Navier-Stokes equations-Millennium Prize Problems

In this work we present final solving Millennium Prize Problems formulated Clay Math. Inst., Cambridge in [1] Before this work we already had first results in [2]-[4]. The Navier-Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier-Stokes equations. These equations describe the motion of a fluid in space. Solutions to the Navier-Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier-Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics. Even much more basic properties of the solutions to Navier-Stokes have never been proven. For the threedimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. This is called the Navier-Stokes existence and smoothness problem. Since understanding the Navier-Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics.


Introduction
In this work we present final solving Millennium Prize Problems formulated Clay Math.Inst., Cambridge in [1] Before this work we already had first results in [2]- [4].The Navier-Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier-Stokes equations.These equations describe the motion of a fluid in space.Solutions to the Navier-Stokes equations are used in many practical applications.However, theoretical understanding of the solutions to these equations is incomplete.In particular, solutions of the Navier-Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics.Even much more basic properties of the solutions to Navier-Stokes have never been proven.For the threedimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.This is called the Navier-Stokes existence and smoothness problem.Since understanding the Navier-Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics.
In this paper, we introduce important explanations results presented in the previous studies in [2]- [4].We therefore reiterate the basic provisions of the preceding articles to clarify understanding them.First, we consider some ideas for the potential in the inverse scattering problem,and this is then used to estimate of solutions of the Cauchy problem for the Navier-Stokes equations.
A similar approach has been developed for onedimensional nonlinear equations [5]- [8], but to date, there have been no results for the inverse scattering problem for three-dimensional nonlinear equations.This is primarily due to difficulties in solving the three-dimensional inverse scattering problem.
This paper is organized as follows: first, we study the inverse scattering problem, resulting in a formula for the scattering potential.Furthermore, with the use of this potential, we obtain uniform time estimates in time of solutions of the Navier-Stokes equations, which suggest the global solvability of the Cauchy problem for the Navier-Stokes equations.
Essentially, the present study expands the results for one-dimensional nonlinear equations with inverse scattering methods to multi-dimensional cases.In our opinion, the main achievement is a relatively unchanged projection onto the space of the continuous spectrum for the solution of nonlinear equations, that allows to focus only on the behavior associated with the decomposition of the solutions to the discrete spectrum.In the absence of a discrete spectrum, we obtain estimations for the maximum potential in the weaker norms, compared with the norms for Sobolev' spaces.
Consider the operators ) in the space L 2 (R 3 ), and let q be a bounded fast-decreasing function.The We consider the three-dimensional inverse scattering problem for Schrödinger's operator: the scattering potential must be reconstructed from the scattering amplitude.This problem has been studied by a number of researchers [9], [11], [12] and references therein.

Results
Consider Schrödinger's equation: Let Ψ + (k, θ, x) be a solution of (1) with the following asympotic behavior: where A(k, θ , θ) is the scattering amplitude and Let us also define the solution As is well known [9]: This equation is the key to solving the inverse scattering problem, and was first used by Newton [10], [11] and Somersalo et al. [12].Equation ( 4) is equivalent to the following: where S is a scattering operator with the kernel S(k, ł) and The following theorem was stated in [9]: Theorem 1 (The energy and momentum conservation laws) Let q ∈ R.Then, SS * = I, S * S = I, where I is a unitary operator.

Definition 2
The set of measurable functions R with the norm, defined by is recognized as being of Rollnik class.
As shown in [13], Ψ ± (k, x) is an orthonormal system of H eigenfunctions for the continuous spectrum.In addition to the continuous spectrum there are a finite number N of H negative eigenvalues, designated as −E 2 j with corresponding normalized eigenfunctions ψ j (x, −E 2 j )(j = 1, N ), where We present Povzner's results [13] below: Theorem 3 (Completeness) For both an arbitrary f ∈ L 2 (R 3 ) and for H eigenfunctions, Parseval's identity is valid.
where f and f j are Fourier coefficients for the continuous and discrete cases.
Theorem 4 (Birmann-Schwinger estimation).Let q ∈ R.Then, the number of discrete eigenvalues can be estimated as: This theorem was proved in [14].
Let us introduce the following notation: where We define the operators T ± , T for f ∈ W 1 2 (R) as follows: Consider the Riemann problem of finding a function Φ, that is analytic in the complex plane with a cut along the real axis.Values of Φ on the sides of the cut are denoted as Φ + , Φ − .The following presents the results of [15]: Theorem 6 Let q ∈ R, g = (Φ + − Φ − ).Then , The proof of the above follows from the classic results for the Riemann problem.
The proof of the above follows from the definitions of g, Φ ± , Ψ ± .
Lemma 8 Let q ∈ R, Then The proof of the above again follows from the definitions of the functionsg, Φ ± , Ψ ± .
Lemma 9 Let q ∈ R.Then, The proof of the above follows from the definitions of g, Φ ± , Ψ ± and Theorem 1.
The proof of the above follows from the definitions of g, Φ ± , Ψ ± and Lemma 9 and dispersions relations for analytics functions.The proof of the above follows from the definitions of D, T − and the conditions of Lemma 13.
Lemma 14 Let q ∈ R, and assume that (I −T ± D) −1 exists.Then, The proof of the above follows from the definitions of g, Φ ± , Ψ ± and equation (4).
Lemma 15 Let q ∈ R, and assume that (I −T ± D) −1 exists.Then , where Q 3 represents terms of highest order of T − D.
Proof: Using and ( 18) we get proof.
Lemma 16 Let q ∈ R.Then, The lemma can be proved by substituting Ψ ± into equation (1).
Lemma 17 Let q ∈ R, and assume that (I −T − D) −1 exists.Then, The proof of the above follows from the definitions of N, Ψ ± and Lemma 14.
The proof of the above follows from the definition of D and the unitary nature of S.
The proof of the above follows from the definitions of E 2 j , ψ j and (1).
Lemma 20 Let q ∈ R ∩ L 2 (R 3 ), and Then, To prove this result, one should calculate Ψ ± using (18) Using the notation that: Then, To prove this result, one should Ψ ± using Lemma14 Lemma 22 Let q ∈ R ∩ L 2 (R 3 ), and Then, To prove this result, one should calculate A using Lemma14.
Lemma 23 Let q ∈ R, max k |q| < ∞.Then, A proof of this lemma can be obtained using Plancherels theorem.

