On the Existence of a Global Solution of the Modified Navier–Stokes Equations
HTML articles powered by AMS MathViewer
- by
G. M. Kobel’kov
Translated by: V. E. Nazaikinskii - Trans. Moscow Math. Soc. 2016, 177-201
- DOI: https://doi.org/10.1090/mosc/258
- Published electronically: November 28, 2016
- PDF | Request permission
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
- 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 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 9-10, 1463–1492. MR 2895550, DOI 10.1070/SM2011v202n10ABEH004195
- V. V. Zhikov, On an approach to the solvability of generalized Navier-Stokes equations, Funktsional. Anal. i Prilozhen. 43 (2009), no. 3, 33–53 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 43 (2009), no. 3, 190–207. MR 2583638, DOI 10.1007/s10688-009-0027-9
- G. M. Kobel′kov, Existence of a solution “in the large” for equations of ocean dynamics, Dokl. Akad. Nauk 407 (2006), no. 4, 457–459 (Russian). MR 2348674
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
- O. A. Ladyženskaja, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 126–154 (Russian). MR 0241832
- R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
- Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
- S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Die Grundlehren der mathematischen Wissenschaften, Band 205, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by John M. Danskin, Jr. MR 0374877
- 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, Russian J. Numer. Anal. Math. Modelling 25 (2010), no. 6, 511–534. MR 2748256, DOI 10.1515/RJNAMM.2010.032
- Chongsheng Cao and Edriss S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2) 166 (2007), no. 1, 245–267. MR 2342696, DOI 10.4007/annals.2007.166.245
- A. V. Drutsa, Existence ‘in the large’ of a solution to primitive equations in a domain with uneven bottom, Russian J. Numer. Anal. Math. Modelling 24 (2009), no. 6, 515–542. MR 2604497, DOI 10.1515/RJNAMM.2009.033
- A. V. Gorshkov, Uniqueness of a solution to the problem of atmosphere dynamics, J. Math. Sci. (N.Y.) 167 (2010), no. 3, 340–357. Problems in mathematical analysis. No. 46. MR 2839025, DOI 10.1007/s10958-010-9923-z
- F. Guillén-González, N. Masmoudi, and M. A. Rodríguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations 14 (2001), no. 11, 1381–1408. MR 1859612
- Georgij M. Kobelkov, Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris 343 (2006), no. 4, 283–286 (English, with English and French summaries). MR 2245395, DOI 10.1016/j.crma.2006.04.020
- Georgij M. Kobelkov, On modifications of the Navier-Stokes equations, Russian J. Numer. Anal. Math. Modelling 30 (2015), no. 2, 87–93. MR 3334106, DOI 10.1515/rnam-2015-0009
- Georgy 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, DOI 10.1007/s00021-006-0228-4
- Roger Lewandowski, Analyse mathématique et océanographie, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 39, Masson, Paris, 1997 (French). MR 3468702
- Jacques-Louis Lions, Roger Temam, and Shou Hong Wang, On the equations of the large-scale ocean, Nonlinearity 5 (1992), no. 5, 1007–1053. MR 1187737
- Jacques-Louis Lions, Roger Temam, and Shou Hong Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5 (1992), no. 2, 237–288. MR 1158375
- Roger Temam and Mohammed Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 535–657. MR 2099038
- Changbing Hu, Roger Temam, and Mohammed 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, DOI 10.3934/dcds.2003.9.97
Bibliographic 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
- Published electronically: November 28, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2016, 177-201
- MSC (2010): Primary 76D05; Secondary 35Q30
- DOI: https://doi.org/10.1090/mosc/258
- MathSciNet review: 3643970