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# Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics

## Abstract

Navier-Stokes and Euler Equations; Vorticity; Laminar Flow; Transition; Turbulence; Diffusion; Kinetic Theory of Gases; Randomness

An effort has been made to solve the Cauchy problem of the Navier-Stokes equations of motion in the whole space . It is shown that vorticity associated with any fluid motion is a direct consequence of conservation of angular momentum, and hence our effort has been concentrated on solving the vorticity equation. It is proved that the sum of the three vorticity components is a time-invariant in fluid motion. Two separate methods of solution have been used. The first one is based on an interpolation theory in Sobolev spaces. It has been proved that, given smooth, localized initial data with finite energy and enstrophy, the vorticity equation admits a global, unique and smooth solution. In our second approach, the vorticity equation has been converted into a non-linear integral equation by means of similarity reduction. The solution of the integral equation has been constructed in a series expansion. The series is shown to converge for initial data of finite size and its analytic properties are extremely intricate. Nevertheless, it has been found that the complete vorticity field is characterized, as an instantaneous description, by a multitude of vorticity constituents. The flow field is composed of vortical elements of broad spatio-temporal scales. Every individual element has its own distinct strength apportioned according to viscous diffusion and influence of the Biot-Savart induction. Specifically, the vorticity constituents are vast in quantity.

The mathematical solutions assert that the Navier-Stokes dynamics is deterministic in nature. The law of energy conservation holds over the entire course of flow evolution. Inference of the solutions leads itself to a satisfactory account for the observed dynamic characteristics of transition process, and of turbulent motion without recourse to instability theory or bifurcation mechanism. In essence, the non-linearity in the equations of motion dictates that every fluid motion ultimately evolves into turbulence as long as the initial data are sufficiently large. The flow evolution is a strong function of the specification of the initial vorticity. In particular, any fluid motion eventually decays in time and restores to its stationary thermodynamic equilibrium state. It is shown that, in the limit of vanishing viscosity, the equations of motion cannot develop flow-field singularities in finite time.

By revisiting the Maxwell-Boltzmann kinetic theory for dilute gases, we consider Maxwellian molecules with cut-off as a generic model. It is found that the density function possesses a phase-space distribution resembling the continuum turbulence provided that molecules’ initial conditions are appropriate. In a qualitative sense, the apparent macroscopic randomness of turbulence can be attributed to a ramification of molecular fluctuations, since the viscous dissipation of the mechanical energy is non-uniform and irregular among the multitudinous vortices of small scales. Consequently, the equations of Navier-Stokes dynamics do not contain a separate entity which has been contrived to account for the stochasticity nature of turbulence.

Global Well-posedness of Navier-Stokes Equations]Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics F. Lam]F. Lam

## 1 Introduction

In the Eulerian description of the motion of an incompressible, homogeneous Newtonian fluid, the principle of mass conservation leads to the continuity equation

 ∇.u=0. (1)

The velocity vector has the components . The space variable is denoted by . The momentum equation for fluid dynamics is derived from Newton’s second law of motion. In the absence of a prescribed force, it reads

 ∂u/∂t+(u.∇)u=νΔu−ρ−1∇p, (2)

where the scalar quantity is the pressure, and is the Laplacian. The density and the viscosity of the fluid are denoted by and respectively. The kinematic viscosity is . The system of the equations (1) and (2) is known as the Navier-Stokes equations (Navier 1823; Stokes 1845). They are derived on the basis of the continuum hypothesis (see, for example, Lamb 1975; Prandtl 1952; Serrin 1959; Ladyzhenskaya 1969; Schlichting 1979; Batchelor 1973). For ideal or inviscid flows (), the system is called the Euler equations (Euler 1755).

We are interested in the global well-posedness of the initial value problem, or the Cauchy problem, for the system in the whole space . The initial condition is given by

 u(x,t=0)=u0(x),x∈R3. (3)

In addition, we assume that . To simplify our analysis, we mainly consider regular, localized initial data:

 ∥∥(1+|x|)k0∂α0u0∂xα0∥∥L∞(R3)<∞ (4)

for any values of and . The smoothness requirement, , is a strong restriction. In practice, we frequently encounter initial flows that are not necessarily infinitely differentiable; the localization may be limited to a few values of and . Instead of specifying the localization, we may rely on the concept of compactness. For instance, we require for initial velocity with compact supports.

