1. Introduction

If to designate components of vectors of speed and external force, as

that corresponding problem Navier-Stokes is represented in a kind

(1.1)

(1.2)

(1.3)

is kinematic viscosity, is density, is Laplace operator. The additional equation is the condition incompressibility fluid (1.2). Unknown are speed and pressure P.

The decision of many problems of theoretical and mathematical physics leads to use of various special weight space. In works [7, 8] for the first time have offered a method which gives solution of problem Navier-Stokes in .

To answer the brought attention to the question, we offer the following method of the decision of a problem Navier-Stokes. For this purpose (1.1) we will transform to a kind

(1.4)

(1.5)

where

without breaking equivalence of system (1.1) and (1.4), (1.5). The received systems (1.4), (1.5) contain unknown functions , and pressure P. Here - known functions because are known .

The developed method of the decision of systems (1.4) and (1.5), is connected with functions , i.e.

A₁) or

А₂) or

А₃) is any functions if, accordingly, as necessary conditions, take place: а₀₁)

а₀₂) а₀₃) is any functions.

The work purpose. The main object of this work the proof - existence and singleness the problem decision Navier-Stokes for an incompressible fluid with viscosity in cases (А₁)-(А₃).

Theoretical and practical value. The Received decisionson the basis of the developed analytical methods proves in the general applicability of the equations of Navier-Stokes. Thus it is proved that the system (1.1) has the single decision in cases (A₁)-(А₃). The received results prove to legality of the requirement [1].

Remarks: 1. The Beale-Kato-Majda regularity criterion, originally derived for solutions to the 3D Euler equations [2] and holds for solutions to the 3D equations Navier-Stokes [5] and at that is proved an inequality of a criterion Beale-Kato-Majda [5] for an this problem, and the criterion can be viewed as a continuation principle for strong solutions. A further generalization was presented in [7] where the regularity condition is expressed in terms of the time integrability.

Let’s note that there are several criteria of a priori estimates. For instance, it is sufficient to prove an estimate [5]

(1.6)

where М is a constant as (1.6) for the proof aforesaid the statement.

2. In a case the current is considered with very small viscosity, i.e. in viscous liquids, when force a friction a very small, than forces of inertia [9, 10]. Here Reynolds's number is very great there is an border layer in which viscosity influence is concentrated. Therefore the analytical methods giving the decisions of a problem Navier-Stokes allow to reach full understanding of physics of turbulence [3, 4, 9, 10].

And in a case the current is considered with average size of viscosity [10]. Therefore in a case when convective acceleration is not equal to zero problems connected with methods of integration of the equations of Navier-Stokes in their general view are arisen.

Our problem does not include a derivation of an equation in a physical meaning, since there is a big amount of works reflecting these questions [3, 4, 6, 9-11].

2. Problem Navier-Stokes for Fluid with a Condition (A1)

In this paragraph in the subsequent points, at the specified restrictions on the entrance data, the strict substantiation of compatibility of systems (1.4), (1.5) will be given with very small viscosity .

2.1. Research with a Condition (A₁)

Let functions satisfy to a condition (a₀₁). Then relatively we suppose a condition (A₁) and

(2.1)

where from system (1.4) and (1.5), accordingly we will receive following systems

(2.2)

(2.3)

Theorem 1. Let conditions (1.2), (1.3), (A₁) and (2.1) are satisfied. Then systems (2.2) and (2.3) it is equivalent will be transformed to a kind

(2.4)

Hence, the problem (1.1) - (1.3) has the unique decision which satisfies to a condition (1.2).

Proof. From system (2.2) it is visible, if the 1-equation (2.2, i=1) it is differentiated on x₁, 2-equation on x₂ (2.2, i=2), 3-equation on x₃ (2.2, i=3), and it is summarised

from here we will receive the equation of Puasson [11]

(2.5)

as

(2.6)

At that it is proved

(2.7)

Algorithm when we will receive the equation of Puasson (2.5) for brevity we name «algorithm puassonization systems».

