RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 1990, Volume 181, Number 4, Pages 521–539 (Mi msb1182)

The Navier–Stokes and Euler equations on two-dimensional closed manifolds

A. A. Ilyin

Hydrometeorological Centre of USSR

Abstract: The Navier–Stokes equations
$$\partial_tu+\nabla_uu+\nu\Lambda u=-\nabla p+f, \qquad \operatorname{div}u=0$$
are considered on a two-dimensional closed manifold $M$ imbedded in $R^3$. Theorems on existence and uniqueness of generalized solutions of steady-state and time-dependent problems are proved. Unique solvability of the Euler equations $(\nu=0)$ is proved by passing to the limit as $\nu\to+0$. The existence of a maximal attractor for the Navier–Stokes system on $M$ is proved, and for the case where the manifold $M$ is the sphere $S^2$ an estimate for the Hausdorff dimension of the attractor is obtained:
$$\dim\mathscr A_{S^2}\leqslant c(\nu^{-8/3}\|f\|^{4/3}+\nu^{-2}\|f\|).$$

Full text: PDF file (1858 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1991, 69:2, 559–579

Bibliographic databases:

MSC: Primary 76D05, 35Q10, 58G20; Secondary 86A10

Citation: A. A. Ilyin, “The Navier–Stokes and Euler equations on two-dimensional closed manifolds”, Mat. Sb., 181:4 (1990), 521–539; Math. USSR-Sb., 69:2 (1991), 559–579

Citation in format AMSBIB
\Bibitem{Ily90} \by A.~A.~Ilyin \paper The Navier--Stokes and Euler equations on two-dimensional closed manifolds \jour Mat. Sb. \yr 1990 \vol 181 \issue 4 \pages 521--539 \mathnet{http://mi.mathnet.ru/msb1182} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1055527} \zmath{https://zbmath.org/?q=an:0713.35074|0724.35088} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991SbMat..69..559I} \transl \jour Math. USSR-Sb. \yr 1991 \vol 69 \issue 2 \pages 559--579 \crossref{https://doi.org/10.1070/SM1991v069n02ABEH002116} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991GB41500015} 

• http://mi.mathnet.ru/eng/msb1182
• http://mi.mathnet.ru/eng/msb/v181/i4/p521

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Ilyin, “The Euler equations with dissipation”, Math. USSR-Sb., 74:2 (1993), 475–485
2. Michael E. Taylor, “Analysis on Morrey Spaces and Applications to Navier–Stokes and Other Evolution Equations”, Communications in Partial Differential Equations, 17:9-10 (1992), 1407
3. A. A. Ilyin, “Partly dissipative semigroups generated by the Navier–Stokes system on two-dimensional manifolds, and their attractors”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 47–76
4. Ilyin A., “Lieb-Thirring Inequalities on the N-Sphere and in the Plane, and Some Applications”, Proc. London Math. Soc., 67:Part 1 (1993), 159–182
5. Volker Priebe, “Solvability of the Navier–Stokes equations on manifolds with boundary”, manuscripta math, 83:1 (1994), 145
6. Coron, JM, “Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary”, Russian Journal of Mathematical Physics, 4:4 (1996), 429
7. Yuri N. Skiba, “On dimensions of attractive sets of viscous fluids on a sphere under quasi-periodic forcing”, Geophysical & Astrophysical Fluid Dynamics, 85:3-4 (1997), 233
8. Takeyuki Nagasawa, “Navier–Stokes flow on Riemannian manifolds”, Nonlinear Analysis: Theory, Methods & Applications, 30:2 (1997), 825
9. Kaitai Li, Aixiang Huang, Wen ling Zhang, “A dimension split method for the 3-D compressible Navier–Stokes equations in turbomachine”, Commun Numer Meth Engng, 18:1 (2002), 1
10. Kai-tai Li, Feng Shi, “Hodograph method of flow on two-dimensional manifold”, Appl Math Mech, 31:3 (2010), 363
11. Ipatova V.M., “On uniform attractors of explicit approximations”, Differ Equ, 47:4 (2011), 571–580
12. Ipatova V.M., “O ravnomernykh attraktorakh yavnykh approksimatsii”, Differentsialnye uravneniya, 47:4 (2011), 574–583
13. Kai-tai Li, Jia-ping Yu, Feng Shi, Ai-xiang Huang, “Dimension splitting method for the three dimensional rotating Navier–Stokes equations”, Acta Math. Appl. Sin. Engl. Ser, 28:3 (2012), 417
14. L. Tophøj, T. Bohr, “Stationary ideal flow on a free surface of a given shape”, J. Fluid Mech, 721 (2013), 28
15. S. V. Zakharov, “Asimptotika obobschennogo resheniya statsionarnoi sistemy Nave–Stoksa na mnogoobrazii, diffeomorfnom sfere”, Tr. IMM UrO RAN, 19, no. 4, 2013, 119–124
16. Chan Ch.H., Czubak M., “Non-Uniqueness of the Leray-Hopf Solutions in the Hyperbolic Setting”, Dyn. Partial Differ. Equ., 10:1 (2013), 43–77
17. Z. Brzeźniak, B. Goldys, Q.T. Le Gia, “Random dynamical systems generated by stochastic Navier–Stokes equations on a rotating sphere”, Journal of Mathematical Analysis and Applications, 2015
18. A. Ilyin, A. Laptev, “Lieb–Thirring inequalities on the sphere”, Algebra i analiz, 31:3 (2019), 116–135
•  Number of views: This page: 846 Full text: 248 References: 68 First page: 2