Taking divergence of (2) and making use of the continuity, we obtain a Poisson equation for the pressure

 Δp(x;t)=−ρ3∑i,j=1(∂uj∂xi∂ui∂xj)(x;t)=−ρ\Uppi(x;t). (5)

We have written the independent variables as in order to emphasize the fact that, at every instant of time, equation (5) is a kinematic equation as opposed to a dynamic one. Because the differential equation does not contain a term like , it does not describe any time-evolution of the pressure. Extra care must be taken in evaluating quantities like , . Due to the incompressibility hypothesis, the pressure is non-local; any variation in the velocity gradients will instantaneously affect the pressure at any other space locations. This apparent deficiency in physics is in parallel to the infinitely fast propagation speed in the pure initial value problem of heat equation (see, for example, Courant & Hilbert 1966; Sobolev 1964; John 1982; Evans 2008).

The vorticity is the curl of the velocity,

 ω=∇×u.

The concept of vorticity is of great assistance as it introduces essential simplifications in the mathematical theory (Helmholtz 1858; Thomson 1869; Lamb 1975). We will demonstrate that the equation governing the vorticity dynamics can be derived from first principles of physics, so that vorticity is no longer an abstraction in mathematics. Compared with velocity, vorticity is more amendable to local and global analyses. Flow development in space-time can best be viewed as the evolution of vorticity field which is characterized by interaction of the shears arising from velocity differences.

To evaluate flow quantities during the evolution, it may be convenient, in certain circumstances, to trace fluid particles by the Lagrangian description. A fluid particle is a fluid material point that moves with the local velocity. Let be particle’s position at time which is at with respect to the reference time . The initial reference position at is usually taken as . By integrating the following relation in time,

 ∂a(x,t)∂t=u(a(x,t),t), (6)

we can calculate, at least in principle, particle’s position for all time once we have the full knowledge of the Eulerian velocity field. In practice it is generally much involved to integrate the deceptively straightforward relation defined in (6) as it largely implies strongly non-linear functions of time, even for simple flows. Furthermore, the Eulerian velocity must be a dynamic quantity which must come from the Navier-Stokes equations or the Euler equations. We accentuate the fact that one cannot simply make use of the velocity derived from the Biot-Savart relation (see (36) below) for substituting the dynamics. The velocity field seconded from the elliptic equation is a result of an infinite-range instantaneous interaction (albeit an anomalous causality) and hence contains no time-wise information.

Introducing a scaling parameter , one can undertake algebraic manipulations of (1), (2) and (11) by means of transformations and . This procedure of dimensional analysis shows that if the triplet solves the Navier-Stokes system, so does . For fixed fluid properties and , the scaled solutions are

 uλ(x,t)=λ−1u(λ−1x,λ−2t),pλ(x,t)=λ−2p(λ−1x,λ−2t),ωλ(x,t)=λ−2ω(λ−1x,λ−2t). (7)

These scaling properties are derived on the basis of a formal procedure as the numerical value of the scale parameter has no specific upper bounds. Hence it may be assigned to an arbitrarily large value. We have a paradoxical situation; we are free to zone in on every small-scale motion, possibly beyond the fundamental dimensions of matters, without solving the equations of fluid dynamics! The suggestion is that, for some values of , the aggregate variation due to fine scale motions would always invalidate the continuum hypothesis.

Over a time interval , the kinetic energy of flow motion is

 ρ2∫R3u2(x,t)\rdx.

Since we are working in , the energy may appear to be infinite for an observer moving with a constant finite speed along any straight path which can be arbitrarily far away from the origin. It is defensible that the energy appears to be unbounded. Such an anomaly is due to our choice of frame of reference. As a remedy, we reformulate our fluid dynamics problem by choosing a Galilean transform so that the observer becomes stationary relative the fluid motion. Thus the energy remains finite unless the velocity is out of bounds during the flow development. In subsequent analysis, we take it for granted that a Galilean transform is effected.

As in many applications, we deal with the Navier-Stokes equations for a fluid enclosed by a smooth, impermeable boundary of finite size, the initial condition must be supplemented by the no-slip Dirichlet boundary condition. For fluid motions in , the “boundary condition” takes the form of decay:

 u→0as|x|→∞.

The decay specification is natural and its physical explanation is evident. However, it is of importance to appreciate that the decay is merely a qualitative statement. In rigorous mathematical analyses, what is crucial are quantitative decay rates, such as . In general, they are not known a priori.

