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The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.

In this paper, we introduce important explanatory results presented in a previous study in [

Consider the operators

Consider Schrödinger’s equation:

Let

where

Let us also define the solution

As is well known [

This equation is the key to solving the inverse scattering problem, and was first used by Newton [

Equation (4) is equivalent to the following:

where S is a scattering operator with kernel

The following theorem was stated in [

Theorem 1. (The energy and momentum conservation laws) Let

Definition 1. The set of measurable functions

is recognized as being of Rollnik class.

As shown in [

where

We present Povzner’s results [

Theorem 2. (Completeness) For both an arbitrary

where

Theorem 3. (Birman-Schwinger estimation). Let

This theorem was proved in [

Let us introduce the following notation:

For

where

We define the operators

Consider the Riemann problem of finding a function

Lemma 1.

Theorem 4. Let

The proof of the above follows from the classic results for the Riemann problem.

Lemma 2. Let

The proof of the above follows from the definitions of

Definition 2. Denote by

Definition 3. Denote by

Lemma 3. Suppose

The proof of the above follows from the definitions of

Lemma 4. Let

The proof of the above follows from the definitions of

Lemma 5. Let

where

Proof: Using

and

We establish the proof.

Lemma 6. Let

Lemma 7. Let

The proof of the above follows from the definitions of

Lemma 8. Let

The proof of the above follows from the definition of

Lemma 9. Let

The proof of the above follows from the definitions of

Lemma 10. Let

The proof of the above follows from the definition of

Lemma 11. Let

To prove this result, one calculates

Using Lemma 5, the first approximation can be obtained in terms of

where

This study has shown once again the outstanding properties of the scattering operator, which, in combination with the analytical properties of the wave function, enable an almost-explicit formula for the potential to be obtained from the scattering amplitude. Furthermore, this approach overcomes the problem of over-determination, resulting from the fact that the potential is a function of three variables, whereas the amplitude is a function of five variables. We have shown that it is sufficient to average the scattering amplitude to eliminate the two extra variables.

Numerous studies of the Navier-Stokes equations have been devoted to the problem of the smoothness of its solutions. A good overview of these studies is given in [

Thus, obtaining uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations means that the principle modeling of complex flows and related calculations will be based on the Fourier transform method. The authors are continuing to research these issues in relation to a numerical weather prediction model; this paper provides a theoretical justification for this approach. Consider the Cauchy problem for the Navier-Stokes equations:

in the domain

The problem defined by (28), (29), (30) has at least one weak solution

The following results have been proved [

Theorem 5. If

there is a single generalized solution of (28), (29), (30) in the domain

Note that

Lemma 12. Let

Our goal is to provide global estimations for the Fourier transforms of the derivatives of the solutions to the Navier-Stokes Equations (28), (29), (30) without requiring the initial velocity and force to be small. We obtain the following uniform time estimation. Using the notation

Assertion 1. The solution of (28) (30) according to Theorem 5 satisfies:

where

This follows from the definition of the Fourier transform and the theory of linear differential equations.

Assertion 2.The solution of (28) (30) satisfies:

and the following estimations:

This expression for

Lemma 13. The solution of (28), (29), (30) in Theorem 5 satisfies the following inequalities:

or

This follows from the Navier-Stokes equations, our first a priori estimation (Lemma 1), and Lemma 2.

Lemma 14. The solution of (28) (30) satisfies the following inequalities:

These estimations follow from (9), Parseval’s identity, the Cauchy-Schwarz inequality, and Lemma 3.

Lemma 15. The solution of (28) (30) according to Theorem 5 satisfies

This follows from our a priori estimation (Lemma 1) and the assertion of Lemma 3.

Lemma 16. The solution of (28) (30) according to Theorem 5 satisfies to the following inequalities:

where

Proof. From (36), we have the inequality:

where

Using the notation

And Hölder’s inequality in

where

Using the estimation for

Lemma 17. Let

Then,

A proof of this lemma can be obtained using Plancherel’s theorem. For

consider the transformation of the Navier-Stokes:

Lemma 18. Let

then

Proof. Using the definitions for

We now obtain uniform time estimations for Rollnik’s norms of the solutions of (28) (30). The following (and main) goal is to obtain the same estimations for

the velocity components of the Cauchy problem for the Navier-Stokes equations. We shall use Lemmas 6 and 11.

Theorem 6. Let

Then, there exists a unique generalized solution of (28) (30) satisfying the following inequality:

where the value of

Proof. It suffices to obtain uniform estimates of the maximum velocity components

Because uniform estimates allow us to extend the local existence and uniqueness theorem over the interval in which they are valid. To estimate the velocity components, Lemma 10 can be used:

Using Lemmas (13)-(17) for

we can obtain

Theorem 6 asserts the global solvability and uniqueness of the Cauchy problem for the Navier-Stokes equations.

Theorem 7. Let

Then, there exists

Proof. A proof of this lemma can be obtained using

Theorem 7 describes the blowup of classical solutions for the Navier-Stokes equations.

Uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations indicate that the principle modeling of complex flows and related calculations can be based on the Fourier transform method. In terms of the Fourier transform, under both smooth initial conditions and right-hand sides, no apparent fluctuations appear in the speed and pressure modes. A loss of smoothness in terms of the Fourier transform can only be expected for singular initial conditions or unbounded forces in

We are grateful to the Ministry of Education and Science of the Republic of Kazakhstan for a grant, and to the System Research “Factor” Company for joint efforts in this project. The work was performed as part of an international project, “Joint Kazakh-Indian studies of the influence of anthropogenic factors on atmospheric phenomena on the basis of numerical weather prediction models WRF (Weather Research and Forecasting)”, commissioned by the Ministry of Education and Science of the Republic of Kazakhstan.