Communications in Mathematical Analysis

Unique solvability of vorticity equation of incompressible viscous fluid on a rotating sphere

Yuri N. Skiba

Full-text: Open access


Ortogonal projectors, fractional derivatives and Hilbert and Banach spaces on a two-dimensional unit sphere are introduced. Unique solvability of nonstationary problem for barotropic vorticity equation on the sphere in classes of generalized functions is proved.

Article information

Commun. Math. Anal., Conference 3 (2011), 209-224.

First available in Project Euclid: 25 February 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Skiba, Yuri N. Unique solvability of vorticity equation of incompressible viscous fluid on a rotating sphere. Commun. Math. Anal. (2011), no. 3, 209--224.

Export citation


  • V. I. Arnold, Ordinary Differential Equations. Nauka, Moscow 1965 (Russian) (Translated from the Russian by Roger Cooke. 2nd edition. Universitext. Springer-Verlag, Berlin 2006).
  • R. Askey and St. Wainger, On the Behaviour of Special Classes of Ultraspherical Expansions. Part 1,2. J. Analyse Math. (1965) 15, pp 193-220, pp 221-244.
  • L. F. F. Bers John and M. Schecter, Partial Differential Equations. Wiley, New York 1964.
  • V. P. Dymnikov and Yu. N. Skiba, Barotropic instability of zonally asymmetric atmospheric flows over topography. Sov. J. Numer. Anal. Math. Modelling 2 (1987), No. 2, pp 83-98.
  • A. D. Gadzhiev, Differential properties of the symbol of a multidimensional singular integral operator. (Russian) Mat. Sbornik 114, (156), (1981), No. 4, pp 483-510.
  • S. Helgason, Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions. Academic Press, Orlando 1984.
  • A. A. Ilyin and A. N. Filatov, On unique solvability of Navier-Stokes equations on two-dimensional sphere. Dokl. Akad. Nauk SSSR 301 (1988), pp 18-22.
  • V. A. Ivanov, On the Bernstein-Nikolskiy and Favard inequalities on compact homogeneous spaces of rank 1. Usp. Mat. Nauk 38 (1983), pp 179-180.
  • A. I. Kamzolov, On approximation of smooth functions on the sphere $S^{n}$ by Fourier method. Math. Zametki 31 (1982), No. 6, pp 847-853 (Russian).
  • A. I. Kamzolov, On the best approximation of classes of functions $ \mathbf{W}_{p}^{\alpha }(S^{n})$ by polynomials in spherical harmonics. Math. Zametki 32 (1982), No. 3, pp 285-293.
  • A. I. Kamzolov, Bernstein's inequality for fractional derivatives of polynomials in spherical harmonics. Uspekhi Mat. Nauk 39 (1984), No. 2 (236), pp 159-160 (Russian).
  • T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, Heidelberg, New York 1980.
  • A. N. Kolmogorov and S. V. Fomin, Elements of the Function Theory and Functional Analysis. Nauka, Moscow 1976 (Russian).
  • O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incomprehensible Flow. Mathematics and its Applications, Vol. 2, Gordon and Breach, New York 1969.
  • O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics. Nauka, Moscow 1973 (Russian).
  • J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. (French) Dunod; Gauthier-Villars, Paris 1969 (French); Certain Methods for the Solution of Nonlinear Boundary Value Problems. Mir, Moscow 1972 (Russian).
  • P. I. Lizorkin and S. M. Nikolskiy, Approximation of functions on the sphere in $\mathbf{L}_{2}$. Dokl. Akad. Nauk SSSR 271 (1983), No. 5, pp 1059-1063.
  • R. D. Richtmyer, Principles of Advanced Mathematical Physics. Springer-Verlag, New York 1978 (Vol. 1); 1981 (Vol. 2).
  • A. J. Simmons, J. M. Wallace, and G. W. Branstator, Barotropic Wave Propogation and Instability, and Atmospheric Teleconnection Patterns. J. Atmos. Sci. 40 (1983), No. 6, pp 1363-1392.
  • Yu. N. Skiba, Mathematical Problems of the Dynamics of Viscous Barotropic Fluid on a Rotating Sphere. VINITI, Moscow, 1989, 178 pp. (Russian); Indian Inst. Tropical Meteorology, Pune, India, 1990, 211 pp. (English).
  • Yu. N. Skiba, Unique solvability of the equation of a barotropic vortex of viscous fluid in classes of generalized functions on a sphere. Depart. Numer. Mathematics, Akad. Nauk SSSR, Moscow, 1988, 194, 56 pp. (Russian).
  • P. Szeptycki, Equations of Hydrodynamics on Compact Reimannian Manifolds. Bull. L'acad. Pol. Sci., Seria: Sci. Math., Astr., Phys., 21 (1973), No. 4, pp 335-339.
  • P. Szeptycki, Equations of Hydrodynamics on Manifold Diffeomorfic to the Sphere. Bull. L'acad. Pol. Sci., Seria: Sci. Math., Astr., Phys., 21 (1973), No. 4, pp 341-344.
  • R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam 1984.