The Navier-Stokes equations are a system of non-linear parabolic partial differential equations. In , the absence of a solid boundary simplifies our problem since there are no external sources for generating vorticity. Once the initial data are specified, there do not exist a characteristic velocity and a reference length scale during the flow evolution. Thence the equations are in a canonical form where the only parameter of dynamic similarity is related to the property of the fluid . It has long been conjectured that a singular behaviour may develop during flow evolution (Oseen 1927; Leray 1934). Allowance must be made for solutions of the equations to be rough functions which have limited regularity in space or in time. The concept of weak solutions was introduced by Leray. In practice, we have to deal with distribution solutions; no a priori assumptions should be made on the integrability and differentiability on the triplet . In three space dimensions, some a priori bounds are known to exist but they do not lend themselves to the solution of the global regularity. Nevertheless, we observe that an equation governing the vorticity dynamics enjoys certain symmetry which in turn imposes a cancellation condition at infinity so that an invariance principle holds over the entire flow evolution.

To attack the Cauchy problem, we introduce two separate courses of analysis. The first one focuses on the method of interpolation for certain Sobolev spaces. A substantial analysis is devoted to deriving coercive a priori bounds. The well-posedness for the smooth initial data follows without difficulty. We then address the global regularity of the Navier-Stokes equations by extending the properties of the smooth solution to initial data .

Since our ultimate aim of solving the Navier-Stokes equations is to elucidate on the nature of turbulence, it is of importance to evoke our second method which solves a non-linear integral equation for the vorticity by construction. The analytical structure of the solution represents a complex vorticity field from which many ingredients for turbulence can be found. On the basis of our solution of the Navier-Stokes equations, we make an attempt to account for the transition phenomenon in Reynolds’ pipe flow experiments (Reynolds 1883). Furthermore, it is a well-established experimental fact that turbulent fluid motions exhibit intensive, irregular fluctuations at high Reynolds numbers. The kinetic energy is supplied on the continuum and dissipated at the microscopic scales due to molecular friction. It is intuitively clear that we ought not be able to adequately understand many observed characters of turbulence without any knowledge of fluid’s microscopic properties. Consequently, an effort is made to inquire the connection between the continuum fluid dynamics and the kinetic theory of gases (Maxwell 1867; Boltzmann 1905).

The study of the Navier-Stokes equations and turbulence is an immense subject; there exists a large collection of literature. It must be admitted that it is almost impractical for one to comprehend all the technical details. So we would like to stress our priority for solving the problems of fluid dynamics; it is not our main concern here to conduct an overview of the past work. To grasp basic developments in the mathematical theory, one may consult recent reviews and monographs (see, for example, Rosenhead 1963; Serrin 1963; Ladyzhenskaya 1969; Temam 1977; Constantin & Foias 1988; Stuart 1991; Doering & Gibbon 1995; Lions 1996; Foias et al 2001; Temam 2001; Lemarie-Rieusset 2002; Cannone 2003; Ladyzhenskaya 2003; Germain 2006; Heywood 2007; Doering 2009). For the development of the equations from a historical prospective, consult Darrigol (2002).

The following list on turbulence is by no means comprehensive. These works and the references cited therein contain detailed accounts on theoretical and practical issues relating to turbulence, notably on the work by Reynolds, Richardson, Prandtl, Taylor, Kolmogorov, Kraichnan and others. Dedicated monographs on theory of turbulence are available (see, for example, Batchelor 1953; Bradshaw 1971; Tennekes & Lumley 1972; Leslie 1973; Hinze 1975; Monin & Yaglom 1975; Townsend 1976; McComb 1990; ; Saffman 1992; Chorin 1994; Pope 2000; Tsinober 2001; Davidson 2004; Lesieur 2008). We recommend the review articles by Prandtl 1925; von Kármán 1948; von Neumann 1949; Lin 1959; Corrsin 1961; Saffman 1978; Liepmann 1979; Frisch & Orszag 1990; Bradshaw 1994; Hunt 2000; Moffatt 2000; Lumley & Yaglom 2001.

