EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 763-785 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Global estimation of the Cauchy problem solution’ and blow up the Navier-Stokes equation Asset Durmagambetov1 1 Faculty of Mathematiks, L.N.Gumilyov Eurasian National University, Kazakhstan Abstract. The paper presents results of the research of gradient catastrophe development during phase change. It shows that classical methods of the function estimation theory do not fit well to study gradient catastrophe problem. The paper presents results, indicating that embedding theorems do not allow to study a process of a catastrophe formation. In fact, the paper justifies Terence Tao’s pessimism about a failure of modern mathematics to solve the Navier-Stokes problem. An alternative method is proposed for dealing with the gradient catastrophe by studying Fourier transformation for a function and selecting a function singularity through phase singularities of Fourier transformation for a given function. The analytic properties of the scattering amplitude are discussed in R3, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided.Describes the loss of smoothness of classical solutions for the Navier-Stokes equations -Millennium Prize Problems. Key Words and Phrases: Schrödinger’s equation; potential, scattering amplitude, Cauchy problem, Navier–Stokes equations, Fourier transform, the global solvability and uniqueness of the Cauchy problem,the loss of smoothness,The Millennium Prize Problems 1. Introduction The research presents a process of gradient catastrophe formation under conditions of phase change. The paper shows that classical methods of the function estimation the- ory in context of Sobolev- Schwartz Space Theory are not suitable for studying gradient catastrophe problem. Results which are presented here show that the embedding theorems do not allow to study a process of a catastrophe formation. Actually, the paper justifies Terence Taos pessimism about a failure of using present mathematical methods for solv- ing the Navier-Stokes problem. An alternative method is proposed for studying gradient catastrophe by applying Fourier transformation to a function and selecting function sin- gularity through phase singularities of Fourier transformation for a given function. We know a general definition of a gradient catastrophe - an unbounded increase of a function derivative upon conditions of boundedness of the function itself. This phenomenon occurs Email address: aset.durmagambet@gmail.com (A. Durmagambetov) http://www.ejpam.com 763 c© 2017 EJPAM All rights reserved. A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 764 in various problems of hydrodynamics, such as a formation of shock waves, weather fronts, hydraulic and seismic fracturing, and others. In modern physics and mathematics, as well as in many other areas of science and technology, this phenomenon is considered as a very difficult problem, both from a theoretical and applied perspective. From a theoretical point of view this is important as we have to know how to describe qualitative changes in processes, which are manifested in appearance of new quality objects during a process of description model evolution, and in the context of applied research, the problem is facing numerical instability in the event of a gradient catastrophe formation. Thus, we approach an important obstacle while using modeling - a barrier created by the gradient catastrophe. Since, on the one hand, the gradient catastrophe is still unknown phenomenon, it is very important from a practical point of view, because the phenomenon is connected with the most interesting and important aspects of reality. Terence Tao formulated and illustrated this in [1] based on the Millennium problem stated by Clay Institute for the Navier-Stokes equations. Our point of view on these issues agrees with one, stated in article [1],[5],[6] but in our research we propose a way for solving these problems. Our point of view is that the modern mathematical methods of the theory of functions dedicated to the function estimation have ignored such an important component of the Fourier transformation as its