Cauchy problem for the Navier-Stokes equation
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 [16]- [20].The spatial differentiability of the solutions is an important factor, this controls their evolution.Obviously, differentiable solutions do not provide an effective description of turbulence.Nevertheless, the global solvability and differentiability of the solutions has not been proven, and therefore the problem of describing turbulence remains open.It is interesting to study the properties of the Fourier transform of solutions of the Navier-Stokes equations.Of particular interest is how they can be used in the description of turbulence, and whether they are differentiable.The differentiability of such Fourier transforms appears to be related to the appearance or disappearance of resonance, as this implies the absence of large energy flows from small to large harmonics, which in turn precludes the appearance of turbulence.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: where div q = 0, in the domain Q T = R 3 × (0, T ), where : The problem defined by (34), ( 35), (36) has at least one weak solution (q, p) in the so-called Leray-Hopf class [16].
The following results have been proved [17]: there is a single generalized solution of (34), ( 35), (36) in the domain satisfying the following conditions: Note that T 1 depends on q 0 and f .
Our goal is to provide global estimations for the Fourier transforms of derivatives of the Navier-Stokes equations' solutions (34), ( 35), (36) without the that the smallness of the initial velocity and force are small.We obtain the following uniform time estimation.
Proposition 27 The solution of (34), ( 35), (36) according to Theorem 25 satisfies: where This follows from the definition of the Fourier transform and the theory of linear differential equations.
Proposition 28 The solution of (34), ( 35), (36) satisfies: and the following estimations: This expression for p is obtained using div and the Fourier transformpresentation.
Lemma 29 Let Then, the solution of (34), (35), (36) in Theorem 25 satisfies the following inequalities: Proof this follows from the a priori estimation of Lemma 26 and conditions of Lemma 29.
Then, the solution of (34), (35), (36) in Theorem 25 satisfies 2 the following inequalities: Proof this follows from the a priori estimation of Lemma 26 and conditions of Lemma 30 Lemma 31 The solution of (34), ( 35), (36) in Theorem 25 satisfies the following inequalities: Proof this follows from the a priori estimation of Lemma 26, conditions of Lemma 31, the Navier-Stokes equations.
Lemma 32 The solution of (34), ( 35), (36) satisfies the following inequalities: Proof this follows from the a priori estimation of Lemma 26, conditions of Lemma 32, the Navier-Stokes equations.
Lemma 36 Weak solution of problem (34), (35), (36) from Theorem 25 satisfies the following inequalities where where F 1 = (q, ∇)q + F. Using the denotation taking into account Holder's inequality in I we obtain , where p, q satisfies the equality 1 p + 1 q = 1.Suppose p = q = 2. Then Taking into consideration the estimate I in (53), we obtain the statement of the lemma.This proves Lemma 36.
Lemma 37 Weak solution of problem (34), ( 35), (36), from Theorem 25 satisfies the following inequalities ∂Qq ∂z where On applying Holder's inequality, we get , where p, q satisfy the equality 1 p + 1 q = 1.For p = q = 2 we have Inserting I 1 , I 2 in to ∂ q ∂z , we obtain the statement of the lemma.This completes the proof of Lemma 37.

Conclusions
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.

Definition 11
Denote by T A the set of functions f (k, θ, θ ) with the norm ||f || T A = sup θ,k,θ (|T f | + |f |) < ∞.Definition 12 Denote by R (I−T − D) the set of functions g such that g = (I − T − D)f for any f ∈ T A. Lemma 13 Suppose ||A|| T A < α < 1.Then, the operator (I − T − D), defined on the set T A has an inverse defined on R (I−T − D) .
Lemma 35 Suppose that q