Most mathematical symbols and notations used in the present work are standard. The parabolic cylinder in space-time is written as for given fixed , where time is finite and given. We use multi-index notation for the spatial partial derivatives of order , , for all multi-indexes with . All ’s are non-negative integers. For integrals over , we write for . It is convenient to work in the space of

 Y=L2(R3)∩C0c(R3),

where denotes the set of continuous functions with compact supports, normed by . The norm has the dimensions of and the same dimensions hold for . When we refer to space , we mean the intersection of two Banach spaces of functions as , assuming that the spaces have been normalized so that equality’s dimensions are consistent. We use shorthand notations and for space and time respectively.

Many mathematical symbols have multiple meanings in different sections but extra care has been taken to ensure that their uses do not cause confusion.

## 2 Vorticity as a physical characteristic

Consider an infinitesimal fluid element whose sides are and . Its mass centre is at which is moving with velocity and rotating with angular velocity /2, see figure 1. The instantaneous rate of deformation of the element can be written as a sum of a symmetric tensor for strains and an antisymmetric tensor for rotations:

 ∂ui∂xj=12(∂ui∂xj+∂uj∂xi)+12(∂ui∂xj−∂uj∂xi)=Sij+Rij. (8)

The deviatoric stress tensor is related to the strain tensor by

 τij=2μ(Sij−δij∇.u/3)=2μSij. (9)

The symbol is the Kronecker delta. The antisymmetric tensor equals to the angular velocity. The vorticity is twice of the instantaneous local angular velocity (Stokes 1845; Truesdell 1954). The total stress in the -direction is given by

 σij=−pδij+τij.

Over an infinitesimal time interval , the velocity changes to and the vorticity to . The changes in the angular momentum can be considered to consist of two parts. The first one is associated with the change of angular velocity as if the element behaves like a rigid body rotating about the axes, see the left sketch in figure 2. Because of symmetry, the angular momentum is about the axes passing through the mass centre of the element. The axes are in fact the instantaneous principal axes. The moment of inertia of the element about one (say ) of its principle axes is given by or , where the element is treated as a particle, and hence its actual geometry is immaterial. Denote the mass of the fluid element by . The moment of inertia is simply given by the diagonal matrix whose entries are

 Iii=m(δxi)2.

The second part of the angular momentum is related to the change of the geometric shape because the element is being deformed during the motion. The result of the deformation leads to the loss of the geometric symmetry so that the mass centre no longer coincides with its geometric centre, see the right sketch in figure 2. To determine the instantaneous angular momentum, we make use of the inertia tensor about point . The tensor has the form

 J=m⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝(δx2)2+(δx3)2−δx1δx2−δx1δx3−δx2δx1(δx3)2+(δx1)2−δx2δx3−δx3δx1−δx3δx2(δx1)2+(δx2)2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

This is the inertia tensor for rigid body: , where is the position vector. This formula is well-known in mathematical physics (see, for example, Byron & Fuller 1969; Landau & Lifshitz 1976). The off-diagonal entries are the products of inertia. The entries, , stand for the moment of inertia around the -axis when the element rotates about the -axis with angular velocity .

Thus the rate of change of angular momentum is found from the following two relations:

 m2I\rdω\rdt∣∣\small{Rigid % body}+m2\rdJ\rdtω∣∣\small{% Deformed body}=m2(LR+LD),

and

 \rdm\rdtJω,

where is vorticity column vector. The rate of change in mass is

 \rdm\rdt=∂ρ∂tδV+∂(ρV)∂t=∂ρ∂tδV+∂(ρui)∂xiδV.

The differential equality reduces to, after simplification,

 ∂ρ/∂t+∇.(ρu)=Dρ/Dt+ρ(∇.u)=0 (10)

in view of the principle of mass conservation. The continuity equation for incompressible flows is recovered according to hypothesis .

The total force per unit volume in the -direction can be calculated from the normal and the shear stresses,

 ∂σij/∂xi=−∂p/∂xj+∂τij/∂xi.

Now we derive the rate of change of angular momentum in the -direction. The contribution relating to the rigid body rotation per unit mass per unit element area has the form

 LR=∂ω1∂t+∂ω1∂u1∂u1∂x1δx1δt+∂ω1∂u2∂u2∂x2δx2δt+∂ω1∂u3∂u3∂x3δx3δt=∂ω1∂t+u1∂ω1∂x1+u2∂ω1∂x2+u3∂ω1∂x3.

Since and , from the first row of the tensor , the contribution due to deformation is given by

 LD=ω1(δx2δx2δt+δx3δx3δt)−ω2δx2δx1δt−ω3δx3δx1δt=ω1(∂u2∂x2(δx2)2+∂u3∂x3(δx3)2)−ω2∂u1∂x2(δx2)2−ω3∂u1∂x3(δx3)2.