phase. Our research is outlined as follows: first, we give examples of the gradient catastro- phe caused by the phase change, and then proceed to an expansion of classes of functions subjected to the gradient catastrophe. Our final results lie in the nonlinear representation of functions showing some new classification of functions through a phase classification. In addition, the notions of discreteness and continuity of functions are naturally merged. And, in our opinon, this leads to understanding of how discrete objects are born under a continuous change of the world. Discrete objects are associated with discrete spectrum of the Liouville- Schrdinger equations. And they, as it is known, reflect the wave nature of things. But here, we abstract away from the quantum formalism, because our goal lies in a purely mathematical approach to the analysis of the arbitrary functions. For the analysis of which, we formally consider a function as a potential of the Schrdinger equation. At the same time we come across the concepts that generated by the Liouville- Schrdinger equations. These concepts allow to classify and estimate functions by a phase generated by discrete spectrum of the Liouville equation. 2. Results for the one-dimensional case Let us consider one-dimensional function f and its Fourier transformation f̃ . Using notions of module and phase, we write Fourier transformation in the following form f̃ = |f̃ | exp(iφ) , where φ is phase. To cite Plancherel equality: ||f ||L2 = Const||f̃ ||L2 . Here we can see that a phase is not contributed to determination of X norm. To estimate a maximum we have a simple estimate as max|f |2 ≤ 2||f ||L2 ||∇f ||L2 .Now we have an estimate of the function maximum in which a phase is not involved. Let us consider a behavior of a progressing wave running with a constant velocity of v = a described by function F (x, t) = f(x+ at). For its Fourier transformation along x variable we have F̃ = f̃ exp(iatk). Again in this case we can see that when we will be studying a module A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 765 of the Fourier transformation, we will not obtain major physical information about the wave, such as its velocity and location of the wave crest because of |F̃ | = |f̃ | . These two examples show w eaknesses of studying Fourier transformation. On the other hand, many researchers focus on the study of functions using embedding theorem, but in the embedding theorems main object of the study is module of function. But as we have seen in given examples, a phase is a main physical characteristic of a process, and as we can see in the mathematical studies, which use embedding theorems with energy estimates, the phase disappears. Along with phase, all reasonable information about physical process disappears, as demonstrated by Terence Tao [1] and other research considerations. In fact, he built progressing waves that are not followed energy estimates. Let us proceed with more essential analysis of influence of the phase on behavior of functions. Theorem 1. There are functions of W 1 2 (R) with a constant rate of the norm for a gradient catastrophe of which a phase change of its Fourier transformation is sufficient. Proof. To prove this, we consider a sequence of testing functions f̃n = ∆/(1+k2),∆ = (i − k)n/(i + k)n. it is obvious that |f̃n| = 1/(1 + k2). max|fn|2 ≤ 2||fn||L2 ||∇fn||L2 ≤ Const.. Calculating the Fourier transformation of these testing functions, we obtain: fn = x(−1)(n−1)2π exp(−x)L1 (n−1)(2x) where L1 (n−1)(2x) is a Laguerre polynomial. Now we see that the functions are equibounded and derivatives of these functions will grow with the growth of n. Thus, we have built an example of a sequence of the bounded functions of W 1 2 (R) which have a constant norm W 1 2 (R) and this sequence converges to a discontinuous function. Thus, we have demonstrated an importance of the phase and that the phase is not involved into energy norms that are inherent to the mathematical arguments used in physical processes analysis. Our next goal is to maximally expand this class of functions in which a phase is important. Our goal is also to use a phase, which appears in the inverse scattering