Geodesic
Global solutions of the Navier–Stokes equations in a uniformly rotating space
Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 2, pp. 196-215
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We consider the Cauchy problem for the Navier–Stokes system of equations in a three-dimensional space rotating uniformly about the vertical axis with the periodicity condition with respect to the spatial variables. Studying this problem is based on expanding given and sought vector functions in Fourier series in terms of the eigenfunctions of the curl and Stokes operators. Using the Galerkin method, we reduce the problem to the Cauchy problem for the system of ordinary differential equations, which has a simple explicit form in the basis under consideration. Its linear part is diagonal, which allows writing explicit solutions of the linear Stokes–Sobolev system, to which fluid flows with a nonzero vorticity correspond. Based on the study of the nonlinear interaction of vortical flows, we find an approach that we can use to obtain families of explicit global solutions of the nonlinear problem.
Keywords: eigenfunction, eigenvalue, curl operator, Stokes operator, Navier–Stokes system of equations, Fourier method, Galerkin method. 
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R. S. Saks. Global solutions of the Navier–Stokes equations in a uniformly rotating space. Teoretičeskaâ i matematičeskaâ fizika, Tome 162 (2010) no. 2, pp. 196-215. http://geodesic.mathdoc.fr/item/TMF_2010_162_2_a2/

[1] V. I. Arnold, Izbrannoe-60, Fazis, M., 1997, 215–236 | MR | Zbl

[2] V. V. Kozlov, Obschaya teoriya vikhrei, RKhD, Izhevsk, 1998 | MR | Zbl

[3] V. V. Pukhnachev, Uspekhi mekhaniki, 4:1 (2006), 6–76

[4] S. Chandrasekhar, P. C. Kendall, J. Astrophys., 126 (1957), 457–460 | DOI | MR

[5] J. B. Taylor, Phys. Rev. Lett., 33:19 (1974), 1139–1141 | DOI

[6] D. Montgomery, L. Turner, G. Vahala, Phys. Fluids, 21:5 (1978), 757–764 | DOI | Zbl

[7] R. S. Saks, “Reshenie spektralnoi zadachi dlya operatora rotor i operatora Stoksa s periodicheskimi kraevymi usloviyami”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 36, Zap. nauchn. sem. POMI, 318, POMI, SPb., 2004, 246–276 | DOI | MR | Zbl

[8] R. S. Saks, Dokl. AN SSSR, 416:4 (2007), 446–450 | MR | Zbl

[9] R. S. Saks, “The solution of spectral problems for the curl and Stokes operators with periodic boundary conditions and some classes of explicit solutions of Navier–Stokes equations”, More Progress in Analysis, Proc. V Int. ISAAC Congress (Italy, 2005), World Sci., Singapore, 2008, 1195–1207 | MR

[10] R. S. Saks, Dokl. RAN, 424:2 (2009), 171–176 | MR | Zbl

[11] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika. T. VI. Gidrodinamika, Nauka, M., 1988 | MR | MR | Zbl

[12] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR | MR | Zbl

[13] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR | MR | Zbl

[14] R. I. Temam, Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Fazis, M., 1997 | MR | Zbl

[15] Kh. P. Grinspen, Teoriya vraschayuschikhsya zhidkostei, Gidrometeoizdat, L., 1975 | MR | Zbl

[16] W. Thomson (Lord Kelvin), Phil. Mag., Proc. Roy. Soc. Edinburgh Sect. 5, 10 (1880), 443–456 | DOI | Zbl

[17] R. R. Long, J. Atmospheric Sci., 8:4 (1951), 207–221 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[18] D. Fultz, J. Atmospheric Sci., 16:2 (1959), 199–208 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[19] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Internat. Ser. Monogr. Phys., The Clarendon Press, Oxford, 1981 | MR

[20] O. M. Phillips, Phys. Fluids, 6:4 (1963), 513–520 | DOI | MR | Zbl

[21] A. Babin, A. Mahalov, B. Nicolaenko, Indiana Univ. Math. J., 48:3 (1999), 1133–1176 | MR | Zbl

[22] A. S. Makhalov, V. P. Nikolaenko, UMN, 58:2(350) (2003), 79–110 | DOI | MR | Zbl

[23] A. G. Khaibullin, R. S. Saks, “O programme nakhozhdeniya koeffitsientov ryada Fure i ee primenenie pri issledovanii sistemy Nave–Stoksa v trekhmernom tore s periodicheskimi kraevymi usloviyami”, Sb. trudov IX mezhdunarodnogo seminara-soveschaniya “Kubaturnye formuly i ikh prilozheniya”, IM VTs UNTs RAN, Ufa, 2007, 175–181

[24] C. L. Sobolev, Izv. AN SSSR. Ser. matem., 18:1 (1954), 3–50 | MR | Zbl

[25] R. S. Saks, Differents. uravneniya, 8:1 (1972), 126–133 | MR | Zbl

[26] R. S. Saks, “Primenenie funktsionalnogo analiza k uravneniyam s chastnymi proizvodnymi”, Izbrannye trudy. T. II. Funktsionalnyi analiz. Differentsialnye uravneniya s chastnymi proizvodnymi, Trudy seminara akademika S. L. Soboleva, 83, 1983, 129–158 | MR | Zbl

[27] R. S. Saks, DAN SSSR, 279:4 (1984), 813–817 | MR | Zbl

[28] V. A. Ilin, Izbrannye trudy, T. 1, Makspress, M., 2008