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References
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.
V. I. Arnold, Small denominators. I. Mappings of the circumference onto itself, Am. Math. Soc. Transl. Ser. 2 46 (1965), p. 213-284.
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer, 1997.
A. Babin, A. Mahalov, and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, Eur. J. Mech. B 15 (1996), 291-300.
A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity and integrability of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, Asymptotic Anal. 15 (1997), 103–150.
A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier–Stokes equations for resonant domains, Indiana Univ. Math. J. 48 (1999), no. 3, 1133-1176.
A. Babin, A. Mahalov, and B. Nicolaenko, 3D Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J. 50 (2001), 1-35.
J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys. 94 (1984), 61-66.
A. S. Besicovitch, Almost Periodic Functions, Dover, New York, 1954.
N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach Sci. Publ., New York, 1961.
J. P. Bourguignon and H. Brezis, Remark on the Euler equations, J. Func. Anal. 15 (1974), 341-363.
Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, Intermittency in the joint cascade of energy and helicity, Phys. Rev. Letters 90 (2003), p. 214503.
C. Corduneanu, Almost Periodic Functions, Wiley-Interscience, New York, 1968.
J. Deng, T. Y. Hou, and X. Yu, Geometric properties and nonblowup of 3D incompressible Euler flow, Commun. Partial Differ. Equations 30 (2005), no. 3, 225-243.
R. J. DiPerna and P. L. Lions, Ordinary differential equations, Sobolev spaces and transport theory, Invent. Math. 98 (1989), 511-547.
C. L. Fefferman, Existence and smoothness of the Navier–Stokes equations, In: The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA (2006), pp. 57-67.
U. Frisch, Turbulence: the Legacy of A. N. Kolmogolov, Cambridge University Press, Cambridge, 1995.
E. Frolova, A. Mahalov, and B. Nicolaenko, Restricted interactions and global regularity of 3D rapidly rotating Navier–Stokes equations in cylindrical domains, J. Math. Sci., New York. [To appear]
E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl. 18 (1973), 216-217.
E. B. Gledzer, F. V. Dolzhanskij, and A. M. Obukhov, Systems of Hydrodynamic Type and Their Application [in Russian], Nauka, Moscow, 1981.
F. Golse, A. Mahalov, and B. Nicolaenko, Infinite dimensional systems of coupled rigid bodies asymptotic to the 3D Euler equations. [In preparation]
J. Guckenheimer and A. Mahalov, Resonant triad interaction in symmetric systems, Physica D 54 (1992), 267-310.
T. Y. Hou and R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3D incompressible Euler equations, J. Nonlinear Sci. 16 (2006), 639–664.
T. Kato, Nonstationary flows of viscous and ideal fluids inR3, J. Func. Anal. 9 (1972), 296-305.
R. M. Kerr, Evidence for a singularity of the three dimensional, incompressible Euler equations, Phys. Fluids 5 (1993), no. 7, 1725-1746.
M. Lesieur, Turbulence in Fluids, 2nd edition, Kluwer, Dortrecht, 1990.
P. L. Lions, Mathematical Topics in Fluid Mechanics: Incompressible Models, Vol 1, Oxford University Press, Oxford, 1998.
A. Mahalov, The instability of rotating fluid columns subjected to a weak external Coriolis force, Phys. Fluids A 5 (1993), no. 4, 891-900.
A. Mahalov, B. Nicolaenko, C. Bardos, and F. Golse, Non blow-up of the 3D Euler equations for a class of three-dimensional initial data in cylindrical domains, Methods Appl. Anal. 11 (2004), no. 4, 605-634.
S. V. Manakov, Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976), no. 4, 328-329.
J. J. Moreau, Une methode de cinematique fonctionelle en hydrodynamicque [in French], C. R. Acad. Sci. Paris 249 (1959), 2156-2158
J. J. Moreau, Constantes d’un ilôt tourbillonaire en fluide parfait barotrope [in French], C. R. Acad. Sci. Paris 252 (1961), 2810-2812
H. K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 106 (1969), 117-129.
H. Poincaré, Sur la précession des corps déformables [in French], Bull. Astronomique 27 (1910), 321-356.
S. L. Sobolev, On one new problem in mathematical physics [in Russian], Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), no. 1, 3–50.
S. M. Visik, On invariant characteristics of quadratically nonlinear systems of cascade type, Sov. Math. Dokl. 17 (1976), 895-899.
J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interactions of Waves in Plasmas, Pergamon, Oxford, 1977.
V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, U.S.S.R. Comput. Math. Math. Phys. 3 (1963), 1407-1456.
V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett. 2 (1995), 27-38.
V. E. Zakharov and S. V. Manakov, Resonant interactions of wave packets in nonlinear media, Sov. Phys. JETP Lett. 18 (1973), 243-245.
V. E. Zakharov and S. V. Manakov, The theory of resonance interaction of wave packets in nonlinear media, Sov. Phys. JETP 42 (1976), 842-850.
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Golse, F., Mahalov, A., Nicolaenko, B. (2008). Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains. In: Bardos, C., Fursikov, A. (eds) Instability in Models Connected with Fluid Flows I. International Mathematical Series, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75217-4_7
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