Global Estimation of the Cauchy Problem Solutions’ the Navier-Stokes Equation ()
1. Introduction
This paper combines the results of studies on the inverse scattering problem with the Cauchy problem for the Navier-Stokes equations. 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 equations [1] -[4] , 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
, 
defined in the dense set 
in the space
, and let 
be a bounded fast-decreasing function. The operator 
is called Schrödinger’s operator.
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 ([5] -[8] and references therein).
2. Results
Consider Schrödinger’s equation:
(1)
Let 
be a solution of (1) with the following asympotic behavior:
(2)
where 
is the scattering amplitude and 
for 
(3)
Let us also define the solution 
for 
as
As is well known [1] :
(4)
This equation is the key to solving the inverse scattering problem, and was first used by Newton [6] [7] and Somersalo et al. [8] .
Equation (4) is equivalent to the following:
(5)
where 
is a scattering operator with the kernel
.
The following theorem was stated in [1] :
Theorem 1 (The energy and momentum conservation laws) Let
. Then, 
where 
is a unitary operator.
Definition 1 The set of measurable functions 
with the norm, defined by 
is recognized as being of Rollnik class.
As shown in [8] , 
is an orthonormal system of 
eigenfunctions for the continuous spectrum. In addition to the continuous spectrum there are a finite number 
of 
negative eigenvalues, designated as
with corresponding normalized eigenfunctions
, where
.
We present Povzner’s results [9] below:
Theorem 2 (Completeness) For both an arbitrary 
and for 
eigenfunctions, Parseval’s identity is valid.
(6)
where 
and 
are Fourier coefficients for the continuous and discrete cases.
Theorem 3 (Birmann-Schwinger estimation). Let
. Then, the number of discrete eigenvalues can be estimated as:
(7)
This theorem was proved in [10] .
Let us introduce the following notation:
(8)
(9)
where
. We define the operators
, 
for 
as follows:
(10)
(11)
(12)
Consider the Riemann problem of finding a function
, that is analytic in the complex plane with a cut along the real axis. Values of 
on the sides of the cut are denoted as
,
. The following presents the results of [11] :
Lemma 1
(13)
Theorem 4 Let
,
. Then,
(14)
The proof of the above follows from the classic results for the Riemann problem.
Lemma 2 Let
. Then,
(15)
The proof of the above follows from the definitions of
.
Lemma 3 Let
,
Then
(16)
The proof of the above again follows from the definitions of the functions
.
Lemma 4 Let
. Then,
(17)
The proof of the above follows from the definitions of 
and Theorem 1.
Definition 2 Denote by 
the set of functions 
with the norm 
Definition 3 Denote by 
the set of functions 
such that
, for any 
Lemma 5 Suppose
. Then, the operator
, defined on the set 
has an inverse defined on
.
The proof of the above follows from the definitions of 
and the conditions of Lemma 5.
Lemma 6 Let
, and assume that 
exists. Then,
(18)
(19)
(20)
The proof of the above follows from the definitions of 
and Equation (4) Let us rewrite (20) using
(21)
Lemma 7 Let
. Then,
(22)
The proof is the same as that in [5] .
Lemma 8 Let
. Then,
(23)
The lemma can be proved by substituting 
into Equation (1).
Lemma 9 Let
, and assume that 
exists. Then,
(24)
The proof of the above follows from the definitions of 
and Lemma 6.
Lemma 10 Let
. Then
.
The proof of the above follows from the definition of 
and the unitary nature of
.
Lemma 11 Let
. Then,
(25)
(26)
The proof of the above follows from the definitions of 
and (1).
Lemma 12 Let
. Then,
(27)
The proof of the above follows from the definition of
.
Lemma 13 Let
, and
. Then,
(28)
To prove this result, one should calculate
(29)
Using Lemma 7, the first approximation can be obtained in terms of
:
(30)
where 
represents terms of highest order of
. The lemma can be proved using obvious estimations for 
and Lemmas 8, 10.
3. 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 approach overcomes the problem of overdetermination, 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.
4. Cauchy Problem for the Navier-Stokes Equation
Numerous studies of the Navier-Stokes equations are devoted to the problem of the smoothness of its solutions. A good overview of these studies is given in [12] -[14] . 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. On the other hand, 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, and this paper is a theoretical justification for this approach.
