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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
##### Keywords:
exponential growth; Sobolev norm
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##### References:
 [1] Bardos, C.; Tartar, L., Sur l’unicité rétrograde des équations parabolique et quelques questiones voisines, Arch. rational mech., 50, 10-25, (1973) · Zbl 0258.35039 [2] Constantin, P.; Foias, C., Navier – stokes equations, Chicago lectures in math., (1988), University of Chicago Press Chicago · Zbl 0687.35071 [3] Constantin, P.; Foias, C.; Kukavica, I.; Majda, A., Dirichlet quotients and 2-d periodic navier – stokes equations, J. math. pures appl., 76, 125-153, (1997) · Zbl 0874.35085 [4] Dascaliuc, R., On the backward-time behavior of Burgers’ original model for turbulence, Nonlinearity, 16, 6, 1945-1965, (2003) · Zbl 1120.76316 [5] C. Foias, B. Nicolaenko, Some estimates on the nonlinear term of the Navier-Stokes equation, Preprint, 2003 [6] Kukavica, I., On the behavior of the solutions of the kuramoto – sivashinsky equations for negative time, J. math. anal. appl., 166, 601-606, (1992) · Zbl 0788.35118 [7] I. Kukavica, M. Malcok, Backward behavior of solutions of Kuramoto-Sivashinsky equation, 2003, submitted for publication · Zbl 1080.35121 [8] Richards, I., On the gaps between numbers which are sums of two squares, Adv. in math., 46, 1-2, (1982) · Zbl 0501.10047 [9] Temam, R., Navier – stokes equations and nonlinear functional analysis, (1983), SIAM Philadelphia · Zbl 0522.35002 [10] Vukadinovic, J., On the backwards behavior of the solutions of the 2D periodic viscous kamassa – holm equations, J. dynam. differential equations, 14, 2, (2002) · Zbl 1007.35076 [11] J. Vukadinovic, On the backwards behavior of the solutions of the Kelvin-filtered 2D periodic Navier-Stokes equations, Ph.D. Thesis, Indiana University, 2002 · Zbl 1007.35076
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