In work of Sobolev [11] it is specified that function (2.7) satisfies to the equation (2.5) and is called Newton’s potential. Therefore, if J - the decision of the equation (2.5), then substituting

(2.8)

in (2.2), we have

(2.9)

i.e. system (2.2) it is equivalent by (2.9) will be transformed to a kind linear the nonuniform equation of heat conductivity. Here the equations (2.5), (2.9) is there are first and second equations of system (2.4).

From system (2.9), follows [7]:

(2.10)

All - is known functions and are defined from system (2.10)

(2.11)

Then, on the basis of (2.3), (2.10) and (2.11), and their private derivatives on x_i, we find

(2.12)

As - is known functions, hence from system (2.12) differentiating 1 equation on x₁ [(2.12): i=1], 2 equations on х₂ [(2.12): i=2], 3 equations on х₃ [(2.12): i=3], and summarising, we will receive

(2.13)

at that

The equation (2.13) is the third equation of system (2.4). Therefore, from the received results, taking into account (2.7), follows

(2.14)

i.e. functions are defined from systems (2.10), (2.13), (2.14), where (2.14) - the equation of type of Bernoulli.

Singleness is obvious, as a method by contradiction from (2.10) singleness of the decision follows. Results (2.10) with a condition ((A₁), (2.1)) are received where smoothness of functions is required only on x_i as the derivative of 1st order is in time has t>0. Then taking into account (2.10), (2.13), (2.14) and the system (2.4) has the single continuous decision.

Further, considering private derivatives of 1st order

(2.15)

and summarising (2.15) with taking into account (1.2), we have

as . Means, the system (2.10) satisfies to the equation (1.2).

2.2. Limitation of Functions in or

I. The limiting case which we will consider concerns results of the theorem 1. Then the decision of system (1.1) is representing in the form of (2.10) with conditions (1.2), (1.3), (A₁), (2.1) and

(2.16)

Really, estimating (2.10) in , we have [7]:

(2.17)

Theorem 2. In the conditions of the theorem 1 and (2.16), (2.17) the problem (1.1) - (1.3) has the single decision in.

II. Alternatively, we can consider, e.g., a class of suitable solutions constructed in As , the decision of problem Navier-Stokes (1.1) - (1.3) belongs in . Then on the basis of lemma K. Friedrichs [13], it is the decision and belongs to space of weight

where

The limiting case which we will consider concerns results of the theorem 1. Then the decision of system (1.1) is representing in the form of (2.10). Then estimating (2.10) in , we have

(2.18)

i.e. in the conditions of (1.2), (1.3), (А₁) and (2.16), (2.18) the problem (1.1) - (1.3) has a single decision in .

Remarks:

1. From the received results follows that on the basis of the developed methods of equation Navier-Stokes it is led to the linear equations of a kind of heat conductivity with a condition of Koshi and for

in a class of the limited functions with smooth enough initial data at is correctly put [11, 12]. Accordingly there is a unique, is conditional-smooth decision of problem Navier-Stokes in or

2. Definition 1. The generalised decision a problems (1.1)-(1.3), (A₁) in area we name any continuous in equation decision (2.10). Therefore, the nonstationary problem of Navier-Stokes (1.1) - (1.3) has the single decision.

If the problem (1.1) - (1.3), (А₁) has the exact classical decision the generalised decision coincides with it [11].

2.3. Inequality Beale-Kato-Majda

Inequality Beale-Kato-Majda. The criterion can be viewed as a continuation principle for strong solutions.

On the basis of results of the theorem 1 the of decision a systems (1.1) it is presented in of a kind (2.10), where global existence of decisions is received in a class (or ) from the point of view of the initial data satisfying (2.10). It is pleasant that results of this theorem leads to such global classical decision Navier-Stokes, besides it is known that in [5] classical decision is received, if the criterion of Beale-Kato-Majda is executed. There are several criteria of a priori estimates. For instance, it is sufficient to prove an estimate (1.6).

Really at performance of conditions of the theorem 1 takes place

Therefore we will receive an estimation of type Beale-Kato-Majda

(2.19)

Let's note that the similar inequality turns out and in Really in conditions, when the decision of system (2.10)

belongs in we will receive

(2.20)

3. Fluid with Average Viscosity with a Condition (A₂)

Area of the fluid with average viscosity where all inertial members contain in the equations of Navier-Stokes theoretically it is not investigated till now [10].

