Unique Measure for the Time-Periodic Navier-Stokes on the Sphere Navier-Stokes on the Sphere

DOI: 10.4236/am.2015.611160   PDF   HTML   XML   2,633 Downloads   3,051 Views   Citations

Abstract

This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of deterministic force satisfying the necessary conditions for a unique invariant measure to exist are given. The support of the measure is examined and given explicitly for several cases.

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Varner, G. (2015) Unique Measure for the Time-Periodic Navier-Stokes on the Sphere Navier-Stokes on the Sphere. Applied Mathematics, 6, 1809-1830. doi: 10.4236/am.2015.611160.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Kuksin, S. and Shirikyan, A. (2000) Stochastic Dissipative PDE’s and Gibbs Measures. Communications in Mathematical Physics, 213, 291-330.
http://dx.doi.org/10.1007/s002200000237
[2] Kuksin, S. and Shirikyan, A. (2001) A Coupling Approach to Randomly Forced Nonlinear PDE’s. I. Communications in Mathematical Physics, 221, 351-366.
http://dx.doi.org/10.1007/s002200100479
[3] Kuksin, S. and Shirikyan, A. (2006) Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions. European Mathematical Society, Zürich.
http://dx.doi.org/10.4171/021
[4] Kuksin, S. and Shirikyan, A. (2002) Coupling Approach to White-Forced Nonlinear PDEs. Journal de Mathématiques Pures et Appliquées, 81, 567-602.
http://dx.doi.org/10.1016/S0021-7824(02)01259-X
[5] Shirikyan, A. (2005) Ergodicity for a Class of Markov Processes and Applications to Randomly Forced PDE’s. I. Russian Journal of Mathematical Physics, 12, 81-96.
[6] Varner, G. (2013) Stochastically Perturbed Navier-Stokes System on the Rotating Sphere. PhD Dissertation, The University of Missouri, Columbia.
[7] Shirikyan, A. (2015) Control and Mixing for 2D Navier-Stokes Equations with Space-Time Localised Noise. Annales Scientifiques de l’ENS, 48, 253-280.
[8] Brzezniak, Z., Goldys, B. and Le Gia, Q.T. (2015) Random Dynamical Systems Generated by Stochastic Navier-Stokes Equation on the Rotating Sphere. Journal of Mathematical Analysis and Applications, 426, 505-545.
http://dx.doi.org/10.1016/j.jmaa.2015.01.054
[9] Il’in, A.A. (1991) The Navier-Stokes and Euler Equations on Two-Dimensional Closed Manifolds. Mathematics of the USSR-Sbornik, 69, 559-579.
http://dx.doi.org/10.1070/SM1991v069n02ABEH002116
[10] Ilyin, A. (2004) Stability and Instability of Generalized Kolmogorov Flows on the Two-Dimensional Sphere. Advances in Differential Equations, 9, 979-1008.
[11] Dymnikov, V. and Filatov, A. (1997) Mathematics of Climate Modeling. Birkhäuser, Boston.
[12] Kuksin, S. and Shirikyan, A. (2012) Mathematics of Two-Dimensional Turbulence. Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9781139137119
[13] Robinson, J. (2001) Infinite-Dimensional Dynamical Systems. Cambridge University Press, New York.
http://dx.doi.org/10.1007/978-94-010-0732-0
[14] Heywood, J. and Rannacher, R. (1986) An Analysis of Stability Concepts for the Navier-Stokes Equations. Journal für die Reine und Angewandte Mathematik, 372, 1-33.
[15] Furshikov, A. and Vishik, M. (1988) Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publishers, Boston.
[16] Lions, J.L. and Magenes, E. (1972) Non-Homogeneous Boundary Value Problems, II. Spring-Verlag, Heidelberg and New York.
[17] Kuksin, S., Piatnitski, A. and Shirikyan, A. (2002) A Coupling Approach to Randomly Forced Nonlinear PDE’s. II. Communications in Mathematical Physics, 230, 81-85.
http://dx.doi.org/10.1007/s00220-002-0707-2
[18] Dymnikov, V. and Filatov, A. (1997) Mathematics of Climate Modeling. Birkh?user, Boston.
[19] Il’in, A.A. (1994) Partly Dissipative Semi-Groups Generated by the Navier-Stokes System on Two-Dimensional Manifolds, and Their Attractors. Russian Academy of Sciences. Sbornik Mathematics, 78, 159-182.
http://dx.doi.org/10.1070/sm1994v078n01abeh003458
[20] Skiba, Y. (2012) On the Existence and Uniqueness of Solution to Problems of Fluid Dynamics on a Sphere. Journal of Mathematical Analysis and Applications, 388, 627-644.
http://dx.doi.org/10.1016/j.jmaa.2011.10.045

  
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