Existence and properties of the Navier-Stokes equations

Abstract: A proof of existence, uniqueness, and smoothness of the Navier–Stokes equations is an actual problem, whose solution is important for different branches of science. The subject of this study is obtaining the smooth and unique solutions of the three-dimensional Stokes–Navier equations for the initial and boundary value problem. The analysis shows that there exist no viscous solutions of the Navier– Stokes equations in three dimensions. The reason is the insufficient capability of the divergence-free velocity field. It is necessary to modify the Navier–Stokes equations for obtaining the desirable solutions. The modified equations describe a threedimensional flow of incompressible fluid which sticks to a body surface. The equation solutions show the resonant blowup of the laminar flow, laminar–turbulent transition, and fluid detachment that opens the way to solve the magnetic dynamo problem.


PUBLIC INTEREST STATEMENT
The Navier-Stokes equations are one of the fundamentals of fluid mechanics. They describe the motion of viscous liquids, gases or plasmas, and are the basis of mathematical modeling of many natural phenomena. Their applications cover virtually all areas of science: aerodynamics and hydrodynamics, astrophysics, engineering, medicine and health, chemistry, biology, geophysics, electro and magnetic hydrodynamics, etc. However, the solutions of the Navier-Stokes equations suitable for analysis are found only for simplified problems that have little or no physical interest. There is no complete understanding of the properties of these equations. An open problem in mathematics is whether their physically reasonable (smooth) solutions always exist in three dimensions. Also it is very important to understand how the Navier-Stokes equations describe the chaotic behavior of fluids (turbulence) that is one of the most difficult problems of mathematical physics. This article is the response to the Navier-Stokes existence and smoothness challenge.

Introduction
A proof of existence, uniqueness, and regularity of three-dimensional fluid flows for an incompressible fluid concerns most mathematically complex unsolved problems (Constantin, 2001;Fefferman, 2000;Ladyzhenskaya, 2003). The positive result is obtained in two space dimensions (Ladyzhenskaya, 1969). In three dimensions, existence is proved for weak solutions (Leray, 1934), but their uniqueness and smoothness are not known. For strong solutions, the three-dimensional problem is not solved till now (Constantin, 2001;Fefferman, 2000;Ladyzhenskaya, 2003).
The solution of the problem can help ensure the development in different branches of science. The significance of the challenge is well described in the work (Fefferman, 2000).
"A fundamental problem in analysis is to decide whether such smooth, physically reasonable solutions exist for the Navier-Stokes equations. … Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behavior of solutions of the Euler and Navier-Stokes equations. … Since we don't even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas." In the work of (Fefferman, 2000), the detailed statement for the Cauchy initial value problem and the problem with the initial periodical conditions of the Navier-Stokes equations is presented. We use this statement as the basic formulation of problem-solving in our research.
The regularity question considering the boundary value problem is posed in the work (Constantin, 2001): "What are the most general conditions for smooth incompressible velocities ⃗ v(0, ⃗ r) that ensure, in the absence of external input of energy, that the solutions of the Navier-Stokes equations exist for all positive time and are smooth? … A specific regularity question, still open, is for instance: Given an arbitrary infinitely differentiable, incompressible, compactly supported initial velocity field in R 3 , does the solution remain smooth for all time? A version of the same question is: Given an arbitrary threedimensional divergence-free periodic real analytic initial velocity, does the ensuing solution remain smooth for all time?" In the work of (Ladyzhenskaya, 2003), there is a more general formulation of the problem, Do the Navier-Stokes equations together with the initial and boundary conditions provide the deterministic description of the dynamics of an incompressible fluid or not?
In our research, we will try to find the answers to all these questions.

Statement of problem
The system of the non-stationary Navier-Stokes equations for an incompressible fluid in three dimension looks as follows: where ρ is the fluid density, ⃗ v(t, ⃗ r) is the vector field of velocities of a fluid at time t and a point with div (⃗ v(t, ⃗ r)) = 0, The problem is formulated in four statements (Fefferman, 2000).
(A) Existence and smoothness of Navier-Stokes solutions on R 3 . Take η > 0 and space dimension n = 3. Let ⃗ v 0 (⃗ r) be any smooth, divergence-free vector field satisfying (5). Take ⃗ f ex (t, ⃗ r) to be identically zero. Then, there exist smooth functions and (8).