The decision method, from where follows of equations integration of Navier-Stokes in a case (А₂), is a major factor of this point. The developed method of the decision of system (1.1) is connected with where these functions will transform (1.1) to systems (1.4), (1.5) with conditions (а₀₂) and

(1.3)*

(3.1)

where the current is considered with average size of viscosity.

Theorem 3. Systems (1.4), (1.5) it is equivalent will be transformed to a kind

(3.2)

when conditions (1.2), (1.3)*, (3.1), (А₂) are satisfied. Hence, the nonstationary problem of Navier-Stokes (1.1)-(1.3)* has the single continuous decision.

Proof. Really, from system (1.4), considering conditions (1.2), (1.3)*, (3.1) and having entered «algorithm puassonization systems», i.e. differentiating the equations of system (1.4) accordingly on х_i and, then summarising, we have the equation

(3.3)

If - the decision of the equation (3.3), then substituting

in system (1.4), we have

(3.4)

The decision of a problem (1.1)-(1.3)* is represented in a kind

(3.5)

where concerning functions , we will receive

(3.6)

Here for example, private derivatives of functions are defined:

(3.7)

Here (3.6) – system of the nonlinear integrated equations of Volterr-Abel of the second sort concerning on a variable t.

Therefore, if operators: compressing with a compression factor ,

(3.8)

then the system (3.6) is solvable at . Hence the solution of this system we can find on the basis of Picard’s method:

where initial estimates. Thus we will receive

(3.9)

Then according to results of the theorem 3, functions are defined from system (3.5)

(3.5)*

here - known functions and

i.e.

Thus the equation (3.5) * satisfies conditions (1.2).

Really considering private derivatives of 1st order the equations (3.5)* and summing, we have:

as

From the received results, on the basis of (3.3) follows

(3.10)

Then according to results of the theorem 3, functions are defined from system (3.5)* and satisfies the equation (1.2). The theorem is proved.

Remark 1. For a problem Navier-Stokes (1.1)-(1.3)*, (A₂), are proved: existence and singleness of the decision in the field of , and we will notice that the received decision (3.5)* continuously depends on the initial data.

Let's notice that the generalised decision a problems (1.1)-(1.3)*, (A₂) in area we name any continuous in equation decision (3.5)*, when

As the problem (1.1)-(1.3)*, (A₂) has the single usual decision the generalised decision coincides with this decision [11].

4. Fluid with Very Small Viscosity with a Condition (A₃)

From the received results of this point follows that system Navier-Stokes (1.1) in the conditions of (1.2), (1.3), (A₃) can have the analytical unique, is conditional-smooth decision. At least, such decision answers a mathematical question, and possibility to construct the decision on a problem Navier-Stokes (1.1)-(1.3) for an incompressible liquid with viscosity with a condition (А₃).

4.1. Fluid with Viscosity with a Condition (A₃)

I. Let initial components of a vector of speed at the moment of time it is set in a kind (1.3):

(4.1)

where known constants. Then speed components are defined by a rule

(4.2)

Hence, the system (1.1) will be transformed to a kind

(4.3)

where new unknown function which defines the decision on problem Navier-Stokes. Here substitution (4.2) it is equivalent will transform system (1.1) to the linear nonuniform equation of heat conductivity of a kind (4.3).

From system (4.3), considering conditions (4.1), (4.2), and having entered «algorithm puassonization systems», i.e. differentiating the equations of system (4.3) accordingly on х_i and, then summarising, we have the equation

(4.4)

Then the decision of a problem (4.2), (4.3) is represented in a kind

(4.5)

here - known function and

(4.6)

From the received results follows that functions are defined on the basis of (4.2), i.e.

(4.7)

Remark 2. Further, considering private derivatives of 1st order systems (4.7) and summing up with acceptance in attention (1.2), (4.2) we have, that the system (4.7) satisfies to a condition (1.2).

II. Let's notice that, as , the decision of problem Navier-Stokes (1.1), (1.2), (4.1) belongs in For this purpose it is enough to show function accessories in

(4.8)

Really, if

(4.9)

that

(4.10)

Therefore in the conditions of (4.2), (4.6) and (4.9) the equation (4.5) has a single decision in

Theorem 4. In the conditions of (1.2), (4.1), (А₃), (4.2), (4.6) and (4.9) the problem (1.1), (1.2), (4.1) has a single decision in , which is defined by a rule (4.7).