Consider the Cauchy problem for the Navier-Stokes equations:
(31)
(32)
in the domain
, where:
(33)
The problem defined by (31), (32), (33) has at least one weak solution 
in the so-called Leray-Hopf class [12] .
The following results have been proved [12] :
Theorem 5 If
(34)
there is a single generalized solution of (31), (32), (33) in the domain
, 
, satisfying the following conditions:
(35)
Note that 
depends on 
and
.
Lemma 14 Let
,
. Then,
(36)
Our goal is to provide global estimations for the Fourier transforms of derivatives of the Navier-Stokes equations’ solutions (31), (32), (33) without the that the smallness of the initial velocity and force are small. We obtain the following uniform time estimation. Using the notation that:
(37)
(38)
Assertion 1 The solution of (31), (32), (33) according to Theorem 5 satisfies:
(39)
where
.
This follows from the definition of the Fourier transform and the theory of linear differential equations.
Assertion 2 The solution of (31), (32), (33) satisfies:
(40)
and the following estimations:
(41)
(42)
This expression for 
is obtained using 
and the Fourier transform. The estimations follow from this representation.
Lemma 15 The solution of (31), (32), (33) in Theorem 5 satisfies the following inequalities:
(43)
(44)
or
(45)
(46)
This follows from the Navier-Stokes equations, our first a priori estimation (Lemma 1) and Lemma 2.
Lemma 16 The solution of (31), (32), (33) satisfies the following inequalities:
(47)
(48)
(49)
These estimations follow from (9), Parseval’s identity, the Cauchy-Schwarz inequality, and Lemma 3.
Lemma 17 The solution of (31), (32), (33) according to Theorem 5 satisfies 
where:
(50)
This follows from our a priori estimation (Lemma 1) and the assertion of Lemma 3.
Lemma 18 The solution of (31), (32), (33) according to Theorem 5 satisfies to the following inequalities:
(51)
(52)
where
(53)
Proof. From (39), we have the inequality:
(54)
where
(55)
Using the notation
(56)
and Hölder’s inequality in
, the following inequality can be obtained:
(57)
where 
satisfy
.
Let 
Then,
(58)
Using the estimation for 
in (57), the assertion in the lemma can be proved. □
Lemma 19 Let
,
. Then,
(59)
A proof of this lemma can be obtained using Plancherel’s theorem.
For 
consider the transformation of the Navier-Stokes:
(60)
Lemma 20 Let
, then 
Proof. Using the definitions for 
и
we get
(61)
(62)
□
We now obtain uniform time estimations for Rollnik’s norms of the solutions of (31), (32), (33). The following (and main) goal is to obtain the same estimations for
—velocity components of the Cauchy problem for the Navier-Stokes equations. We will use Lemmas 8 and 13.
Theorem 6 Let 
Then, there exists a unique generalized solution of (31), (32), (33) satisfying the following inequality: 
where the value of 
depends only on the conditions of the theorem.
Proof. It suffices to obtain uniform estimates of the maximum velocity components
, which obviously follow from
, because uniform estimates allow us to extend the local existence and uniqueness theorem over the interval in which they are valid. To estimate the velocity components, Lemma 12 can be used:
Using Lemmas (15)-(19) for
we can obtain
, where 
is the amplitude of potential 
and
. That is, discrete solutions are not significant in proving the theorem, so its assertion follows the conditions of Theorem 6, which defines uniform time estimations for the maximum values of velocity components.
Theorem 6 asserts the global solvability and uniqueness of the Cauchy problem for the Navier-Stokes equations.
5. Conclusion
Uniform global estimations of the Fourier transform 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. In terms of the Fourier transform, under both smooth initial conditions and right-hand sides, no appear exacerbations appear in the speed and pressure modes. A loss of smoothness in terms of the Fourier transform can only be expected in the case of singular initial conditions, or of unlimited forces in
.
Acknowledgements
We are grateful to the Ministry of Education and Science of the Republic of Kazakhstan for a grant, and to the System Research “Factor” Company for combining our efforts in this project.
The work was performed as part of an international project, “Joint Kazakh-Indian studies of the influence of anthropogenic factors on atmospheric phenomena on the basis of numerical weather prediction models WRF (Weather Research and Forecasting)”, commissioned by the Ministry of Education and Science of the Republic of Kazakhstan.