Idea of proof
According to the Cauchy-Kowalevski theorem, for the local existence and uniqueness of the solution of a Cauchy initial value problem for a linear partially differential equation with constant coefficients, it is enough to prove the expansibility of this solution to a uniformly converging power series in some neighborhood of every point in the solution domain. The solution can be a function of real or complex variables (Courant & Hilbert, 1962;Vladimirov, 1983).
We will extend this statement to the nonlinear Navier-Stokes equations and generalize the result obtained for a local solution to all space R 1 + × R 3 to obtain a global solution. We can do it by the following way.
At first, we carry out the statements for the boundary value problem. We define a compact simply connected domain G ⊂ R 3 with a border S. We accept that The initial step of the solution existence proof is the definition of a class of generalized solutions for which a uniqueness theorem is performed (Ladyzhenskaya, 2003).
Let's designate all functions satisfying the condition (16) as L 2 (G). As is known from the differential equation theory (Vladimirov, 1983), any function from this space can be presented as a Fourier series converging to this function. Due to the condition (15), the Fourier decompositions to converging series can be applied both to space and time variables. So we introduce the function space L 2 (G T ), (Ladyzhenskaya, 2003), whose functions are decomposed to the Fourier series that are constructed from the complete system of functions {⃗ u n (t, ⃗ r)}, n = 0, ∞. If this com- presented as the expansions to this complete system are solutions of these equations.
A Fourier series is based on the use of the trigonometric functions which are presented as the exponential function with the imaginary-valued argument. So a Fourier series can be presented as a complex-valued power series (Fihtengolts, 2003a). For a continuously differentiable function on [0, ∞) × R 3 , the Lipschitz condition is performed for any domain G ′ T , which is wider than G T , so the Fourier series of this function is uniformly converged on G T (Fihtengolts, 2003a). The Fourier coefficients are uniquely determined from the boundary condition. So the local existence and uniqueness theorem can be applied to a general partial differential equation with constant coefficients (Courant & Hilbert, 1962) for the function space G T .
We have the following reasons to suppose that this claim is also valid for the nonlinear Navier-Stokes equations. According to conditions 13-16, the functions ⃗ times differentiable for k = 1, … , ∞, and continuous with all their kth-order partial derivatives on [0, ∞) × R 3 . The nonlinear term of the Cauchy momentum equation is k times differentiable and continuous together with its all derivatives also. Hence, and the nonlinear term can be presented as the uniformly converging Fourier series on G T (Fihtengolts, 2003a). These Fourier series are differentiable and can be multiplied to other Fourier series. The result of multiplication is a Fourier series (Fihtengolts, 2003a). The Fourier coefficients of the nonlinear term are uniquely defined from the boundary condition. Assuming that the nonlinear term contains no unknowns, we obtain the unique solution of the Navier-Stokes equations as for the system of two linear equations, one of which contains the additional source presented as the Fourier series.
We can obtain the unique global solution of the initial value problem (1), (2), and (3) satisfying the statement (A) of the Clay Mathematics Institute if we take the limit of the Fourier series for G T = [0, T] × G → [0, ∞) × R 3 . As a result, the Fourier series turns to the Fourier transforms which are presented by improper integrals. Under conditions (5)-(8), these integrals exist and converge. The similar statements can be carried out for the periodic functions satisfying (9)-(12).
The use of a trigonometric Fourier series is not the only option to obtain a solution of the Stokes-Navier equations. It is possible that for some variables (e.g. time variable), the solution is decomposed to function series based on exponential functions with a real-valued argument instead of a complex-valued one. These series are real-valued power one. Under the condition of the uniform convergence, we can reduce the nonlinear problem to a linear one and transform the series to improper integrals the same way as using the trigonometric Fourier series.
Let's verify whether the functions expandable to the uniformly converging Fourier series are really solutions of the Navier-Stokes equations. We will obtain the eigensolutions of the Navier-Stokes equations and study their relation to the Fourier series.
If p(t, ⃗ r) ∈ L 2 (G T ) for the proof of Navier-Stokes equation resolvability, it is enough to find the solutions of the mass continuity equation in L 2 (G T ). This is a constant coefficient homogeneous differential equation with private derivatives of the first order. It is linear and only one required un- The Cauchy momentum equation is also linear with respect to the function p(t, ⃗ r). If we take the divergence of the right and left parts of Equation (1), we obtain that the pressure is a solution of the Poisson equation The solution p(t, ⃗ r) ∈ L 2 (G T ) can be easily obtained at substitution of ⃗ v(t, ⃗ r) in this equation.
We should take into account that, generally, the velocity vector does not coincide with a direction of pressure gradient in the mass continuity equation. Therefore, we are interested in the resolvability of the Cauchy momentum equation in the absence of the pressure gradient. Thus, it is necessary to solve the equation (2), (3), and (5)-(8).

