Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

On the Existence of a Global Solution of the Modified Navier-Stokes Equations


Author: G. M. Kobel’kov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 77 (2016), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2016, 177-201
MSC (2010): Primary 76D05; Secondary 35Q30
DOI: https://doi.org/10.1090/mosc/258
Published electronically: November 28, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove global existence theorems for initial-boundary value problems for the modified Navier-Stokes equations used when modeling ocean dynamic processes. First, the case of distinct vertical and horizontal viscosities for the Navier-Stokes equations is considered. Then a result due to Ladyzhenskaya for the modified Navier-Stokes equations is improved, whereby the elliptic operator is strengthened with respect to the horizontal variables alone and only for the horizontal momentum equations. Finally, the global existence and uniqueness of a solution is proved for the primitive equations describing the large-scale ocean dynamics.


References [Enhancements On Off] (What's this?)

  • 1. A. V. Drutsa, Existence ``in the large'' of a solution to a system of equations of large-scale ocean dynamics on a manifold, Mat. Sb. 202 (2011), no. 10, 55-86; English transl., Sb. Math. 202 (2011), no. 9-10, 1463-1492. MR 2895550
  • 2. V. V. Zhikov, On an approach to the solvability of generalized Navier-Stokes equations, Funktsional. Anal. i Prilozhen. 43 (2009), no. 3, 33-53; English transl., Funct. Anal. Appl. 43 (2009), no. 3, 190-207. MR 2583638
  • 3. G. M. Kobel'kov, Global existence of a solution to the ocean dynamics equations, Dokl. Akad. Nauk 407 (2006), no. 4, 457-459; English transl., Dokl. Math. 73 (2006), no. 2, 296-298. MR 2348674
  • 4. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961; Nauka, Moscow, 1970; English transl., Gordon and Breach, New York, 1969. MR 0254401
  • 5. O. A. Ladyzhenskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968) 126-154. (Russian) MR 0241832
  • 6. G. I. Marchuk and A. S. Sarkisyan, eds., Mathematical models of circulation in the ocean, Nauka, Novosibirsk, 1980. (Russian) MR 86A05
  • 7. S. A. Nazarov and B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, Nauka, Moscow, 1991; English transl., De Gruyter, Berlin, 1994. MR 1283387
  • 8. S. M. Nikol'skii, Approximation of functions of several variables and imbedding theorems, Nauka, Moscow, 1977, 2nd rev. aug. ed.; English transl. of 1st ed., Springer, Berlin, 1975. MR 0374877
  • 9. V. M. Ipatova, V. I. Agoshkov, G. M. Kobelkov, and V. B. Zalesny, Theory of solvability of boundary value problems and data assimilation problems for ocean dynamics equations, Russ. J. Numer. Anal. Math. Model. 25 (2010), no. 6, 511-534. MR 2748256
  • 10. C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math. 166 (2007), no. 1, 245-267. MR 2342696
  • 11. A. V. Drutsa, Existence ``in the large'' of a solution to primitive equations in a domain with uneven bottom, Russ. J. Numer. Anal. Math. Model. 24 (2009), no. 6, 515-542. MR 2604497
  • 12. A. V. Gorshkov, Uniqueness of a solution to the problem of atmosphere dynamics, J. Math. Sci. 167 (2010), no. 3, 340-357. MR 2839025
  • 13. F. Guillén-González, N. Masmoudi, and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Diff. Int. Equat. 14 (2001), no. 11, 1381-1408. MR 1859612
  • 14. G. M. Kobelkov, Existence of a solution ``in the large'' for the 3D large scale ocean dynamics equations, C. R. Acad. Sci. Paris. Ser. II 343 (2006), no. 4, 283-286. MR 2245395
  • 15. G. M. Kobelkov, Modifications of the Navier-Stokes equations, Russ. J. Numer. Anal. Math. Model. 30 (2015), no. 2, 87-94. MR 3334106
  • 16. G. M. Kobelkov, Existence of a solution ``in the large'' for ocean dynamics equations, J. Math. Fluid Mech. 9 (2007), no. 4, 588-610. MR 2374160
  • 17. R. Lewandowski, Analyse mathématique en océanographie, Masson, Paris, 1997. MR 3468702
  • 18. J. L. Lions, R. Temam, and S. Wang, On the equations of the large-scale ocean, Nonlinearity 5 (1992), 1007-1053. MR 1187737
  • 19. J. L. Lions, R. Temam, and S. Wang, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity 5 (1992), 237-288. MR 1158375
  • 20. R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook mathematical fluid dynamics, vol. 3, Elsevier, Amsterdam, 2004, pp. 535-658. MR 2099038
  • 21. C. Hu, R. Temam, and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst. 9 (2003), no. 1, 97-131. MR 1951315

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 76D05, 35Q30

Retrieve articles in all journals with MSC (2010): 76D05, 35Q30


Additional Information

G. M. Kobel’kov
Affiliation: Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia; Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
Email: kobelkov@dodo.inm.ras.ru

DOI: https://doi.org/10.1090/mosc/258
Keywords: Navier--Stokes equations, primitive equations, large-scale ocean dynamics, modification of Navier--Stokes equations, global existence.
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society