Global Estimation of the Cauchy Problem Solutions ’ Fourier Transform Derivatives for the Navier-Stokes Equation Asset

The uniform estimation of the derivatives of Fourier transforms of the Cauchy problem for the Navier-Stokes equations was obtained.


Introduction
Numerous studies of the Navier-Stokes equations are devoted to the problem of its solution's smoothness.A good overview of the studies was made in [1,2].Spatial differentiability of the solutions is one of the most important aspect of the study, as it is mostly responsible for an evolution of the solutions.
Obviously differentiable solutions do not provide an effective description of turbulence.On the other hand, the global solvability and differentiability of the solutions are not proven, and therefore the turbulence description problem remains to be open.
It is interesting to study the properties of the Fourier transform of solutions of the Navier-Stokes equation: how they can be used in the description of turbulence, and whether they are differentiable.Differentiability of Fourier transform of the Navier-Stokes equation's solution seems to be related to the appearance or disappearance of the resonances, as it will mean the absence of large energy flows from small to large harmonics, which in turn means the inability of turbulence.
Thus, obtaining the uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations will mean that the principle of modeling of complex flows and related calculations shall be based on the Fourier transform method.
The authors continue to research these issues in application to the numerical weather prediction model and this paper is a theoretical justification for this approach.

Cauchy Problem for the Navier-Stokes Equation
Consider Cauchy problem for the Navier-Stokes equations:   in the domain The problem (1), ( 2), (3) has at least one weak solution   , q p in the so-called Leray-Hopf class [1].
Here are the known results proved in [1]: there is a single generalized solution of (1), ( 2), (3) in the domain , satisfying to the following conditions: Our goal is to provide global estimatimation of Fourier transforms of the Navier-Stokes equations' soutions derivatives (1), ( 2), (3) without assumption of the smallness of the initial velocity and force.
We get this time-uniform estimation.Let use the following notation: Assertion 1 The solution of ( 1), ( 2), (3) according to the Theorem 1 staisfies to: where F p f    .This follows from the definition of the Fourier transform and the theory of linear differential equations.Assertion 2 The solution of (1), ( 2), (3) staisfies to: and the following estimations: Submission for obtained by using and the Fourier transform.The estimations follow from this representation.
p div
Provement of the lemma can be obtained using Plancherel theorem.

Conclusion
M are finite.Let prove the first estimation.By definition of and Lemm 3,4 assertions follow the inequlities: can be proved in a similar way.Lemma 5 The solution of (1), (2), (3) according to the Theorem 1 staisfies to: o n s t (16) Thus, the uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations indicate Open Access IJMNTA A. A. DURMAGAMBETOV, L. S. FAZYLOVA Open Access IJMNTA 234that the principle of modeling of complex flows and related calculations shall be based on the Fourier transform method.In terms of the Fourier transform, both smooth initial conditions and right-hand sides do not appear exacerbations modes for speed and pressure.Loss of smoothness in terms of the Fourier transform can be expected only in the case of singular initial conditions, or of unlimited forces in the