problem; moreover we will be interested mainly in a phase generated by a discrete spectrum of the Liouville equation. Thereby, we come now to an important subject of our research, such as an occurrence of discontinuities, fronts and other instable states in numerical modeling and which are at the same very stable physical objects. Theorem 2. There are functions of W 1 2 (R) with a constant rate of the norm for a gradient catastrophe of which a phase change of its Fourier transformation is sufficient. Proof. To prove this, we consider a sequence of testing functions f̃n = ∆/(1+k2),∆ = (i − k)n/(i + k)n. it is obvious that |f̃n| = 1/(1 + k2). max|fn|2 ≤ 2||fn||L2 ||∇fn||L2 ≤ Const.. Calculating the Fourier transformation of these testing functions, we obtain: fn = x(−1)(n−1)2π exp(−x)L1 (n−1)(2x) where L1 (n−1)(2x) is a Laguerre polynomial. Now we see that the functions are equibounded and derivatives of these functions will grow with the growth of n. Thus, we have built an example of a sequence of the bounded functions of W 1 2 (R) which have a constant norm W 1 2 (R) and this sequence converges to a discontinuous function. A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 766 Thus, we have demonstrated an importance of the phase and that the phase is not involved into energy norms that are inherent to the mathematical arguments used in physical processes analysis. Our next goal is to maximally expand this class of functions in which a phase is important. Our goal is also to use a phase, which appears in the inverse scattering problem; moreover we will be interested mainly in a phase generated by a discrete spectrum of the Liouville equation. Thereby, we come now to an important subject of our research, such as an occurrence of discontinuities, fronts and other instable states in numerical modeling and which are at the same very stable physical objects. As we think, our arguments are very important in issues of plasma stability in nuclear fusion technology, since the gradient catastrophe formation serves as a preamble to a process of nuclear fusion stop. To build more in-depth analysis we apply results of scattering theory to our problem. For this, we consider a spectral problem for the Liouville equations with a potential q that satisfies and belongs to M space of functions with the following norm ||q||M = +∞∫ −∞ |q(x)|(1 + |x|)dx As it is known from −Ψ” + qΨ = |k|2Ψ, k ∈ C (1) with the following asymptotics: lim x→−∞ Ψ1(k, x) = eikx + s12(k)e−ikx, lim x→+∞ Ψ1(k, x) = s11(k)e−ikx (2) lim x→−∞ Ψ2(k, x) = s22(k)e−ikx, lim x→+∞ Ψ2(k, x) = e−ikx + s11(k)eikx (3) It is also known from the theory of equations [2], that any solution is a combination of some fundamental solutions satisfying certain boundary conditions. lim x→∞ f1(k, x)e−ikx = 1, lim x→−∞ f2(k, x)eikx = 1. (4) It is known [2], that they satisfy the following equations: f1(k, x) = eikx − +∞∫ −∞ G1(k, x, t)q(t)f1(k, t)dt, (5) f2(k, x) = e−ikx + +∞∫ −∞ G2(k, x, t)q(t)f1(k, t)dt, (6) (7) E+(k, x) = eikx E−(k, x) = e−ikx, (8) G1(k, x, t) = −θ(x− t)sin(k(x− t)) k , G2(k, x, t) = θ(x− t)sin(k(x− t)) k , (9) f1 = E+ − ∞∑ j=1 Gj1E+, f2 = E− + ∞∑ j=1 Gj2E−, (10) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 767 Let us also provide known results for the scattering coefficients and fundamental solutions outlined in [2]. u+ 1 (k, x) = s12f2(k, x), u+ 2 (k, x) = s11f1(k, x), u−1 (k, x) = u+ 1 (k, x), u+ 2 (k, x) = u+ 2 (k, x), (11) s11s ∗ 12 + s12, s ∗ 22 = 0, s2 11 + s2 12 = s2 22 + s2 21 = 1, si,j(−k) = s∗i,j(k), (12) lim |k|→∞ s12 = s21 = 1 +O(1/|k|), , lim |k|→∞ s11 = s22 = O(1/|k|), (13) s11 = exp( 1 2πi +∞∫ −∞ ln(1− |s12|) k′ − k dk ′ n∏ j=1 ( iEj + k k − iEj ) dk ′ , (14) s11(k) = lim ε→0 = s11(k + iε), s21(k) = −s12(−k)s11(k) s11(−k) (15) s21(k) = 1 2ki +∞∫ −∞ exp(ikt)q(t)f2(k, t)dt 1− 1 2ki +∞∫ −∞ exp(ikt)q(t)f2(k, t)dt , s12(k) = 1 2ki +∞∫ −∞ exp(−ikt)q(t)f1(k, t)dt 1− 1 2ki +∞∫ −∞ exp(ikt)q(t)f1(k, t)dt , (16) b(k) = 1 2ki +∞∫ −∞ exp(−ikt)q(t)f1(k, t)dt, a(k) = 1− 1 2ki +∞∫ −∞ exp(ikt)q(t)f1(k, t)dt. (17) Now we are able to return to our question of the gradient catastrophe for more general class of functions. For this we consider Liouville equation and a sequence of inverse