Let the net shear force on the plane be , see figure 3.

The infinitesimal change in torque due to shear per unit mass between the two opposite planes is

 δx3∂(F2δx1δx2)∂x3.

As the pressure acts in the normal direction to the surfaces of the element, the resultant force in the direction is a pure shear force

 F2=∂τ12∂x1+∂τ22∂x2+∂τ32∂x3.

Similarly, the shear in the direction is

 F3=∂τ13∂x1+∂τ23∂x2+∂τ33∂x3.

Hence the net torque per volume about the -axis is given by

 12(∂F3∂x2−∂F2∂x3)=12μΔω1,

where the last quantity has been obtained by direct calculation using (8), (9) and (10). If an external force is present, the additional torque due to this force must be included. In the limit , the sides of the fluid element are indistinguishable so that . Newton’s second law for the rate of change of angular momentum gives rise to an identity: . In view of the continuity, this relation yields the equation for the -component of vorticity:

 (∂/∂t+u.∇)ω1−(ω.∇)u1=νΔω1.

Two similar equations for the other two vorticity components can be derived by cyclic permutations. Thus the vorticity equation is written as

 ∂ω/∂t−νΔω=(ω.∇)u−(u.∇)ω. (11)

The vorticity field is solenoidal

 ∇.ω=0. (12)

For the Cauchy problem, the initial vorticity is specified as

 ω(x,0)=∇×u0(x)=ω0(x). (13)

We assume that the localization requirements in (4) apply.

The vorticity equation for incompressible flows can be derived from the first principle of conservation of angular momentum. Thus the vorticity must be regarded as a physical characteristic in fluid motions governed by the Navier-Stokes dynamics.

## 3 The vorticity Cauchy problem and a priori bounds

With the vector identity,

 ∇(F.G)=F×(∇×G)+G×(∇×F)+(F.∇)G+(G.∇)F,

for two vectors and , the Navier-Stokes momentum equation can be rewritten as

 ∂u/∂t−νΔu=u×(∇×u)−∇χ, (14)

where denotes the Bernoulli-Euler pressure. This form of the equation takes advantage of the symmetry in quantity that is a spatial invariant in . We take divergence of (14) to get an analogous equation to (5):

 Δχ(x;t)=(ω.ω−u.(∇×ω))(x;t). (15)

As allowance must be made for the presence of a possible singularity in the velocity-pressure field, we begin our analysis by using a standard mathematical device. We introduce a sequence of regularizations on the solutions of the momentum equation. For fixed , we put . The symbol is referred as the mollification parameter, and we wish to examine the behaviour of the flow field when the parameter becomes arbitrarily small. We use notation to denote the mollification of the velocity . In essence, the mollification is a convolution in space and in time

 ϕδ(u)(x,t)=δ−4∫R∫R3˜ϕ(yδ,sδ)u(x−y,t−s)\rdy\rds. (16)

The kernel function, , is a mollifier which is a smooth function in space and in time with compact supports. In addition, it has the properties of , and

 ∫R∫R3˜ϕ(x,t)\rdx\rdt=1.

In view of the well-known properties of mollifiers (see, for example, Adams & Fournier 2003; Majda & Bertozzi 2002; Brezis 2011), the mollification preserves the solenoidal property of :

 ∇.ϕδ(u)=ϕδ(∇.u)=0. (17)

Similarly, the mollification of the vorticity implies that

 ∇.ϕδ(ω)=ϕδ(∇.ω)=0. (18)

When it is necessary, the initial data may also be mollified:

 ψδ(u0)(x)=δ−3∫R3˜ψ(yδ)u0(x−y)\rdy, (19)

where the mollifier function is non-negative, and its space integral equals to unity. Let and . The mollified momentum equation reads

 ∂u/∂t−νΔu=ϕδ(u)×(∇×u)−∇χ. (20)

The continuity constraint (1) remains unchanged. The values of at time depend solely on the values of at positive time in the interval . This procedure of mollification has been used to construct weak or generalized solutions of the Navier-Stokes equations in the work of Leray 1934; Scheffer 1976; Caffarelli et al 1982; Constantin 1990. The orthodox Leray mollification refers to smoothing the momentum equation in the form

 ∂u/∂t−νΔu=−(ϕδ(u).∇)u−ρ−1∇p.

