Application of analytic functions to the global solvabilty of the Cauchy problem for equations of Navier-Stokes

The interrelation between analytic functions and real-valued functions is formulated in the work. It is shown such an interrelation realizes nonlinear representations for real-valued functions that allow to develop new methods of estimation for them. These methods of estimation are approved by solving the Cauchy problem for equations of viscous incompressible liquid.


INTRODUCTION
The work of L. Fadeyev dedicated to the many-dimensional inverse problem of scattering theory inspired the author of this article to conduct this research.The first results obtained by the author are described in the works [1][2][3].This problem includes a number of subproblems which appear to be very interesting and complicated.These subproblems are thoroughly considered in the works of the following scientists: R. Newton [4], R. Faddeyev [5], R. Novikov and G. Khenkin [6], A. Ramm [3] and others.The latest advances in the theory of SIPM (Scattering Inverse Problem Method) were a great stimulus for the author as well as other researchers.Another important stimulus was the work of M. Lavrentyev on the application of analytic functions to Hydrodynamics.Only one-dimensional equations were integrated by SIPM.The application of analytic functions to Hydrodynamics is restricted only by bidimensional problems.The further progress in applying SIPM to the solution of nonlinear equations in R3 was hampered by the poor development of the three-dimensional inverse problem of scattering in comparison with the progress achieved in the work on the one-dimensional inverse problem of scattering and also by the difficulties the researchers encountered building up the corresponding Lax' pairs.It is easy to come to a conclusion that all the success in developing the theory of SIPM is connected with analytic functions, i.e., solutions to Schrodinger's equation.Therefore we consider Schrodinger's equation as an interrelation between real-valued functions and analytic functions, where real-valued functions are potentials in Schrodinger's equation and analytic functions are the corresponding eigenfunctions of the continuous spectrum of Schrodinger's operator.The basic aim of the paper is to study this interrelation and its application for obtaining new estimates to the solutions of the problem for Navier-Stokes' equations.We concentrated on formulating the conditions of momentum and energy conservation laws in terms of potential instead of formulating them in terms of wave functions.As a result of our study, we obtained non-trivial nonlinear relationships of potential.The effectiveness and novelty of the obtained results are displayed when solving the notoriously difficult Chauchy problem for Navier-Stokes' equations of viscous incompressible fluid.

BASIC NOTIONS AND SUBSIDIARY STATEMENT
Let us consider Shrödingerse equation where  is a bounded fast-decreasing function, Definition 1. Rolnik's Class  is a set of measurable functions , ||||  = �  6
It is considered to be a general definition ( [7]).
The proof of this theorem is in [7].Consider the operators  = − Δ  + (),  0 = − Δ  defined in the dense set  2 2 ( 3 ) in the space  2 ( 3 ).The operator  is called Schrodinger's operator.Povzner [8] proved that the functions  ± (, ) form a complete orthonormal system of eigenfunctions of the continuous spectrum of the operator , and the operator fills up the whole positive semi-axis.Besides the continuous spectrum the operator  can have a finite number  of negative eigenvalues Denote these eigenvalues by −  2 and conforming normalized egenfunctions by   (, −  2 )( = 1, ), where   (, −  2 ) ∈  2 ( 3 ).

Theorem 2 (About Completeness).
For any vector-function  ∈  2 ( 3 ) and eigenfunctions of the operator , we have Parseval's identity 2 , where   and  ̅ are Fourier coefficients in case of discrete of and continuous spectrum respectively.
The proof of this theorem is in [8].
As well as in [7] we formulate.
The proof of the theorem is in [7].

Theorem 5. (Conservation Law of Impulse and Energy). Assume that 𝑞𝑞 ∈ 𝑹𝑹, then
where  is anunit operator.
The proof is in [5].

REPRESENTATION OF FUNCTIONS BY ITS SPHERICAL AVERAGES
Let us consider the problem of defining a function by its spherical average.This problem emerged in the course of our calculation and we shall consider it hereinafter.Let us consider the following integral equation where ,  ∈
Theorem 9. Suppose that  ∈ ,  ± | =0,=0 = 0; then the functions 2 =  ± | =0 are coincided according to the class of analytical functions, coincide with bounded derivatives all over the complex plane with a slit along the positive semi axis.Lemma 2. There exists 0 < || < ∞ such that it satisfies the following condition  + | =0,=0 = 0 holds for the potential of the form  = , where  ∈ .Now, we can formulate Riemann's problem.Find the analytic function  ± that satisfies (5), (6) and its solution is set by the following theorem.

NONLINEAR REPRESENTATION OF POTENTIAL
Let us proceed to the construction of potential nonlinear representation.

