Abstract
The incompressible Navier–Stokes equations are considered in the two-dimensional strip \({\mathbb{R} \times [0,L]}\), with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, it is shown that the solution remains uniformly bounded for all time, and that the vorticity distribution converges to zero as \({t \to \infty}\). This implies, after a transient period, the emergence of a laminar regime in which the solution rapidly converges to a shear flow described by the one-dimensional heat equation in an appropriate Galilean frame. The approach is constructive and provides explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number.
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Afendikov A., Mielke A.: Dynamical properties of spatially non-decaying 2D Navier–Stokes flows with Kolmogorov forcing in an infinite strip. J. Math. Fluid. Mech. 7(suppl. 1), S51–S67 (2005)
Ambrose, D., Kelliher, J., Lopes Filho, M., Nussenzveig Lopes, H.: Serfati solutions to the 2D Euler equations on exterior domains. (2014, preprint) arXiv:1401.2655
Anthony, P., Zelik, S.: Infinite-energy solutions for the Navier–Stokes equations in a strip revisited. Commun. Pure Appl. Anal. 13, 1361–1393 (2014)
Arrieta J., Rodriguez-Bernal A., Cholewa J., Dlotko T.: Linear parabolic equations in locally uniform spaces. Math. Models Methods Appl. Sci. 14, 253–293 (2004)
Aronson D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)
Bergh J., Löfström J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)
Constantin P., Foias C.: Navier–Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)
Coulhon Th.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141, 510–539 (1996)
Coulhon Th., Grigoryan A., Levin D.: On isoperimetric profiles of product spaces. Comm. Anal. Geom. 11, 85–120 (2003)
Fabes E.B., Stroock D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)
Gallay Th., Slijepčević S.: Energy flow in formally gradient partial differential equations on unbounded domains. J. Dyn. Differ. Equ. 13, 757–789 (2001)
Gallay, Th., Slijepčević, S.: Distribution of energy and convergence to equilibria in extended dissipative systems. J. Dyn. Differ. Equ. (2014, in press)
Gallay, Th., Slijepčević, S.: Energy bounds for the two-dimensional Navier–Stokes equations in an infinite cylinder. Commun. Partial Differ. Equ. 39, 1741–1769 (2014)
Giga Y., Matsui S., Sawada O.: Global existence of two dimensional Navier–Stokes flow with non-decaying initial velocity. J. Math. Fluid. Mech. 3, 302–315 (2001)
Grigoryan A.: Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10, 395–452 (1994)
Grigor’yan A.: Heat kernel and analysis on manifolds. AMS/IP Studies in Advanced Mathematics, vol. 47. AMS, Providence (2009)
Kato J.: The uniqueness of nondecaying solutions for the navier–stokes equations. Arch. Ration. Mech. Anal. 169, 159–175 (2003)
Maekawa Y., Terasawa Y.: The Navier–Stokes equations with initial data in uniformly local L p spaces. Diff. Int. Equ. 19, 369–400 (2006)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Sawada O., Taniuchi Y.: A remark on L ∞ solutions to the 2-D Navier–Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)
Serfati Ph.: Solutions C ∞ en temps, n-log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris Sér. I Math. 320, 555–558 (1995)
Zelik S.: Spatially nondecaying solutions of the 2D Navier–Stokes equation in a strip. Glasg. Math. J. 49, 525–588 (2007)
Zelik S.: Infinite energy solutions for damped Navier–Stokes equations in \({\mathbb{R}^{2}}\). J. Math. Fluid Mech. 15, 717–745 (2013)
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Gallay, T., Slijepčević, S. Uniform Boundedness and Long-Time Asymptotics for the Two-Dimensional Navier–Stokes Equations in an Infinite Cylinder. J. Math. Fluid Mech. 17, 23–46 (2015). https://doi.org/10.1007/s00021-014-0188-z
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DOI: https://doi.org/10.1007/s00021-014-0188-z