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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.

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge in [

Consider the operators

Consider Schrödinger’s equation:

Let

where

Let us also dene 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

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:

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

Then,

The proof of the above follows from the definitions of

Lemma 3. Let

Then,

The proof of the above again follows from the definitions of the functions

Lemma 4. Let

The proof of the above follows from the definitions of

Lemma 5. Let

The proof of the above follows from the definitions of

Definition 2. Denote by TA the set of functions

Definition 3. Denote by

Lemma 6. Suppose

The proof of the above follows from the definitions of

Lemma 7. Let

The proof of the above follows from the denitions of

Lemma 8. Let

where

The proof. Using

and (18) we get proof.

Lemma 9. Let

The lemma can be proved by substituting

Lemma 10. Let

The proof of the above follows from the definitions of

Lemma 11. Let

The proof of the above follows from the definition of D and the unitary nature of S.

Lemma 12. Let

The proof of the above follows from the definitions of

Lemma 13. Let

To prove this result, one should calculate

Using the notation that:

For

Lemma 14. Let

To prove this result, one should

Using Lemma 7.

Lemma 15. Let

To prove this result, one should calculate A using Lemma 7.

Lemma 16. Let

Then,

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

Lemma 17. Let

Then,

To prove this result, one should calculate

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 [

in the domain

The problem defined by (34), (35), (36) has at least one weak solution

The following results have been proved [

Theorem 5. If

there is a single generalized solution of (34), (35), (36) in the domain

Note that

Lemma 18. Let

Our goal is to provide global estimations for the Fourier transforms of the derivatives of the solutions to the Navier-Stokes Equations (34)-(36) without requiring the initial velocity and force to be small. We obtain the following uniform time estimation.

Statement 1. The solution of (34), (35), (36) according to Theorem 5 satisfies:

where

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

Statement 2. The solution of (34), (35), (36) satisfies:

and the following estimations:

This expression for

Lemma 19. The solution of (34), (35), (36) in Theorem 5 satisfies the following inequalities:

Proof this follows from the a priory estimation of Lemma18 and conditions of Lemma 19.

Lemma 20. Let

Proof this follows from the a priory estimation of Lemma18 and conditions of Lemma 20.

Lemma 21. The solution of (34), (35), (36) in Theorem 5 satisfies the following inequalities:

or

Proof this follows from the a priory estimation of Lemma18, conditions of Lemma 19, the Navier-Stokes equations.

Lemma 22. The solution of (34), (35), (36) satisfies the following inequalities:

Proof this follows from the a priory estimation of Lemma 18, conditions of Lemma 22, the Navier-Stokes equations.

Lemma 23. The solution of (34), (35), (36) according to Theorem 5 satisfies

Proof this follows from the a priory estimation of Lemma18, the Navier-Stokes equations.

Lemma 24.Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities:

where

Let is prove the first estimate. These inequalities

where

Proof now this follows from the a priori estimation of Lemma 18, conditions of Lemma 24, the Navier-Stokes equations.

The rest of estimates are proved similarly.

Lemma 25. Suppose that

Then,

Proof. Using Plansherel’s theorem, we get the statement of the lemma.

This proves Lemma 25.

Lemma 26. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities

where

Proof. From (40) we get

where

Using the notation

taking into account Holder’s inequality in I we obtain:

where

Taking into consideration the estimate I in (53), we obtain the statement of the lemma.

This proves Lemma 26.

Lemma 27. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities

Proof. The underwritten inequalities follows from representation (40)

Let us introduce the following denotation

then

Estimate I_{1} by means of

where

On applying Holder’s inequality, we get

where p, q satisfies the equality

For

Inserting

we obtain the statement of the lemma.

This completes the proof of Lemma 27.

Lemma 28. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities

where

Lemma 25. Let

Then,

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

We now obtain uniform time estimations for Rollnik’s norms of the solutions of (34), (35), (36).The following (and main) goal is to obtain the same estimations for

Let’s consider the influence of the following large scale transformations in Navier-Stokes’ equation on

Statement 3. Let

Proof. By the definitions

This proves Statement 3.

Theorem 6. Let

and

Then, there exists a unique generalized solution of (34), (35), (36) satisfying the following inequality:

where the value of

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

Using Lemmas (25)-(29) 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 loss of smoothness of classical solutions for the Navier-Stokes equations.

Theorem 7 describes the time blow up of the classical solutions for the Navier-Stokes equations arises, and complements the results of Terence Tao [

New uniform global estimations 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.

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 combining our 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.