Abstract
We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the ℓ 1-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping from a global-in-time unique solution to the limit equation.
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References
Babin A., Mahalov A., Nicolaenko B.: Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids. Asymptot. Anal. 15, 103–150 (1997)
Babin A., Mahalov A., Nicolaenko B.: Global regularity of the 3D Rotating Navier–Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999)
Chemin, J.Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical geophysics. An introduction to rotating fluids and the Navier–Stokes equations. Oxford Lecture Series in Mathematics and its Applications, vol. 32. The Clarendon Press, Oxford University Press, Oxford, 2006
Fuentes O.U.V.: Kelvin’s discovery of Taylor columns. Eur. J. Mech. B/Fluids 28, 469–472 (2009)
Giga Y., Inui K., Mahalov A., Matsui S.: Uniform local solvability for the Navier–Stokes equations with the Coriolis force. Methods Appl. Anal. 12, 381–393 (2005)
Giga Y., Inui K., Mahalov A., Saal J.: Global solvability of the Navier–Stokes equations in spaces based on sum-closed frequency sets. Adv. Diff. Eq. 12, 721–736 (2007)
Giga Y., Inui K., Mahalov A., Saal J.: Uniform global solvability of the rotating Navier–Stokes equations for nondecaying initial data. Indiana Univ. Math. J. 57, 2775–2791 (2008)
Giga Y., Jo H., Mahalov A., Yoneda T.: On time analyticity of the Navier–Stokes equations in a rotating frame with spatially almost periodic data. Phys. D 237, 1422–1428 (2008)
Giga Y., Mahalov A., Nicolaenko B. et al.: The Cauchy problem for the Navier–Stokes equations with spatially almost periodic initial data. In: Bourgain, J. (eds) Mathematical Aspects of Nonlinear Dispersive Equations 163, pp. 213–222. Princeton Press, New Jersey (2007)
Giga, Y., Mahalov, A., Yoneda, T.: On a bound for amplitudes of Navier–Stokes flow with almost periodic initial data. J. Math. Fluid Mech. (2010) (to appear)
Hough S.S.: On the application of harmonic analysis to the dynamical theory of the tides. Part I. On Laplaces oscillations of the first species, Eand on the dynamics of ocean currents. Phil. Trans. R. Soc. Lond. A 189, 201–257 (1897)
Koba, H., Mahalov, A., Yoneda, T.: Global Well-posedness for the rotating Navier–Stokes–Boussinesq equations with stratification effects (preprint)
Poincaré H.: Sur la précession des corps déformables. Bull. Astronomique 27, 321 (1910)
Proudman J.: On the motion of solids in a liquid possessing vorticity. Proc. R. Soc. Lond. 92, 408–424 (1916)
Taylor G.I.: Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. 93, 92–113 (1917)
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Communicated by V. Šverák
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Yoneda, T. Long-time Solvability of the Navier–Stokes Equations in a Rotating Frame with Spatially Almost Periodic Large Data. Arch Rational Mech Anal 200, 225–237 (2011). https://doi.org/10.1007/s00205-010-0360-4
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DOI: https://doi.org/10.1007/s00205-010-0360-4