NS  Vol.7 No.2 , February 2015
Navier-Stokes Equations—Millennium Prize Problems
ABSTRACT
In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.

Cite this paper
Durmagambetov, A. and Fazilova, L. (2015) Navier-Stokes Equations—Millennium Prize Problems. Natural Science, 7, 88-99. doi: 10.4236/ns.2015.72010.
References
[1]   Fefferman, C.L. (2006) Existence and Smoothness of the Navier-Stokes Equation. The Millennium Prize Problems, Clay Mathematics Institute, Cambridge, 57-67.

[2]   Durmagambetov, A.A. and Fazilova, L.S. (2013) Global Estimation of the Cauchy Problem Solutions’ Fourier Transform Derivatives for the Navier-Stokes Equation. International Journal of Modern Nonlinear Theory and Application, 2, 232-234.
http://www.scirp.org/journal/IJMNTA/

[3]   Durmagambetov, A.A. and Fazilova, L.S. (2014) Global Estimation of the Cauchy Problem Solutions’ the Navier-Stokes Equation. Journal of Applied Mathematics and Physics, 2, 17-25.
http://www.scirp.org/journal/JAMP/

[4]   Durmagambetov, A.A. and Fazilova, L.S. (2014) Existence and Blowup Behavior of Global Strong Solutions Navier-Stokes. International Journal of Engineering Science and Innovative Technology, 3, 679-687.
http://ijesit.com/archivedescription.php?id=16

[5]   Russell, J.S. (1844) Report on Wave. Report of the Fourteenth Meeting of the British Association for the Advancement of Science, York, Plates XLVII-LVII, 90-311.

[6]   Russell, J.S. (1838) Report of the Committee on Waves. Report of the 7th Meeting of British Association for the Advancement of Science, John Murray, London, 417-496.

[7]   Ablowitz, M.J. and Segur, H. (1981) Solitons and the Inverse Scattering Transform. SIAM, 435-436.

[8]   Zabusky, N.J. and Kruskal, M.D. (1965) Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States. Physical Review Letters, 15, 240-243.
http://dx.doi.org/10.1103/PhysRevLett.15.240

[9]   Faddeev, L.D. (1974) The Inverse Problem in the Quantum Theory of Scattering II. Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, VINITI, Moscow, 93-180.

[10]   Newton, R.G. (1979) New Result on the Inverse Scattering Problem in Three Dimensions. Physical Review Letters, 43, 541-542.
http://dx.doi.org/10.1103/PhysRevLett.43.541

[11]   Newton, R.G. (1980) Inverse Scattering. II. Three Dimensions. Journal of Mathematical Physics, 21, 1698-1715.
http://dx.doi.org/10.1063/1.524637

[12]   Somersalo, E., et al. (1988) Inverse Scattering Problem for the Schrodinger’s Equation in Three Dimensions: Connections between Exact and Approximate Methods. Journal of Mathematical Physics, 21, 1698-1715.

[13]   Povzner, A.Y. (1953) On the Expansion of Arbitrary Functions in Characteristic Functions of the Operator. Russian, Sbornik Mathematics, 32, 56-109.

[14]   Birman, M.S. (1961) On the Spectrum of Singular Boundary-Value Problems. Russian, Sbornik Mathematics, 55, 74-125.

[15]   Poincare, H. (1910) Lecons de mecanique celeste, t. Math. & Phys. Papers, 4, 141-148.

[16]   Leray, J. (1934). Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica, 63, 193-248.
http://dx.doi.org/10.1007/BF02547354

[17]   Ladyzhenskaya, O.A. (1970) Mathematics Problems of Viscous Incondensable Liquid Dynamics. Science, 288.

[18]   Solonnikov, V.A. (1964) Estimates Solving Nonstationary Linearized Systems of Navier-Stokes’ Equations. Transactions Academy of Sciences USSR, 70, 213-317.

[19]   Huang, X., Li, J. and Wang, Y. (2013) Serrin-Type Blowup Criterion for Full Compressible Navier-Stokes System. Archive for Rational Mechanics and Analysis, 207, 303-316.
http://dx.doi.org/10.1007/s00205-012-0577-5

[20]   Tao, T. (2014) Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation.

 
 
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