### Bounds derived from vorticity equation

Taking curl of equation (20) and taking the solenoidal constraints (17) and (18) into account, we obtain

 ∂ω/∂t−νΔω=(ω.∇)ϕδ(u)−(ϕδ(u).∇)ω, (21)

where the vorticity . This dynamic equation has an important property of being symmetric with respect to the velocity mollification.

We consider the components of (21) and let

 Ri(x,t,u,ω)=((ω.∇)ϕδ(ui)−(ϕδ(u).∇)ωi)(x,t). (22)

Carrying out an integration over space on the sum of three components, integrating by parts and taking the continuity constraints (12) and (17) into account, we deduce the summability condition,

 3∑i=1∫Ri(x,t,u,ω)\rdx=0. (23)

We notice that this identity is independent of the mollification parameter . This observation suggests that it is advantageous to work in terms of the total vorticity,

 \upomega=ω1+ω2+ω3. (24)

By Duhamel’s principle, the total vorticity satisfies the following scalar integral equation :

 \upomega(x,t)=∫Z(x−y,t)\upomega0(y)\rdy+∫t0∫Z(x−y,t−s)3∑i=1Ri(y,s,u,ω)\rdy\rds, (25)

where is the initial total vorticity. The integral kernel is the fundamental solution of the heat equation in ,

 Z(x−y,t)=(4πνt)−3/2exp(−|x−y|24νt),t>0. (26)

A further integration followed by a trivial application of Fubini’s theorem enables us to infer the invariance of the total vorticity:

 \rd\rdt∫\upomega(x,t)\rdx=0 (27)

thanks to the well-known properties of the heat kernel.

The total vorticity is shown to be Lebesgue integrable in space and in time provided that the initial total vorticity is an integrable function on . The invariance principle (27) states that, in any fluid motion, the total vorticity is conserved during the entire course of flow evolution. In particular, if the initial total vorticity is zero, the total vorticity remains zero at all subsequent time.

Consider any subset . Since there exists a non-zero subset such that , it implies that does not vanish altogether. It follows that the invariance principle (27) implies

 ∫\upomega(x,t)\rdx=∫\upomega0(x)\rdx≤∥\upomega0∥L1(R3).

(If the initial velocity has compact support, then the initial energy is finite. Thus the invariance principle renders the left-hand side to zero.) Consider as a measurable, real-valued function, we write , where both and are measurable, non-negative and finite. Hence the set has measure zero. By the Archimedean property of the real numbers (see, for example, Royden & Fitzpatrick 2010), we deduce that every component of vorticity satisfies the bound,

 ∥∥ωi(x)∥∥L1(R3)≤N∥∥\upomega0(x)∥∥L1(R3),

except possibly on a set of measure zero. The symbol denotes a natural number and it can be suitably chosen for any . Making use of the elementary identity

 ω21+ω22+ω23≤(|ω21|+|ω22|+|ω23|)2,

and of the fact that , the integrability bound can be upgraded to

 ∥∥ω(x)∥∥L1(R3)≤N∥∥ω0(x)∥∥L1(R3) (28)

for some natural number . The vorticity invariance is well-known for the Euler equations (see, for example, Batchelor 1969; Majda & Bertozzi 2002). The existence of vorticity -bound in terms of initial energy norm was shown by Constantin (1990) in a periodic domain and by Qian (2009) on a 3D torus.

Differentiating the mollified vorticity components in (21) yields

 ∂(∂xωi)/∂t−νΔ(∂xωi)=∂xRi(x,t,u,ω).

We validate the following integral condition analogous to (23):

 3∑i=1∫∂xRi(x,t,u,ω)\rdx=0.

Thus

 \rd\rdt∫∂x\upomega(x,t)\rdx=0, (29)

because the operators and the gradient operator commute. By analogy, we see that the invariance (29) implies the summability of :

 Extra open brace or missing close brace (30)

for some natural number . Since the vorticity and its gradient are integrable in , we establish the a priori bound

 ω∈L3/2(R3) (31)

by virtue of the Sobolev embedding theorem (see, for example, Adams & Fournier 2003; Brezis 2011). The norm number is just the Sobolev conjugate of unity. In view of the scaling properties given in (7), this bound is scale-invariant and hence it is a critical bound. In fact, we can generalize the integrability procedure as follows. For any integer , we derive the following equation for the vorticity derivatives:

 ∂(∂αxωi)/∂t−νΔ(∂αxωi)=∂αxRi(x,t,u,ω).

