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

This article is cited in 18 scientific papers (total in 18 papers)

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:
$$ \dim\mathscr A_{S^2}\leqslant c(\nu^{-8/3}\|f\|^{4/3}+\nu^{-2}\|f\|). $$

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

Bibliographic databases:

MSC: Primary 76D05, 35Q10, 58G20; Secondary 86A10
Received: 03.01.1989

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
\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
\jour Math. USSR-Sb.
\yr 1991
\vol 69
\issue 2
\pages 559--579

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    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    5. Volker Priebe, “Solvability of the Navier–Stokes equations on manifolds with boundary”, manuscripta math, 83:1 (1994), 145  crossref  mathscinet  zmath  isi
    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  mathscinet  zmath  isi  elib
    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  crossref  mathscinet
    8. Takeyuki Nagasawa, “Navier–Stokes flow on Riemannian manifolds”, Nonlinear Analysis: Theory, Methods & Applications, 30:2 (1997), 825  crossref  mathscinet  zmath
    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  crossref  mathscinet  zmath  isi
    10. Kai-tai Li, Feng Shi, “Hodograph method of flow on two-dimensional manifold”, Appl Math Mech, 31:3 (2010), 363  crossref  mathscinet  zmath  isi
    11. Ipatova V.M., “On uniform attractors of explicit approximations”, Differ Equ, 47:4 (2011), 571–580  crossref  mathscinet  zmath  isi
    12. Ipatova V.M., “O ravnomernykh attraktorakh yavnykh approksimatsii”, Differentsialnye uravneniya, 47:4 (2011), 574–583  mathscinet  elib
    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  crossref  mathscinet  zmath
    14. L. Tophøj, T. Bohr, “Stationary ideal flow on a free surface of a given shape”, J. Fluid Mech, 721 (2013), 28  crossref  mathscinet  zmath
    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  mathnet  mathscinet  elib
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet
    18. A. Ilyin, A. Laptev, “Lieb–Thirring inequalities on the sphere”, Algebra i analiz, 31:3 (2019), 116–135  mathnet
  • Математический сборник - 1989–1990 Sbornik: Mathematics (from 1967)
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