scattering problems with constant in module scattering coefficients si,j , where discrete eigenvalues Ei, 0 < i < n+ 1 such that lim n→∞ = E∞ Theorem 3. There are potentials from W 1 2 (R)M with the constant norm of W 1 2 (R)M for the gradient catastrophe for which existence of limit point for the discrete spectrum with given potential in the Liouville equation is sufficient. Proof. Following notations [2], we introduce functions A+, B+,Ω+ according to the formulas: s21(k) = +∞∫ −∞ A+(t) exp(2ikt)dt, Ω+(t) = n∑ i=1 M1 j exp(−Ejt) +A+(t) (18) B+(x, y) + +∞∫ 0 B+(x+ y + t)Ω+(x+ y + t)dt+ Ω+(x+ y) = 0 (19) where M1 j are normalized numbers. In other words, we will consider inverse problems of the potential recovery, and for the n-th potential we will consider a case with an accuracy A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 768 up to n discrete eigenvalues. It is sufficient to consider a first approximation of these equations. In a first approximation, the n-th potential is recovered by the equation for B+(x, y) and also in a first approximation. We have the following arguments for the first approximation d dx B+(x, x) = − d dx Ω+(2x). (20) d dx Ω+(2x) = n∑ i=1 −EjM1 j exp(−Ejt) + d dx A+(x) (21) For the last term, we also consider a first approximation d dx A+(x) = +∞∫ −∞ q̃+(2t) exp(2ixt)δ(k)2dt, (22) δ(k) = n∏ j=1 ( iEj + k k − iEj ) ∗ exp( 1 2πi V p +∞∫ −∞ ln(1− |s12|)) k′ − k dk ′ ) (23) To prove this, let us consider a sequence of d dxA+(x)with n going to infinity and under a proper selection of scattering coefficients, we fall into conditions of the Theorem 1. Let us come down from specific obvious examples to more systematic analysis of the gradient catastrophe. In given below all our arguments will be based on well-known equation: s21(k) = −s12(−k)s11(k)/s11(−k) Let us consider s21, s12 in the following form: 2iks12(k) = q̃(2k) + I1(k), 2iks21(k) = q̃(−2k) + I2(k). (24) Let us conceive q̃(2k) = u(k) + iv(k). Then we will have the following equation for u, v u(k) + iv(k) = 2iks12(k)− I1(k), u(k)− iv(k) = 2iks21(k)− I2(k), (25) s11(k) s11(−k) = exp(2iδ(k)), δ(k) = arg(s11(k)). (26) Now we can formulate the following theorem. Theorem 4. The following equations are true u(k) = (1 + cos(2δ(k)))R1 + sin(2δ(k))R2 sin(2δ(k)) , v = (−1 + sin(2δ(k)))R1 + (1− cos(2δ(k)))R2 sin(2δ(k)) (27) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 769 Proof. Using equation (26) and representation for Fourier transformation we obtain Whence, solving the equation for u and v, we obtain u(k) + iv(k) + I1(k) = (u(k)− iv(k) + I2(k))(cos(2δ(k)) + i sin(2δ(k))), (28) from last equation we have u(k)(1− cos(2δ(k)))) + v(k)(1− sin(2δ(k))) = R1, (29) −u(k) sin(2δ(k)) + v(k)(1 + cos(2δ(k))) = R2 (30) where R1 = Real(−I1 + I2 cos(2δ(k)) + i(I1 sin(2δ(k)))), (31) R2 = Im(−I1 + I2 cos(2δ(k)) + i(I1 sin(2δ(k)). (32) Theorem 5. If δ(k)(k) = 0, |q̃(k)| < C <∞ then R1(k) = 0. Proof. using equation (30-31) and conditional theorem we obtain proof. Theorem 6. The following estimates are true for Fourier transformation |u| < C(|R1|+ |R2|+ |∇R1|), (33) |v| < C(|R1|+ |R2|+ |∇R1|), (34) |̃q| < C(|R1|+ |R2|+ |∇R1|). (35) Proof. follows from the representation of u, v. Here, we just point out this as a separate theorem in order to emphasize the significance of this result. We note separately the terms with a derivative ∇R1. Obviously, these terms are appeared due to points of the phase nulling. Theorem 7. For estimation of a maximum of the potential the following estimates are true |q| < C +∞∫ −∞ (C(|I1|+ |I2|+ |∇I1|+ |∇I2|))dk (36) Proof. follows from the estimation of u, v and use ofR1, R2 which are simple arguments. Here we outline the theorem in order to emphasize importance of this result for 3- dimentional case. Analyzing the last formula, we see again an effect of the phase on the function behavior. In addition, a finiteness of the discrete spectrum is the main requirement of the gradient catastrophe nonoccurrence. And from other hand, in case of A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 770 unconstrained growth of points in discrete spectrum, we fall into the terms of theorems 1 and 2. The last theorem expands a class of functions described