Solutions of mass continuity equation
Let's consider the mass continuity equation div(⃗ v(t, ⃗ r)) = 0. This equation defines a velocity vector field as a divergence-free one.
For a divergence-free field, where Δ⃗ v(t, ⃗ r) = 0 is a vector potential of a vector velocity field.
Let's admit that

For this function,
This expression is a necessary condition of resolvability of Equation (20) (Fihtengolts, 2003a).
Let's obtain the general solution of the heterogeneous system of the equations: We use the solution of the inverse problem of the vector analysis (Fihtengolts, 2003a).
At first, we obtain the general solution of homogeneous system of the equations: The first equation testifies that a divergence-free field is a potential one: where Φ(t, ⃗ r) is a scalar potential of a vector velocity field.
As a result, we obtain the Laplace equation Let's take a gradient of this expression. We consider that We obtain the Laplace equation for the velocity: .
The solutions of this equation are harmonic functions.
Let's obtain the private solution of the heterogeneous system (22). We substitute expression (19) for the vector potential in Equation (20).

Then,
where ⃗ H(t, ⃗ r) is a vector potential of a vector velocity field.
Using the correlations curl(∇(div( ⃗ H(t, ⃗ r))) = 0 and curl(Δ ⃗ The vector field ⃗ v(t, ⃗ r) is completely defined by its scalar and vector potentials. Let's investigate the ability of potentials Φ(t, ⃗ r) and H(t, ⃗ r) to build the complete systems of functions in three-dimensional space R 3 if these potentials are harmonic functions.
The harmonic functions can be presented as converging power series in three dimensions. For two space variables, these power series have the form of the trigonometric Fourier series. For the third variable, the power series is based on the exponential function with the real-valued argument.
The harmonic functions possess the property of completeness only for two variables. For the third variable, it is impossible to construct the complete system of functions.
complex-valued vector. In this solution, the value of one wave vector projection always depends on two others. The correlation between projections does not allow constructing the orthonormal basis for one of the three spatial coordinates. The complete system can be constructed only for the functions of two spatial variables.
In the cylindrical coordinates, the solutions of the Laplace equation are functions like where R n (k ρ ⋅ ρ) is the cylindrical Bessel function and k ρ and k ϕ are the real eigenvalues of the problem. As well as in the previous case, we have only two independent eigenvalues for the function of three variables.
In the spherical coordinates, the solution aspiring zero at infinity looks like where Y n (ϑ, φ) is the spherical superficial harmonics possessing the property of completeness.
The functions 1/r n don't possess this property. The correlation between the eigenvalues in the Cartesian coordinate problem is kept also in the curvilinear one. The solutions depending on the curvilinear coordinates can be obtained by the expansion of the exponential function to the series on the complete system of the functions corresponding defined geometry (e.g. the cylindrical and spherical functions). Under the condition of ⃗ k 2 = 0, these functions lose the completeness for the three variables.
But it is much more important that the harmonic functions represent a non-viscous fluid motion. According to formula (27), there is no term with the viscosity in the Cauchy momentum equation. A viscous fluid motion is defined by not a potential (curl-free) divergence-free field but exclusively a divergence-free one which is determined by heterogeneous Equation (30). Let's consider the case when the velocity ⃗ v(t, ⃗ r) is defined only by the vector potential ⃗ H(t, ⃗ r). We substitute the functions in Equation (29). We obtain It means that We have the same result if we substitute the function (33) in the equation div(⃗ v(t, ⃗ r)) = 0. We obtain the expression The case when leads to the expression ⃗ k 2 = 0. It is performed for the potential velocity ⃗ v POT (t, ⃗ r) satisfying Equation (27). (29). In this case, there exists correlation defined by the necessity to choose the vector ⃗ k direction that is perpendicular to the vector ⃗ v(t, ⃗ r). For the problem-solving in the Cartesian or curvilinear coordinates, this correlation reduces the number of uncertain eigenvalues just as it has occurred for the Laplace equation. If one axis of the Cartesian system of coordinates is directed in parallel to the vector ⃗ v(t, ⃗ r), the solution becomes the function of only two spatial variables. The transformation to an arbitrary Cartesian system of coordinates and a curvilinear coordinate system leads to the result that there are only two independent spatial variables.