Remark 3. At performance of conditions of remark 2 and

takes place

(4.11)

And in a case , we will receive a inequality

i.e. we will receive an estimation of type Beale-Kato-Majda [2, 5].

4.2. Modified Variant of a Method (4.2) with the Condition (A₃), When

Here we will show that at certain mathematical transformations of the equation Navier-Stokes is led to a linear kind. Thus once again visually confirming [10], i.e. that really equations Navier-Stokes in a case of very small viscosity will have of the decision the obvious form really turns out.

Let's consider updating of the basic method §4.2: (4.2). Here we will see that components of speed the more any, than in rules (4.2). Therefore in this case the problem (1.1) - (1.3) does not contain in itself restriction in kinds (А₁), (А₂). And in it urgency of research in a case (А₃) consists.

Let’s for incompressible streams with a friction with a condition (А₃) also it is prospective

(4.12)

Then functions is represented in a kind

(4.13)

where known constants.

Further, supposing (4.12), (4.13) and

(4.14)

Then for incompressible currents with a friction the equations of Navier-Stokes (1.1) become simpler as take place (4.12)-(4.14). Therefore the problem (1.1)-(1.3), is led to a kind

(4.15)

From system (4.15), considering conditions (4.12)-(4.14) and having entered «algorithm puassonization systems», i.e. differentiating the equations of system (4.15) accordingly on and, then summarising, we have the equation:

(4.16)

as takes place

Then on a basis (4.16) system (4.15) it is equivalent, will be transformed to a kind:

(4.17)

or from (4.17), follows:

(4.18)

Further, differentiating (4.18) on and having entered designations

(4.19)

from (4.18) we will receive

(4.20)

Or we will transform (4.20) to a kind

(4.21)

Here (4.21) – system of the nonlinear integrated equations of Volterr-Abel of the second sort concerning on a variable t. Therefore, if takes place:

(4.22)

that on a basis (4.22) and (4.21) it is had

i.e.:

(4.23)

Therefore the solution of this system we can find on the basis of Pikard’s method

(4.24)

at that

(4.25)

From here follows

(4.26)

Theorem 5. Under conditions (1.2), (1.3), (A₃), (4.12), (4.13), (4.22), (4.26) problem Navier-Stokes has the single decision in in a kind:

(4.27)

Remarks:

I. If takes place that In a case then

(4.28)

Let's note, as (4.22) Concerns is to the equations of Volterr-Abel on a variable , discussing in language of the Volterrov equations we can find the decision. But such way from the practical point of view when it is not applicable. Therefore, an offered variant more universal in sense of the theory of operators [11, 12].

II. From the received results follows that a problem Navier-Stokes (1.1)-(1.3) in a case (4.12), (A₃) with smooth enough initial data has the conditional-smooth and single decision in a kind (4.27).

4.3. Method (4.13), When

The algorithm (4.13) also is applicable in a case, if

(4.29)

That on a basis (4.14) we will receive (4.15). Hence, we have

(4.30)

Then on a basis (4.30) system (4.15) it is equivalent, will be transformed to a kind:

(4.31)

or

(4.32)

Further, differentiating (4.32) on and having entered designation:

from (4.32) we will receive (4.21):

(4.33)

Hence, as takes place (4.22), that the solution of system (4.33) we can find on the basis of Pikard’s method. Then on a basis (4.24), (4.25) we have

(4.34)

Further, we will receive similar results in the conditions of the theorem 5, i.e. under conditions (1.2), (1.3), (A₃), (4.29), (4.34) problem Navier-Stokes has the single decision in in a kind (4.27).

4.4. Method (4.13), When

Results §4.2, §4.3 it is modified in a case when condtions:

(4.35)

are satisfied. Then considering results §4.2, §4.3 we will receive similar conclusions of theorems 5, accordingly.