THE CAUCHY PROBLEM FOR NAVIER-STOKES' EQUATIONS
Let us apply the obtained results to estimate the solutions of Cauchy problem for Navier-Stokes' set of equations in the domain of   =  3 × (0, ).With respect to  0 , assume Problem (11), ( 12), (13) has at least one weak solution (q,p) in the so-called Leray-Hopf class, see [3].
Let us mention the known statements proved in [10].
Theorem 16.Suppose that then there exists a unique weak solution of problem (11), ( 12), (13), in   Our goal is to prove the global unicity weak solution of (11), ( 12), (13) irrespective of initial velocity and power smallness conditions.Therefore let us obtain uniform estimates.
Proof.The proof follows from the definition of Fourier transformation and the formulas for linear differential equations.
Lemma 17.The solution of the problem (11), ( 12), (13) from Theorem 16, satisfies the following equation and the following estimates Proof.We obtain the equation for  using  and Fourier transformation.The estimates follow from the obtained equation.
This completes the proof of Lemma 17.
Proof.Let us prove the first estimate.These inequalities where  1 = .Follows from definition (2) for the average of  and from Lemmas 18, 19.
The rest of estimates are proved similarly.
This completes the proof of Lemma 21.

Using the denotation
Taking into account Holder's inequality in  we obtain where ,  satisfies the equality Taking into consideration the estimate  in (16), we obtain the statement of the lemma.
This proves Lemma 23.Now, we have the uniform estimates of Rolnik norms for the solution of problems (11), ( 12), (13).Our further and basic aim is to get the uniform estimates |  � |  1 ( 3 ) , a component of velocity components in the Cauchy problem for Navier-Stokes' equations.In order to achieve the aim, we use Theorem 8 it implies to get estimates of spherical average.Lemma 24.Weak solution of problem (11), ( 12), (13), from Theorem 16 satisfies the following inequalities (15) in the integral, we obtain Let us use the notation Let us use the notation and obtain  0 .Since where  is the angle between the unit vectors   ,   , it follows that Using  0 in the estimate | � mv |  1 ( 3 ) , we obtain the statement of the lemma.
This completes the proof of Lemma 24.
Proof.By the definitions  and  0 , we have This proves Statement 2.
Lemma 25.Weak solution of problem (11), ( 12), (13), from Theorem 16 satisfies the following inequalities where Proof.The underwritten inequalities follows from representation (14) Let us introduce the following denotation where  > 0 we obtain On applying Holder's inequality, we get , where ,  satisfy the equality For  =  = 2 we have Inserting  1 ,  2 in to � ∂ � ∂ �, we obtain the statement of the lemma.
This completes the proof of Lemma 25.
Proof.From the statement of Theorem 14 we get the following estimate .
Let us introduce the following denotation .
The estimate of  1 was obtained in theorem 16, therefore from (15), (18), it follows that .
Inserting inequality (19) into  2 , we get Let us take into account the estimate of  0 obtained in Lemma 25, Inserting this value in  2 , we obtain | � mv (, )|.
Let us introduce the following denotation where Using inequality (16) in  3 , we get Similarly as we estimated  2 , obtain , where , we obtain the statement of the theorem.This completes the proof of Theorem 18.
Lemma 26.Weak solution of problem (11), ( 12), ( 13), from Theorem 16 satisfies the following inequalities where Proof.From (14) we have the following inequalities Let us introduce the following denotation Using Holder's inequality , where ,  satisfy the equality For  =  = 2 we get Taking into account Holder's inequality for  3 , we get ∂ 2 �, we get the statement of the lemma.
This completes the proof of Lemma 26.
Proof.From the statement of Theorem 14 we have the estimate .
Let us use the estimates for  1 ,  2 Let us use inequality (19) to estimate  3 , then we get where and  is an exponent of .From this recurrence formula, as  = 0,  = −1, we get estimates (17) and (18) accordingly.
For  = 1 we have Considering estimates (17), (18) and the last estimate, we obtain the statement of the lemma.
Proof.From (22), we obtain the following estimate .
Using estimates (23)-(28) in the last inequality, we obtain the statement of the lemma.
This proves Lemma 29.Then there exists a unique weak solution of (11), ( 12), (13), satisfying the following inequalities We have (  ) < 1, i.e., it is not necessary to take into account normalization numbers when proving the theorem.Now the statetement of the theorem follows from Theorems 20, 17, Lemmas 21, 30, 31 and the conditions of Theorem 21, that give uniform of velocity maxima at a specified interval of time.
1 (3 ) ≤ , where  depends only on the theorem conditions.Proof.It is sufficient to get uniform estimates of the maximum   to prove that the theorem .These obviously follow from the estimate |  � |  1 ( 3 ) .Uniform estimates allow to extend the rules of the local existence and unicity local to an interval, where they are correct.To estimate the component of velocity, we use statement 2 Let us consider the problem of estimating the maximum of amplitude, i.e., max ∈ 3 |(, )|.Let us estimate the  term of Born's series |  (, )|.