in in Theorem 1, as we planned at the beginning. Now, studying the behavior of a gradient depending on q we come to the conclusion that its unconstrained growth will be dictated by the phase cluster point, which, in its turn, is due to discrete spectrum acquisition. Hence we get the most important conclusion - we get information about the catastrophe with discrete jumps! Theorem 8. For a potential the following representation is true q = Q(q, E1, ...En); (37) Proof. just consists in calculating I1(k), I2(k) in a form of series of q and substitution of a result of the calculation into the formula for u, v moreover a right side of the obtained formula contains second-order terms only. This representation, in contrast to the classical inverse problems, allows using arbi- trary information on the potential for closure of these equations, because a skeleton of this integral equation is represented by sets of constants in the form of eigenvalues. One of the surprising properties of this representation and all this research is discreteness in continuity. Since a value of the phase, as we can see, changes discontinuously, while a potential-function itself may vary continuously. This implies an important conclusion about the instability in numerical methods, i.e. it is necessary to control phase jumps in numerical modeling to avoid falling into a state of instability. A conclusion of non- scalability of such models is critically important since eigenvalues may appear or may disappear under changes in the potential scale, whereupon a model will be changed signif- icantly. This theorem shows that we have obtained fundamentally new nonlinear integral relations that allow taking a fundamentally fresh look at the problem of estimating func- tions. Now, instead of integral representations, that generate embedding theorem in the Sobolev spaces and by which numerous outstanding achievements in modern mathematics have been gained, we turn to the newest non-linear integral relations and hope thereby opening up new pages of mathematics that will take us further into the wonderful world of mathematics. 3. Introduction for the three-dimensional case In this work we present final solving Millennium Prize Problems formulated Clay Math. Inst., Cambridge in [3] Before this work we already had first results in [4]-[6]. The Navier- Stokes existence and smoothness problem concerns the mathematical properties of solu- tions to the NavierStokes equations. These equations describe the motion of a fluid in space. Solutions to the NavierStokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the NavierStokes equations often include turbulence, which remains one of the greatest unsolved problems in physics. Even much more basic properties of the solutions to NavierStokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 771 smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. This is called the NavierStokes existence and smoothness problem. Since under- standing the NavierStokes 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 [4]-[6] . 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 one-dimensional nonlinear equa- tions [7,8,9,10], 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 equa- tions, 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 es- timations for the maximum potential in the weaker norms, compared with the norms for Sobolev’ spaces. Consider the operators H = −∆x+ q(x), H0 = −∆x defined in the dense set W 2 2 (R3) in the space L2(R3), and let q be a bounded fast-decreasing function. The operator H is called Schrödinger’s operator. We consider the three-dimensional inverse scattering problem for Schrödinger’s operator: the scattering potential must be recon- structed from the scattering amplitude. This problem has been studied by a number of researchers [ 9,11,12] and references therein] 4. Results for the three-dimensional case Consider Schrödinger’s equation: −∆xΨ + qΨ = |k|2Ψ, k ∈ C (38) Let Ψ+(k, θ, x) be a solution of (38) with the following asympotic behavior: Ψ+(k, θ, x) = φ0(θ, x) + ei|k||x| |x| A(k, θ ′ , θ) + 0 ( 1 |x| ) , |x| → ∞, (39) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 772 where A(k, θ ′ , θ) is the