Allow a condition (4.35) it is satisfied. Then from (4.13) follows

(4.36)

Therefore from system (1.1) follows

(4.37)

From system (4.37), considering conditions (4.35), (4.36) and having entered «algorithm puassonization systems», as a result we will receive

(4.38)

as takes place

Then on a basis (4.38) system (4.37) it is equivalent, will be transformed to a kind:

(4.39)

Hence

or

(4.40)

If concerning system (4.40) takes place (4.22), that the solution of system (4.40) we can find on the basis of Pkard’s method

(4.41)

Further, we will receive similar results in the conditions of the theorem 5.

Remark 4. Actually at use of offered transformations occurs linearization of equations Navier-Stokes in the integrated form without the requirement of additional conditions. Thus, the decision of the received integrated equations possesses that and properties, as the decision of initial problems. The received obvious analytical decision is regular concerning viscosity factor and in many respects simplifies carrying out of the analysis and in mathematical and physical sense. Therefore there is no necessity to copy known results of fundamental works, and it is enough to refer to them.

5. Problem Navier - Stokes with Average Viscosity with a Condition (A₃)

Area of the fluid with average viscosity with condition (A₂), when it is studied in point 3. The decision method, from where follows of equations integration of Navier-Stokes (1.1)-(1.3) in a case (А₂), it is not applicable to a problem (1.1) - (1.3) with a condition (А₃). Hence, here we will consider a methods of the equations integration of Navier-Stokes with a condition (А₃), when:

5.1. Fluid with Average Viscosity with a Condition (A₃)

Therefore, in this case, if the initial data is set in a kind:

(5.1)

we enter for definition a component of speeds:

(5.2)

at that

(5.3)

Then on the basis (5.2), (5.3) we will receive

(5.4)

Hence on a basis (5.4), we have

(5.5)

as

Then system (5.4) it is equivalent will be transformed to a kind

(5.6)

Or for consideration of unknown function we have

(5.7)

Hence, differentiating (5.7) on and having entered designation:

(5.8)

from (5.7) we will receive

(5.9)

If takes place:

(5.10)

that

(5.11)

Then the solution of this system we can find on the basis of Pikard’s method

(5.12)

at that

(5.13)

From here follows

(5.14)

Theorem 6. Under conditions (1.2), (1.3), (A₃), (5.1), (5.10) problem Navier-Stokes has the single continuous decision, which it is found by a rule (5.2).

Remarks:

I. Singleness is obvious, as a method by contradiction. Results (5.14) with a condition ((A₃), (5.1), (5.9)) are received where smoothness of functions is required only on x_i as the derivative of 1st order is in time has t>0. Then taking into account (5.14) the system (5.9) has the single continuous decision .

II. The algorithm (5.2) also is applicable in a case, if

(5.15)

That on a basis (5.15), (5.2) we will receive (5.4). Hence, we have

(5.16)

here (5.16) differs from (5.5), as Then considering (5.6)-(5.8) we will receive system (5.9):

(5.17)

Therefore, as takes place (5.10) in a consequence (5.12), (5.13) we will receive

(5.18)

Further, we will receive similar results in the conditions of the theorem 6.

5.2. Method (5.2), When

If the initial data is set in a kind

(5.19)

that on a basis (5.2) and (5.19), we have

(5.20)

Then from (1.1) follows

(5.21)

Hence on a basis (5.21), we have

(5.22)

as

Therefore system (5.21) it is equivalent will be transformed to a kind

(5.23)

where

Or considering (5.7) - (5.9) for of unknown functions

we have

(5.24)

Further, at the condition (5.10) in a consequence (5.11)-(5.13) takes place

Hence, we will receive similar results in the conditions of the theorem 6.

6. Conclusions

1. From the received results follows that system Navier-Stokes (1.1) in the conditions of (1.2), (1.3), (A₁)-(A₃) can have single analytical is conditional-smooth decision, i.e. we will receive the solution of a millennium problem [1]. At least, such decision answers to a question, and possibility to construct the decision of a problem of Navier-Stokes (1.1)-(1.3) for an incompressible fluid with viscosity [1].

2. The received results is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid equations of incompressible fluids [2, 5].

3. Results of the theorems 1, 2 and 4 are applicable in a case . And results of the theorems 3, 5, 6 can be applied to a problem of Navier-Stokes of an incompressible fluid with viscosity, when