## Abstract

Higher moments of the vorticity field *Ω*_{m}(*t*) in the form of *L*^{2m}-norms () are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier–Stokes equations on the domain . It is found that the set of quantities
provide a natural scaling in the problem resulting in a bounded set of time averages 〈*D*_{m}〉_{T} on a finite interval of time [0, *T*]. The behaviour of *D*_{m+1}/*D*_{m} is studied on what are called ‘good’ and ‘bad’ intervals of [0, *T*], which are interspersed with junction points (neutral) *τ*_{i}. For large but finite values of *m* with large initial data (*Ω*_{m}(0)≤*ϖ*_{0}*O*(*Gr*^{4})), it is found that there is an upper bound
which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray (Leray 1934 *Acta Math.* **63**, 193–248 (doi:10.1007/BF02547354)) and Scheffer (Scheffer 1976 *Pacific J. Math.* **66**, 535–552),— this estimate for *Ω*_{m} corresponds to a length scale well below the validity of the Navier–Stokes equations.

## 1. Introduction

The challenge that analysts have faced in the last 75 years has been to prove the existence and uniqueness of the three-dimensional Navier–Stokes equations for arbitrarily long times (Leray 1934; Ladyzhenskaya 1963; Serrin 1963; Constantin & Foias 1988; Temam 1995; Foias *et al.* 2001). Its inclusion in the American Mathematical Society Millennium Clay Prize list (Fefferman 2000) has widely advertised the nature of the problem, but the elusiveness of a rigorous proof^{1} and the severe resolution difficulties encountered in computational fluid dynamics, even at modest Reynolds numbers, are puzzles that have grown as the years progress.

Nevertheless, there is a long-standing belief in many scientific quarters, on the level of a folk-theorem, that the three-dimensional Navier–Stokes equations ‘must’ be regular. Mathematicians are more cautious and still take seriously the possibility that singularities may occur. Leray (1934) and Scheffer (1976) proved that the (potentially) singular set in time has zero half-dimensional Hausdorff measure; see also (Robinson & Sadowski 2007). Scheffer then introduced the idea of suitable weak solutions and showed that the dimension of the singular set in space–time has a dimension bounded by 5/3 (Scheffer 1980). Caffarelli *et al.* (1982) then developed the idea of partial regularity using suitable weak solutions and reduced this bound to unity. Thus, if space–time singularities exist, then they must be relatively rare events. These ideas have spawned a growing literature in which more efficient routes to the construction of suitable weak solutions are in evidence (Lin 1998; Ladyzhenskaya & Seregin 1999; Tian & Xin 1999; Choe & Lewis 2000; Escauriaza *et al.* 2003; Seregin 2005, 2006; Gallavotti 2006; Vasseur 2007; Kukavica 2008*a*,*b*; Robinson & Rodrigo 2009).

It is worth remarking that the wider issue regarding the formation of singularities has been obscured by the very great difficulty that exists in distinguishing them from rough intermittent data. Intermittency is characterized by violent surges or bursts away from averages in the energy dissipation, resulting in the spiky data that is now recognized as a classic hallmark of turbulence (Batchelor & Townsend 1949; Kuo & Corrsin 1971; Douady *et al.* 1991; Meneveau & Sreenivasan 1991; Zeff *et al.* 2003). At least three options are possible:

solutions are always smooth with only mild excursions away from space and time averages,

solutions are intermittent, but, despite their apparent spikiness, remain smooth for arbitrarily long times when examined at very small scales, and

solutions are intermittent, but spikes may be the manifestation of true singularities.

Options (ii) and (iii) are impossible to distinguish using known computational methods. The Leray–Scheffer result, in simple terms, shows that potential singularities in time must be distributed as no more than points on the time axis, but it contains little other information; for instance, these may not be countable or isolated. Both for analytical and computational reasons, it would be desirable to understand the potential structure of the solution in more detail. The aim of this paper is to address this issue.