We confirm that the space-wise integral renders the sum () to zero. Thus we have the invariance,

 \rd\rdt∫∂αx\upomega(x,t)\rdx=0,α>0, (32)

and the bound,

 ∥∥∂αxω(x)∥∥L1(R3)≤Nα∥∥∂αxω0(x)∥∥L1(R3),

for some natural number . Although the invariance of the total vorticity holds for the space derivative of arbitrary order, it is sufficient to restrict our derivations to cases . Now the vorticity belongs to the Sobolev space or . We conclude that the vorticity is a priori bounded for fixed

 ∥∥ω(x)∥∥L∞(R3)≤N∥∥ω0(x)∥∥L∞(R3)a.e. (33)

Evidently, this bound is independent of the mollification parameter . Interpolation of Lebesgue spaces and furnishes the estimate

 ∥∥ω(x)∥∥Lp(R3)≤C∥∥ω0(x)∥∥Lp(R3),1≤p≤∞, (34)

where constant . One of the important consequences is that the enstrophy, , which controls the dissipation of kinetic energy, is bounded a.e. in space – a presumption stipulating the related dynamic bound .

In general, the invariance (32) implies the spatial smoothness,

 ω(x)∈CαB(R3),

because the procedures for integrability consistently enable us to verify .

The following equation of Poisson type defines a kinematic relationship between the vorticity and the solenoidal velocity:

 Δψ(x;t)=−ω(x;t), (35)

where denotes a solenoidal stream-function vector (see, for example, Lamb 1975). Since , there exists a distribution solution . Consequently, the Laplacian has an inverse (see, for example, Folland 1995; Gilbarg & Trudinger 1998). The velocity is recovered by computing at every instant of time, where the curl operation is interpreted as a distributional derivative of the stream-function. Explicitly, the velocity is given by the Biot-Savart law

 u(x;t)=14π∫(x−y)|x−y|3×ω(y;t)\rdy=K∗ω(x;t). (36)

We put special emphasis on our use of notation (cf. (5)) in order to remind ourselves of the fact that velocity-vorticity relation (36) alone does not define a function of time . The gradient in the Biot-Savart law is related to which defines a singular operator of classical Calderón-Zygmund type. Thus a variant of the Calderón-Zygmund theorem for solenoidal velocity can be readily derived:

 ∥∥∇u(x)∥∥Lp(R3)≤C∥∥ω(x)∥∥Lp(R3),1

Applying the Hardy-Littlewood-Sobolev inequality, we have

 ∥∥u(x)∥∥Lq(R3)≤C∥∥∇u(x)∥∥Lp(R3),1/q=1/p−1/3,

where denotes a constant and can be evaluated sharply (see, for example, §4 of Lieb & Loss 1997). Hence a relation between the vorticity and the velocity can be deduced:

 ∥∥u(x)∥∥Lq(R3)≤C∥∥ω(x)∥∥Lp(R3),1/q=1/p−1/3 (38)

for any and .

Next we introduce a device so as to partition domain into two parts: a ball centred at the origin with finite radius , and its complement . In view of Young’s inequality for convolution, the velocity represented in (36) can be estimated according to

 ∥∥u(x)∥∥Lr(R3)≤(∥∥K∥∥Lp(B)∥∥ω(x)∥∥Lq(R3)+∥∥K∥∥Lp′(Bc)∥∥ω(x)∥∥Lq′(R3)), (39)

where , , and . We readily find that the energy must be bounded, viz

 ∥∥u(x)∥∥L2(R3)≤C1∥∥ω0(x)∥∥L1(R3)+C2∥∥ω0∥∥L2(R3),

where and are constants. By the same token, we establish

 ∥∥u(x)∥∥L∞(R3)≤C1∥∥ω0(x)∥∥L∞(R3)+C2∥∥ω0∥∥L2(R3). (40)

So far we have not yet evoked the incompressibility hypothesis, , which in fact defines a finite sum or a zero sum of three (extended) real numbers. Intuition suggests that every member, , must be finite at every time , or . The finiteness may be inferred from the properties of the reals (see, for instance, §3F of Beals 2004). Instructively the expressions, and , are undefined. If one member becomes infinite while the other two are bounded, their sum violates the hypothesis in one direction to infinity (either or ). Similar arguments holds if two members are unbounded. Should the trace of the velocity Jacobian be out of bounds, the immeasurable manifold would insinuate either a vacuum or an infinity plenum.

