the 2-D space periodic Navier-Stokes equations. (English) Zbl 1076.76021
Summary: P. Constantin, C. Foias, I. Kukavica and A. J. Majda [J. Math. Pures Appl., IX. Sér. 76, No. 2, 125–153 (1997; Zbl 0874.35085)] had shown that the 2-D space periodic Navier-Stokes equations have a rich set of the solutions that exist for all times \(t\in \mathbb R\) and grow exponentially in Sobolev \(H^1\) norm when \(t\to -\infty\). In the present note we show that these solutions grow exponentially (when \(t\to-\infty\)) in any Sobolev \(H^m\) norm \((m\geqslant 2)\) provided the driving force is bounded in \(H^{m-1}\) norm.

MSC:
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
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