Solutions of Cauchy momentum equation
Under the condition η > 0, the three-dimensional Navier-Stokes equations are reduced to a two-dimensional one. Besides this, the nonlinear term is equal to 0 in the Cauchy momentum equation if there are no external sources of a fluid motion in R 3 . Using expression (33), we have It means that there is no term with velocity in Equation (17): and Equation (18) is a linear one: As a result, we obtain the eigensolution of the Navier-Stokes equations for velocity in the Cartesian coordinate system with unit vectors ⃗ e x , ⃗ e y , and ⃗ e z under the condition η > 0 as where ⃗ k = (k x , k y , k z ) and ⃗ r = (x, y, z) are the real-valued vectors, ⃗ k⊥⃗ v(t, ⃗ r). This formula is obtained using the expressions: where f 1 , f 2 , and f 3 are linear functions.
We obtain the global solution of the problem for ⃗ f ex (t, ⃗ r) ≡ 0 if we take the improper integral of the function (42) over the continuous spectra of k y and satisfy the initial condition. The solution is a smooth function at any time moment (0 ≤ t < ∞).
For pressure, we have the general solution of the two-dimensional Poisson Equation (40).
We should say that the nonlinear term isn't equal to 0 if we use the curvilinear coordinates (e.g. spherical or cylindrical) instead of the Cartesian one. In this case, we implicitly use a source of a fluid motion at some points of space R 3 (e.g. a point or line source). But div (⃗ v(t, ⃗ r)) ≠ 0 at the source points.

Alternative
The divergence-free velocity field defines viscous two-dimensional solutions of the Navier-Stokes equations. It is necessary to update the Navier-Stokes equations to consider a potential field which is not divergence-free for solving a viscous three-dimensional problem.
The necessity of updating the equations is most evident at the consideration of fluid movement near body surfaces. Sticking fluid molecules to a surface means inelastic collisions of these molecules with a body. The molecules colliding with the stuck one are also involved in the inelastic collisions since they stick to other molecules or lose kinetic energy equal to the binding energy to release the attached molecules. Therefore, there are the inelastic collisions in the whole boundary layer.
The mass continuity equation does not include the inelastic processes. It means that the field of velocities is divergence-free. It is natural to expect that the divergence-free field does not possess sufficient capabilities to describe the fluid motion in the boundary layer. We suppose that the inelastic intermolecular interactions can be taken into account in the equations by the use of the potential field where velocity divergence is not equal to zero. It distinguishes it from the velocity potential field which is already used in fluid dynamics. That field is a kind of a divergence-free one.