By the differential form of the conservation, we have

 0=∇.u=∇.u0≤N1∥∂xu0(x)∥L∞(R3),∀x∈R3,

where is a natural number. Conceptually, it is impossible for a gradient on the left to be unbounded. We now show that it is indeed the case.

Applying Gauss’ divergence theorem, we obtain that, from the integral form of the mass conservation,

 ∫(∇.u)(x)\rdx=∫∂S(u⋅→n)(x)\rdx=0,

where denotes a spherical surface whose radius , and the outward normal on the surface. This apparently simple integral relation enables us to derive two important properties of the velocity. The first one is the rate of the velocity decay

 |u|∼O(|x|−2)as|x|→∞.

This condition is consistent with the bound . The second is the summability of the velocity gradient because

 ∫(∇.u)(x)\rdx≤∥∇u0∥L1(R3).

It follows that every component, , where is a natural number. Combining this result with bound (34), we assert that

 ∥∥∇u(x)∥∥L1(R3)≤C∥∥∇u0(x)∥∥L1(R3), (41)

where is constant. The Archimedean principle for the reals suggests that , where denotes some natural numbers. To be specific, we may write the bound as , where is a constant. For convenience, we also denote

 ∥ω(x)∥q≤C(∥ω0(x)∥Y),1≤q≤∞. (42)

Similarly, can be specifically bounded in this way, as implied in (32).

The identity, , shows that the velocity Laplacian,

 Δu=−∇×ω,

is just a kind of vorticity. This relation is a direct consequence of the continuity. The solenoidal quantity, , is known as divorticity and satisfies the following dynamic equation:

 ∂(∇×ω)i/∂t−νΔ(∇×ω)i=∇×(Ri(x,t,u,ω)).

As the curl is a differential operator, by analogy, the invariance and integrability properties established for vorticity apply equally to the divorticity. The strength of the double curl is measured by palinstrophy, . It is clear that

 ∥∥∇×ω(x)∥∥Lq(R3)≤∥∥∇ω(x)∥∥Lq(R3),1≤q≤∞,

as all the derivatives of are in which has a total of velocity derivatives. This simple rule can be applied to higher differentiations of the divorticity. Essentially, the velocity Jacobian can be found from the expression

 ∇u(x;t)=−∫∇xN(x,y)(∇×ω)(y;t)\rdy=M∗(∇×ω)(x;t),

where is the Newtonian potential. Following the idea of partition (cf. (39)), we readily establish that

 Extra open brace or missing close brace

By interpolation, we conclude that

 ∥∥∇u(x)∥∥Lq(R3)≤C(∥ω0∥Y),1≤q≤∞. (43)

Evidently, bounds (41) and (43) supplement (37).

To determine whether solutions of the Navier-Stokes equations are smooth, we need a priori bounds for the time variable. We observe that

 ∂t(∂βt\upomega)−νΔ(∂βt\upomega)=∂βt(3∑i=1Ri(x,t,u,ω)) (44)

for any . It is clear that the right-hand side is still in . Thus

 \rd\rdt∫∂βt\upomega(x,t)\rdx=0,t∈[TL,T], (45)

where . Smooth and unique solutions of the Navier-Stokes equations over time interval are known to exist, as a result of local in-time analyses (see, for example, Leray 1934; Hopf 1951; Ladyzhenskaya 1969; Temam 1977; Heywood 1980). Specifically, the local time depends on the norm size of initial data (3). The delicate issue of how the smooth solutions assume the initial data as has been fully vindicated by Heywood (1980, 2007). Consequently, invariance (45) must be valid from or

 \rd\rdt∫∂βt\upomega(x,t)\rdx=0,t∈[0,T]. (46)

Clearly, for any value of ; this bound is independent of the mollification parameter . Once again, we emphasize that bound (46) alone does not imply any time-wise a priori bound for and with exception of the classical time interval , because neither the pressure relation (5) nor the Biot-Savart law (36) contains temporal information.

Let in (46). In view of the local in-time solutions, we deduce that , where . (This choice of the lower time bound is to avoid unnecessary complications in specifying the initial data.) Obviously we may continue this process of upgrading regularity. In conclusion, we assert that

 ω(x,</