scattering amplitude and θ ′ = x |x| , θ ∈ S 2 for k ∈ C̄+ = {Imk ≥ 0} φ0(θ, x) = eikθx A(k, θ ′ , θ) = − 1 4π ∫ R3 q(x)Ψ+(k, θ, x)e−ikθ ′ xdx. (40) Let us also define the solution Ψ−(k, θ, x) for k ∈ C̄− = {Imk ≤ 0} as Ψ−(k, θ, x) = Ψ+(−k,−θ, x) . As is well known[9] : Ψ+(k, θ, x) − Ψ−(k, θ, x) = − k 4π ∫ S2 A(k, θ ′ , θ)Ψ−(k, θ ′ , x)dθ ′ , k ∈ R. (41) This equation is the key to solving the inverse scattering problem, and was first used by Newton [11,12] and Somersalo et al. [13]. Equation (41) is equivalent to the following: Ψ+ = SΨ−, (42) where S is a scattering operator with the kernel S(k, l), S(k, l) = ∫ R3 Ψ+(k, x)Ψ∗−( l, x)dx. The following theorem was stated in [2]: Theorem 9. (The energy and momentum conservation laws) Let q ∈ R. Then, SS∗ = I, S∗S = I, where I is a unitary operator. Definition 1. The set of measurable functions R with the norm, defined by ||q||R =∫ R6 q(x)q(y) |x−y|2 dxdy <∞ is recognized as being of Rollnik class. Let us take into consideration a series for A : A(k, k′) = ∞∑ n=0 An(k, k′), A0(k, k′) = 1 (2π)3 ∫ R3 ei(k−k ′,x)q(x)dx, (43) An(k, k′) = 1 (2π)3 (−1)n (4π)n ∫ R3(n+1) ei(k,x0)q(x0) ei|k||x0−x1| |x0 − x1| q(x1)× ...× ×...× q(xn−1) ei|k||xn−1−xn| |xn−1 − xn| q(xn)e−i(k ′,xn)dx0...dxn. As well as in [8], p.120 we formulate. Definition 2. Series (43) is called Born’s series. Theorem 10. Let q ∈ L1(R3) ∩ R . If ‖q‖2R ≤ 4π, then Born’s series for A(k, k′) converges as k, k′ ∈ R3. A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 773 Proof. is in [8], 121. Let us introduce the following notation: Qf = ∫ S2 Q(k, θ ′ , θ)f(k, θ ′ )dθ ′ , T+ Q f = ∫ ∞ 0 ∫ S2 Q(k, θ ′ , θ)f(k, θ ′ )dθ ′ |k|2 − s2 − i0 s2ds, f = f(k, θ ′ ), Q(k, θ ′ , θ) = ˜q(k − k′), θ′ = k ′ |k′ | , θ = k |k| , θ, θ ′ ∈ S2, Sφx0f = ∫ S2 f(k, θ)ei(x0,k)dθ, for f = f(k, θ ′ , x), Df = k ∫ S2 A(k, θ ′ , θ)f(k, θ ′ , x)dθ ′ , (44) Lemma 11. Suppose that q ∈ R, maxk,k′ |T+ Q (k, k′| < 1/c0, Csupek,ek′ ,k|Q(k, k′)| < 1, then A(k, k′) = c0q̃(k − k′) + c2 0 ∫ R3 ∫ R3 q̃(k + p)q̃(p− k′) (|p|2 − |k|2 − i0) dk + .... A(k, k′) = c0Q(k, k′) + c2 0T + QQ+ c2 0T + Q T + QQ... supek,ek′ ,k|A(k, k′)| < Csupek,ek′ ,k|Q(k, k′)|+ Csupek,ek′ ,k|TQ(k, k′)|, supek,ek′ ,k|TA(k, k′)| < Csupek,ek′ ,k|TQ(k, k′)|+ Csupek,ek′ ,k|Q(k, k′)| Proof. folows from the definition A(k, k′), T+ Q f and the formula for a geometric pro- gression As shown in [14], Ψ±(k, x) is an orthonormal system of H eigenfunctions for the con- tinuous spectrum. In addition to the continuous spectrum there are a finite number N of H negative eigenvalues, designated as −E2 j with corresponding normalized eigenfunctions ψj(x,−E2 j )(j = 1, N), where ψj(x,−E2 j ) ∈ L2(R3). We present Povzner’s results [14] below: Theorem 12. (Completeness) For both an arbitrary f ∈ L2(R3) and for H eigenfunc- tions, Parseval’s identity is valid. |f |2L2 = (PDf, PDf) + (PAcf, PAcf). PDf = N∑ j=1 fjψj(x,−Ej). PAcf = ∫ ∞ 0 ∫ S2 s2f̄(s)Ψ+(s, θ, x)dθds, (45) where f̄ and fj are Fourier coefficients for the continuous and discrete cases. Theorem 13. (Birmann–Schwinger estimation). Let q ∈ R. Then, the number of discrete eigenvalues can be estimated as: N(q) ≤ 1 (4π)2 ∫ R3 ∫ R3 q(x)q(y) |x− y|2 dxdy. (46) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 774 This theorem was proved in [14]. We define the operators T±, T for f ∈ W 1 2 (R) as follows: T+f = 1 2πi lim Imz→0 ∞∫ −∞ f(s) s− z ds, Im z > 0, T−f = 1 2πi lim Imz→0 ∞∫ −∞ f(s) s− z ds, Im z < 0, (47) Tf = 1 2 (T+ + T−)f. (48) 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 [16]: Lemma 14. TT = 1 4 I, TT+ = 1 2 T+, TT− = −1 2 T−, T+ = T + 1 2 I, T− = T − 1 2 I, T−T− = −T− (49) Theorem 15. Let q ∈ R, N(q) < 1, g = (Φ+ − Φ−). Then , Φ± = T±g. (50) Proof. The proof of the above follows from the classic results for the Riemann problem. Lemma 16. Let q ∈ R, N(q) < 1 g+ = g(k, θ, x), g− = g(k,−θ, x), ). Then, Ψ+(k, θ, x) = (T+g+ + eikθx), Ψ−(k, θ, x) = (T−g− + e−ikθx). (51) Proof. The proof of the above follows from the definitions of g,Φ±,Ψ± . Lemma 17. Let , N(q) < 1, sup k ∣∣∣∣∣∣ ∞∫ −∞ ∫ S2 pA(p, θ ′ , θ)dθ ′ 4π(p− k + i0) dp ∣∣∣∣∣∣ < α < 1 sup k ∣∣∣∣∣∣ ∞∫ −∞ ∫ S2 pA(p, θ ′ , θ)φ0dθ ′ 4π(p− k + i0) dp ∣∣∣∣∣∣ < α < 1 Then T−g− = (I − T−D)−1T−Dφ0, Ψ− = (I − T−D)−1T−Dφ0 + φ0, |T−Dφ0| < α 1− α (52) Proof. using equation Ψ+(k, θ, x) − Ψ−(k, θ, x) = − k 4π ∫ S2 A(k, θ ′ , θ)Ψ−(k, θ ′ , x)dθ ′ , k ∈ R. (53) we can rewrite T+g+ − T−g− = D(T−g− + φ0) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 775 Applying the operator T− last equation we have T−g− = T−D(T−g− + φ0) (I − T−D)T−g− = T−Dφ0, T−g− = ∑ n≥0 (−T−D)n φ0 Estimating the terms of the series, we obtain |T−g−| ≤ ∑ n≥0 |T−Dnφ0| ≤ ∑ n≥0 ∣∣∣∣∣ ∫ ∞ −∞ .... ∫ ∞ −∞ φ0 ∏ ∫ S2 kjA(kj , θ ′ kj , θkj )dθ ′ kj 4π(kj+1)− kj + i0) dk1...dkn ∣∣∣∣∣ ≤ ≤ ∑ n≥0 sup k ∣∣∣∣∣∣ ∞∫ −∞ ∫ S2 pA(p, θ ′ , θ)φ−∞dθ ′ 4π(p− k + i0) dp ∣∣∣∣∣∣ ∏ 0≤j0 αn = α 1− α Using operator Λ = ∆k = 3∑ i=1 ∂2 ∂k2i we can formulate folows results: Lemma 18. Let q ∈ R, N(q) < 1, and assume that (I − T−D)−1 exists. Then, T−DT−Λg− = T−Λg− + T−(∇D,∇T−g) + T−ΛDφ0 T−Λg− = (I − T−D)−1 (T−(∇D,∇T−g)− T−ΛDφ0) (54) Proof. The proof of the above follows from the definitions of g,Φ±,Ψ− and equation (41) Lemma 19. Let q ∈ R, N(q) < 1. Then, q = lim z→0 H0Ψ−/Ψ−, (55) q = lim z→0 ΛH0Ψ−/ΛΨ− (56) Proof. The lemma can be proved by substituting Ψ− into equation (38). 5. Conclusions for the three-dimensional inverse scattering problem This study has shown once again the outstanding properties of the scattering operator, which , in combination with the analytical properties of the wave function,allow to obtain an almost- explicit formulas for the potential to be obtained from the scattering amplitude. Furthermore, this appro. The estimations follow from this reach overcomes the problem of over-determination, resulting from the fact that the potential is a function of three variables, whereas the amplitude is a function of five variables. We have shown that it is sufficient to average the scattering amplitude to eliminate the two extra variables. A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 776 6. 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 [17]-[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 in- teresting 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 turbu- lence, 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 pre- diction model; this paper provides a theoretical justification for this approach. Consider the Cauchy problem for the Navier-Stokes equations: qt − ν∆q + (q,∇q) = −∇p+ f(x, t), div q = 0, (57) q|t=0 = q0(x) (58) in the domain QT = R3 × (0, T ),where : div q0 = 0. (59) The problem defined by (57), (58), (59) has at least one weak solution (q, p) in the so-called Leray–Hopf class [16]. The following results have been proved [17]: Theorem 20. If q0 ∈W 1 2 (R3), f ∈ L2(QT ), (60) there is a single generalized solution of (57), (58), (59) in the domain QT1, T1 ∈ [0, T ], satisfying the following conditions: qt,∇2q, ∇p ∈ L2(QT ). (61) Note that T1 depends on q0 and f . Lemma 21. Let q0 ∈W 1 2 (R3), f ∈ L2(QT ).Then, sup 0≤t≤T ||q||2L2(R3) + t∫ 0 ||∇q||2L2(R3)dτ ≤ ||q0||2L2(R3) + ||f ||L2(QT ). (62) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 777 Our goal is to provide global estimations for the Fourier transforms of derivatives of the Navier–Stokes equations’ solutions (57), (58), (59) without the that the smallness of the initial velocity and force are small. We obtain the following uniform time estimation. Lemma 22. The solution of (57), (58), (59) according to Theorem 20 satisfies: q̃ = q̃0 + t∫ 0 e−ν|k| 2|(t−τ)( ˜[(q,∇)q] + F̃ )dτ, (63) where F = −∇p+ f . Proof. This follows from the definition of the Fourier transform and the theory of linear differential equations. Lemma 23. The solution of (57), (58), (59) satisfies: p̃ = ∑ i,j kikj |k|2 q̃iqj + i ∑ i ki |k|2 f̃i (64) and the following estimations: ||p||L2(R3) ≤ 3||∇q|| 3 2 L2(R3) ||q|| 1 2 L2(R3) , (65) |∇p̃| ≤ |q̃ 2| |k| + |f̃ | |k|2 + 1 |k| ∣∣∣∇f̃ ∣∣∣+ 3 ∣∣∇q̃2 ∣∣ . (66) Proof. This expression for p is obtained using div and the Fourier transform presenta- tion. Lemma 24. The solution of (57), (58), (59) in Theorem 20 satisfies the following in- equalities: ∫ R3 |x|2|q|2dx+ t∫ 0 ∫ R3 |x|2|∇q|2dxdτ ≤ const, ∫ R3 |x|4|q|2dx+ t∫ 0 ∫ R3 |x|4|∇q|2dxdτ ≤ const, (67) or ||∇q̃||L2(R3) + t∫ 0 ∫ R3 |k|2|∇̃q|2dkdτ ≤ const, ∣∣∣∣∇2q̃ ∣∣∣∣ L2(R3) + t∫ 0 ∫ R3 |k|2|∇̃2q|2dkdτ ≤ const. (68) This follows from