Modification of equations
Let's update the Navier-Stokes equations using a curl-free potential vector field. The refined mass continuity equation looks as follows where Φt, ⃗ r) is the scalar potential and ⃗ H(t, ⃗ r) is the vector potential of the vector velocity field.

We use the general form of the Cauchy momentum equation
where ζ is the second viscosity considering the intermolecular interactions. The compressibility of a fluid is not used in the derivation of this equation (Landau & Lifshitz, 1987), i.e. we can accept that ρ = const.
We substitute expression (44) in the term ( ∕3 + ) ⋅ ∇(∇⃗ v(t, ⃗ r))of the Cauchy momentum Equation (46). We take into account that We obtain the Cauchy momentum equation in the form Any vector field can be presented as a sum of a divergence-free and a curl-free vector field (Fihtengolts, 2003a). We suppose that , where Φ ex (t, ⃗ r) is the scalar potential and ⃗ H ex (t, ⃗ r) is the vector potential of the vector external source field.The vector field ∇p(t, ⃗ r) is potential. So expression (48) can be separated into two equations We consider Equation (50). We use the expressions

We obtain the solution for pressure where f (t) is an arbitrary function of time and p(t, ⃗ r) is the solution of the equation
In presence of the divergence-free and curl-free fields, the system of the modified Navier-Stokes equations is These equations should be complemented by conditions (3) and (4).
The existence of the general solution of Equation (56) is proved in the next part. The whole system (54-57) with conditions (3) and (4) is solvable in three dimensions. The Fourier method allows obtaining the smooth solutions of these equations.

Incompressibility of fluid
If we accept that div(⃗ v(t, ⃗ r)) ≠ 0, a crucial problem is whether the fluid is incompressible. According to the work of (Batchelor, 2000), the influence of pressure variations on the value of fluid density is negligible if div(⃗ v(t, ⃗ r)) = ΔΦ(t, ⃗ r). where L is the length scale (⃗ v(t, ⃗ r) varies slightly over distances small compared with the scale), U is the value of the variations of with respect to both position and time. It means that the velocity distribution is only approximately divergence-free.
In our research, we suppose that assumption (58) is valid, i.e. a fluid is incompressible. We only accept the existence of a curl-free velocity field besides a divergence-free one on the condition that It is performed for a non-stationary flow if (Landau & Lifshitz, 1987) c s is the speed of sound in the fluid and τ is the time scale. The last condition means that the time during which the sound passes the distance equal to the length scale has to be much less that of the time scale τ during which the motion of the fluid significantly changes, i.e. propagation of interactions in the fluid is instantaneous.
The condition (59) is necessary if we want to investigate the nonlinear phenomena of an incompressible fluid motion. Small terms in nonlinear equations can have a significant influence on behavior of a physical system in space and time as a result of a cumulative effect (Bogoliubov & Mitropolski, 1961). The equations of the viscous fluid motion become more accurate at the account of curl-free vector field.
At first, let's consider this equation without ⃗ f ex,SOL (t, ⃗ r): = . (66) We obtained the infinite set of the inhomogeneous linear heat equations. The existence and uniqueness theorem is proved for them (Tikhonov & Samarskii, 2013). The infinite series of private solutions of the heat equations converges uniformly on [0, ∞) × R 3 . So we can obtain the unique solution of Equation (56) if the initial and boundary conditions are defined.
The presence ⃗ f ex,SOL (t, ⃗ r) in the right part of the equation doesn't change the reasoning. But we will show that there exists a turbulent solution of Equation (56) besides the unique solution for a laminar flow.
At the beginning, we obtain the solution of Equation (56) for a laminar flow which can blow up as a result of a resonance effect.

