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On Existence “In the Large” of a Solution to Modified Navier-Stokes Equations

  • George KobelkovEmail author
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

Abstract

The problem on existence “in the large” of a solution to the 3D Navier-Stokes equations is open up to now. Nevertheless, for some modifications of the Navier-Stokes equations describing practical problems this problem has been successfully solved. For instance, for the system of Primitive equations describing large-scale ocean dynamics, existence and uniqueness of a strong solution for any time interval and arbitrary initial conditions and viscosity coefficient was proved (Kobelkov J Math Fluid Mech 9(4):588–610, 2007) [1]. O.A. Ladyzhenskaya proposed (Trudy MIAN SSSR 102:85–104, 1967) [2] a modification of the Navier-Stokes equations allowing to prove existence of a solution “in the large”, but this modification was not “physical”. Here we improve the Ladyzhenskaya result modifying not all the three motion equations, but only two of them and only in two (horizontal) variables (not three). Such kind of problems arises in ocean dynamics models. We also consider the case when the viscosity coefficients in vertical and horizontal directions are different. For all these cases existence “in the large” of a solution is proved. Unfortunately, these results cannot be extended to the case of 3D Navier-Stokes equations as well as in the case of Ladyzhenskaya modification.

References

  1. 1.
    Kobelkov GM (2007) Existence of a solution “in the large” for ocean dynamics equations. J Math Fluid Mech 9(4):588–610MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ladyzhenskaya OA (1967) On modifications of the Navier-Stokes equations for large gradients. Trudy MIAN SSSR 102:85–104 (in Russian)Google Scholar
  3. 3.
    Ladyzhenskaya OA (1969) The mathematical theory of viscous incompressible flow. Gordon and Breach, New YorkzbMATHGoogle Scholar
  4. 4.
    Temam R (1984) Navier-Stokes equations: theory and numerical analysis. North-Holland, AmsterdamzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Russian Academy of SciencesMoscowRussia

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