the a priori estimation of Lemma 21, conditions of Lemma 24,the Navier–Stokes equations. A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 778 Lemma 25. The solution of (57), (58), (59) satisfies the following inequalities: max k |q̃| ≤ max k |q̃0| + T 2 sup 0≤t≤T ||q||2L2(R3) + t∫ 0 ||∇q||2L2(R3)dτ, (69) max k |∇q̃| ≤ max k |∇q̃0| + T 2 sup 0≤t≤T ||∇q̃||L2(R3) + t∫ 0 ∫ R3 |k|2|∇̃q|2dkdτ, (70) max k ∣∣∇2q̃ ∣∣ ≤ max k ∣∣∇2q̃0 ∣∣ + T 2 sup 0≤t≤T ∣∣∣∣∇2q̃ ∣∣∣∣ L2(R3) + t∫ 0 ∫ R3 |k|2|∇2q̃|2dkdτ. (71) Proof. This follows from the a priori estimation of Lemma 21, conditions of Lemma 25,the Navier–Stokes equations. Lemma 26. The solution of (57), (58), (59) according to Theorem 20 satisfies Ci ≤ const, (i = 0, 2, 4), where: C0 = t∫ 0 |F̃1|2dτ, F1 = (q,∇)q + F, C2 = t∫ 0 ∣∣∣∇F̃1 ∣∣∣2 dτ, C4 = t∫ 0 ∣∣∣∇2F̃1 ∣∣∣2 dτ. (72) Proof. This follows from the a priori estimation of Lemma 21, the Navier–Stokes equations. Lemma 27. Suppose that q ∈ R, max k |q̃| <∞, then ∫ R3 ∫ R3 q(x)q(y) |x− y|2 dxdy ≤ C(|q|L2 + max k |q̃|)2. Proof. Using Plansherel’s theorem, we get the statement of the lemma. This proves Lemma 27. Let’s consider the influence of the following large scale transformations in Navier- Stokes’ equation on K = ν 1 2 ν 1 2 − 4πCC 1 2 0 . t′ = tA, ν ′ = ν A , v′ = v A , F ′0 = F0 A2 . A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 779 Lemma 28. Let A = 4 ν 1 3 (CC0 + 1) 2 3 , then K ≤ 8 7 . Proof. By the definitions C and C0, we have K = ( ν A ) 1 2 (( ν A ) 1 2 −4πCC0 A2 )−1 = ν 1 2 ( ν 1 2 − 4πCC0 A 3 2 )−1 < 8 7 . This proves Lemma Let us introduce operator Fkk′, as Fkk′f = ∫ R3 e i(k,x)−i(x,k′)f(x)dx Lemma 29. Let Q ∈ W 1 2 (R3), Q ∈ L2(QT ), νk(k, k ′) = ν|k − k′|2.Then, the solution of (57), (58), (59) in Theorem 20 satisfies the following inequalities: sup (ek,ek′ )∈S2 |Q(k, k′)| < C, sup (ek,ek′ )∈S2 k|Q(k, k′)| < C√ (1− cos(θ)) , sup (ek,ek′ )∈S2 |A(k, k′)| < C, sup (ek,ek′ )∈S2 k|A(k, k′)| < C√ (1− cos(θ)) , (73) Proof. This follows from Q̇ = −Fkk′[(q,∇)q] + Fkk′(ν∆q + ∇p) + Fkk′F (74) After the transformations we obtain Q̇ = −Fkk′[(q,∇)q] + (νkFkk′q + Fkk′∇p) + Fkk′F, (75) Q = Q0 + ∫ t 0 e−|k| 2(1−cos(θ))(t−τ) (−Fkk′[(q,∇)q] + Fkk′∇p+ Fkk′F ) . from last equation we have |Q| ≤ |Q0|+ CT Integrating by θ and carrying out the coordinate transformations, we obtain Q = Q0 + ∫ t 0 e−|k| 2(1−cos(θ))(t−τ) (−Fkk′[(q,∇)q] + Fkk′∇p+ Fkk′F ) . |Q| ≤ |Q0|+ C k √ (1− cos(θ)) A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 780 Lemma 30. Let Q ∈ W 1 2 (R3), Q ∈ L2(QT ), νk(k, k ′) = ν|k − k′|2.Then, the solution of (57), (58), (59) in Theorem 20 satisfies the following inequalities: sup (ek,ek′ )∈S2 ∣∣TQ(k, k′) ∣∣ < C, sup (ek,ek′ )∈S2 ∣∣ΛTQ(k, k′) ∣∣ < C, sup (ek,ek′ )∈S2 |TA(k, k′)| < C, sup (ek,ek′ )∈S2 ∣∣ΛTA(k, k′) ∣∣ < C, (76) Proof. This follows from TQ = TQ0 + T ∫ t 0 e−|k| 2(1−cos(θ))(t−τ) (−Fkk′[(q,∇)q] + Fkk′∇p+ Fkk′F ) . from last equation we have |TQ| ≤ |TQ0|+ CT Using operator Λ = 3∑ i=1 ∂2 ∂k2i ΛTQ = ΛTQ0 + ΛT ∫ t 0 e−|k| 2(1−cos(θ))(t−τ) (−Fkk′[(q,∇)q] + Fkk′∇p+ Fkk′F ) . |ΛTQ| = |ΛTQ0|+ CT Lemma 31. Let Q ∈ W 1 2 (R3), Q ∈ L2(QT ), νk(k, k ′) = ν|k − k′|2, X(x) = x.Then, the solution of (57), (58), (59) in Theorem 20 satisfies the following inequalities: sup (ek,ek′ )∈S2 ∫ ∞ 0 |Sφx0Q(k, k′)|dk < C ∫ t 0 sup x∈R3 |q(x)| ∥∥(1 +X2)∇q ∥∥ L2(R3) dτ, sup (ek,ek′ )∈S2 ∫ ∞ 0 |ΛSφx0Q(k, k′)|dk < C ∫ t 0 supx∈R3 |q(x)| ∥∥(1 +X2)∇q ∥∥ L2(R3) dτ (77) Proof. This follows from from last equation we have |Sφx0Q| ≤ |Sφx0Q0|+ sup (ek,ek′ )∈S2 ∫ ∞ 0 |Sφx0Q(k, k′)|dk < C ∫ t 0 sup x∈R3 |q(x)| ∥∥(1 +X2)∇q ∥∥ L2(R3) dτ Using operator Λ = 3∑ i=1 ∂2 ∂k2i A. Durmagambetov / Eur. J. Pure Appl. Math, 10 (4) (2017), 763-785 781 |ΛSφx0Q| ≤ |Sφx0Q0|+ sup (ek,ek′ )∈S2 ∣∣∣∣ΛSφx0 ∫ t 0 e−|k| 2(1−cos(θ))(t−τ) (−Fkk′[(q,∇)q] + Fkk′∇p+ Fkk′F ) ∣∣∣∣ < C ∫ t 0 sup x∈R3 |q(x)| ∥∥(1 +X2)∇q ∥∥ L2(R3) dτ Lemma 32. Let q ∈ R ∩ L2(R3), and Csupek,ek′ ,k|TQ(k, k′)|+ Csupek,ek′ ,k|Q(k, k′)| < 1. Then, |ΛΨ±q| |x=x0, k=0 ≥ x2 − C. (78) Proof. |ΛΨ−|k=0 = |T−Λg− + ΛΦ0|k=0 ≥ x 2 − ∣∣(I − T−D)−1 (T−(∇D,∇Tg) + TΛDφ0) ∣∣ k=0 ≥ x2 − ∣∣(I − T−D)−1 ( T−(∇D,∇T ((I − T−D)−1T−Dφ0))− TΛDφ0 )∣∣ k=0 ≥ x2 − C (79) Lemma 33. The following permutation formulas hold true xn∏ 0