Resonant blowup of laminar flow
We consider the equation Let's suppose that to simplify the expressions. We obtain the system We define the solutions in the form We obtain the linear system of the equations Δ⃗ u 3 (⃗ r) The solution of the first equation is .
Let's obtain the solutions of the second and the third equations. We suppose that We study the nonlinear effect which is described as the interaction between the plane wave functions having the initial wave vectors ⃗ k and ⃗ k 0 . The third and higher order approximations of the solution obtain the term which wave vector coincident with the initial wave vector ⃗ k. It is the term which has the function cos( ⃗ k⃗ r). It means that there is a resonance which increases the amplitude of the solution with time ).
Let's study this effect more carefully.
Let's obtain the next order of solutions to establish the general form of the resonance amplitude. We are interested in the solutions only with cos( ⃗ k⃗ r). We define them as ⃗ u 1 2j+1 (⃗ r). We need also the solutions ⃗ u 1 2j (⃗ r)with sin(2 ⃗ k⃗ r)to obtain ⃗ u 1 2j+1 (⃗ r).

Let's investigate the behavior of the solution if
For example, let's solve the system of Equation (72) if We have the second equation as We find the solution as We obtain 3 2

The next equation is
We separate it into two equations: and We find the solution of the first equation as where a is an unknown constant.
We find the solution of the second equation as where a is an unknown constant.
We obtain the solution of Equation (110) as the sum of the solutions of Equations (111) and (112): We have obtained the secular terms increasing the solution for some period of time. The time of growth (the blowup time) is not infinite due to the presence of the exponential function. This time is proportional to the value 1 ⋅ 2 0 . It can be very large if ⋅ 2 0 << 1 and becomes infinitesimal if 2 0 → 0.
The maximum of the time distribution of the solution approaches infinity.

Laminar flow without resonance
The increase of the solution as the result of the resonance of two fields is possible if a physical nature of the values ⃗ v(t, ⃗ r) and ⃗ u(t, ⃗ r)is different. For example, one is a fluid velocity, another is a magnetic field. If ⃗ v(t, ⃗ r) and ⃗ u(t, ⃗ r) describe a divergence-free and curl-free fluid velocity field, the sum of them is a velocity of a fluid. It is considered that there exists no spontaneous growth of a velocity (kinetic energy) for a closed physical system in the absence of an external source . Actually, there is a change of basic values of solution parameters to be compared with their unperturbed values in higher approximations of nonlinear equations. This leads to the suppression of the resonance effect. We can remove the resonance terms using the method of Poincare .

We suppose that
Our purpose is to delete the resonance term with cos( ⃗ k⃗ r).

Laminar-turbulent transition
Let's consider the inhomogeneous equation ⃗ r) is the divergence-free flow velocity, and ⃗ v POT (t, ⃗ r) is the curl-free flow velocity.
We suppose that ⃗ is the constant vector and ⃗ v(t, ⃗ r) = ⃗ + ⃗ V(t, ⃗ r) is the flow velocity vector.
Δ⃗ u 2 (⃗ r) Δ⃗ u 3 (⃗ r) ⃗ v(t, ⃗ r) = 2 ⋅ ⃗ + ⃗ V(t, ⃗ r), right part of the mass continuity equation. In the last case, the solutions are obtained in curvilinear coordinates. The examples are a viscous fluid rotating by a disk (an infinite disk source), a fluid flow in a diffuser and confuser (an infinite linear source), and the Landau problem for a flooded jet (a point source) (Landau & Lifshitz, 1987).
However, we suppose that only a source geometrical factor is not enough for the description of nonlinear processes considering a viscous fluid flow. It is necessary to consider the molecular inelastic interactions in the mass continuity equation. These interactions change the intermolecular binding energy on the expense of the kinetic energy of a fluid. Their presence in the equation is carried out by the use of the curl-free potential velocity field. Due to this field, the mass continuity equation can be reduced to the Helmholtz equation, whose solutions give a complete system of functions in R 3 .
The modified equations describe an incompressible fluid flow. Due to the inelastic interactions, an incompressible fluid sticks to a body surface. The analysis of the equation solutions shows the blowup of the laminar flow at the resonance, transition of the laminar flow to the turbulent one, and fluid detachment. This opens the way to solve the magnetic dynamo problem.