1. Introduction

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

that for each value turns out the corresponding scalar equation of Navier-Stokes

(1.1)

with conditions

(1.2)

(1.3)

> 0 - kinematic viscosity, - density, Δ - Laplas’s operator. The additional equation is the condition incompressibility fluid (2). Unknown are speed and pressure P.

The work purpose. The main object of this work - existence and proofs of single and conditional smoothness of the decision of a problem Navier-Stokes for an incompressible fluid with viscosity.

Theoretical and practical value. 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 [2-4, 8-10]. The Received decisions on the basis of the developed analytical methods proves in the general applicability of the equations of Navier-Stokes.

In a case the current is considered with very small viscosity. When the current is considered with very small viscosity i.e. when Reynolds's number is very great [8,9] there is an border layer in which viscosity influence is concentrated. In many works in this area of the decision of the equations Navier-Stokes received by the numerical analysis, also confirm these conclusions.

And in a case the current is considered with average size of viscosity. At very slow currents, or in currents of is strong-viscous liquids of force of a friction much more, than forces of inertia. Hence convective the acceleration doing the equations nonlinear, everywhere are supposed identically equally to a zero [9]. 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.

The decision of many problems of theoretical and mathematical physics leads to use of various special weight spaces. In works [5-7] for the first time have offered a method which gives solution of problem Navier-Stokes in

is continuous and limited functions on

and

(1.4)

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

(1.5)

(1.6)

(1.7)

(1.8)

without breaking equivalence of system (1.1) and (1.5), (1.6). The received systems (1.5), (1.6) contain unknown persons , and pressure P. Here - known functions because are known .

The developed method of the decision of systems (1.5) and (1.6), is connected with functions , i.e.

A) ; or

B) - any functions if, accordingly, as necessary conditions, take place:

а₀) b₀) - any functions.

2. A Problem of Navier-Stokes with a Condition (А)

In this paragraph in the subsequent points, at the specified restrictions on the entrance data, the strict substantiation of compatibility of systems (1.5), (1.6) will be given.

2.1. Research With a Condition (A)

Let functions satisfy to a condition (a₀). Then relatively we suppose a condition (A) and

(2.1)

where from system (1.5) and (1.6), accordingly we will receive following systems

(2.2)

(2.3)

Theorem 1. Let conditions (1.2), (1.3), (A) 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, we will receive the equation of Puasson [10]

(2.5)

as

At that it is proved

(2.6)

(2.7)

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

In work of Sobolev [10] it is specified that function (2.6) satisfies to the equation (2.5) and is called Newtons’ 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. The equations (2.5), (2.9) is there are first and second equations of system (2.4).

The system (2.9) is solved by S.L.Sobolev’s method:

(2.10)

where

or

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 (2.6), follows

(2.14)

i.e. functions are defined from systems (2.10), (2.13), (2.14).

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

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), (A), (2.1) and

(2.16)

Really, estimating (2.10) in ,we have

(2.17)

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

3. The Decision of a Problem of Navier-Stokes with a Condition (B)

Here we investigate a case (B) when containing convective members of a problem of Navier-Stokes are any. Results of the theorem 1 are not applicable.

Therefore, for the decision of a problem (1.1) - (1.3) we offer following algorithms.

3.1. Problem Navier- Stokes with Average Viscosity

Let conditions (1.2), (1.3) are satisfied and:

(3.1)

at that

(3.2)

Then on the basis of functions and

system (1.1) it is equivalent will be transformed to a kind

(3.3)

In the specified systems unknown persons contain .

Remark 1. Under regular in

the decision we understand the decision the equation (1.1) in, which has a continuous derivative on to the third order inclusive and continuous derivative on t(t>0).

From system (3.3), considering conditions (3.2), and having entered «algorithm puassonization systems», i.e. differentiating the equations of system (3.3) accordingly on and, then summarising, we have the equation:

(3.4)

Therefore, we will receive

(3.5)

Really on a basis [(3.3): ], we have

Then we have the following (3.4).

Further, we have

(3.6)

or for consideration of unknown functions we have

(3.7)

as here consider a method integration in parts

and

If functions are system decisions (3.7) thus takes place (3.2) and

(3.8)

that :

(3.9)

Then the solution of this system (3.7) we can find on the basis of Pikard’s method

(3.10)

where - initial estimates and at that

(3.11)

Theorem 3. Under conditions (2), (3), (3.2), (3.8) problem Navier-Stokes has the unique continuous decision.

Definition 1. The generalised decision a problems (1.1)-(1.3), (3.2) in area we name any continuous in

equation decision (3.7), when

3.2. We will Consider a Fluid with Very Small Viscosity

Let conditions (1.2), (1.3), (3.1) are satisfied and:

(3.12)

at that

Then on the basis of functions , system (1.1) it is equivalent will be transformed to a kind:

(3.13)

in the specified systems unknown persons contain

Therefore

(3.14)

as

Theorem 4. Let functions are system decisions

(3.15)

Then

(3.16)

where known functions and

Theorem 5. If functions are system decisions (3.14), (3.15), (3.10) that (3.1) is the decision of system (1.1) in :

(3.17)

Remark 2. In particular, if conditions (1.2), (1.3), (3.12) and

(3.18)

are satisfied. Then takes place

(3.19)

From here, considering (3.14), (3.15) we have

(3.20)

Hence all conditions of the theorem 4 and 5 are satisfied.

Example 1. The specified method of the theorem 5 can be used in par ticular and for the decision on a problem (1.1)-(1.3) as a test example, when takes place:

that

(3.21)

or

Then

Remark 3. As , that limitedly the decision of a problem of Navier-Stokes (1.1) - (1.3) with a condition (B) it is possible to prove limitation and in - weight space of type of Sobolev:

as

4. Conclusions

I. From the received results follows that system Navier-Stokes (1.1) in the conditions of (1.2), (1.3), (A₁)-(A₃) can have the analytical unique is conditional-smooth decision. At least, such decision answers to a mathematical question, and possibility to construct the decision of a problem of Navier-Stokes (1.1)-(1.3) for an incompressible liquid with viscosity.

II. Results of the theorem 1 and 5 can be applied to a problem of Navier-Stokes of an incompressible fluid with viscosity, when .