In the past generation, physicists have used Kolmogorov’s theory to examine intermittent events by studying anomalies in the scaling of velocity structure functions. This theory is based on a set of statistical axioms, not directly on the Navier–Stokes equations. Nevertheless, to make a comparison, the intermittent dynamics discussed above would lie deep in the dissipation range of the energy spectrum. The book by Frisch (1995) and the recent review by Boffetta *et al.* (2008) contain readable accounts of these ideas.

### (a) General strategy

The main idea of this paper is to use higher moments of the vorticity field ** ω** instead of derivatives. Scaled by a system volume term

*L*

^{−3}, a set of moments with the dimension of a frequency are defined such that, for

*m*≥1, 1.1 where

*ϖ*

_{0}=

*νL*

^{−2}is the basic frequency of the domain of side

*L*.

*Ω*

_{1}is synonymous with the

*H*

_{1}-norm and sits within the sequence of inequalities 1.2 so control from above over

*Ω*

_{m}for any value of

*m*>1 implies control over the

*H*

_{1}-norm which, in turn, controls from above all derivatives of the velocity field

^{2}(Ladyzhenskaya 1963; Serrin 1963; Constantin & Foias 1988; Temam 1995; Foias

*et al.*1981, 2001).

A technical problem lies in how to differentiate the *Ω*_{m}(*t*) and manipulate them without the existence of strong solutions for arbitrarily large *t*. This difficulty can be circumvented by restricting estimates to a finite interval of time [0,*T*] and then pursuing a contradiction proof in the following standard manner. Assume that there exists a *maximal* interval of time on which solutions exist and are unique; that is, strong solutions are assumed to exist in this interval. If is indeed maximal, then . The ultimate aim of such a calculation would then be to show that is finite for any *m*≥1; if this turned out to be the case, it would lead to a contradiction because would not be maximal. Thus, must either be zero or infinity; it cannot be zero because it is known that there exists a short interval [0, *t*_{0}) on which strong solutions exist, so .

The results in §2 have been estimated using this strategy. It turns out that there exists a natural scaling within the Navier–Stokes equations, which makes the variable
1.3
the most natural to choose. Then, theorem 2.3 shows that the time average—see equation (1.17) for a definition of 〈·〉_{T}—is given by
1.4
with a uniform constant *c*_{av}. Two remarks are in order. Firstly, it is not difficult to extract an estimate for a set of length scales from equation (1.4). Defining , this shows that
1.5
and therefore^{3}
1.6
The exponent *α*_{m} within the definition of *D*_{m} appears to be a natural scaling consistent with that of the Sobolev inequalities. This paper suggests that the breaking of this scaling through stretching between *D*_{m+1} and *D*_{m} may be required to make progress. This is gauged more specifically in theorem 3.1 in §3 where it is shown that a finite interval [0, *T*] of the time axis can be potentially broken down into three classes, denoted by good and bad intervals with a set of junction points (or intervals) {*τ*_{i}} designated as *neutral*. In §3, it is found that the direction of the inequality is reversed on the good and bad intervals, that is
1.7
In equation (1.7), *p*(*T*) is a *T*-dependent exponent (>2) of the Grashof number *Gr* and *μ*_{m} is a parameter in the range 0<*μ*_{m}<1. The universal inequality *Ω*_{m}≤*Ω*_{m+1} ultimately shows that, on good and neutral intervals,
1.8
where is a function of *p*(*T*), *Gr*, *α*_{m} and *μ*_{m}. The main question lies in the nature of the transition from the good to the bad intervals through the neutral points *τ*_{i}. On bad intervals, the application of the reverse inequality in equation (1.7) to the differential inequality for *D*_{m} in proposition 2.1 results in regions smaller in amplitude than in which solution trajectories remain bounded by
1.9
The bad regions are not absorbing: solutions remain inside these regions if they enter inside, but they are not attracted into them if they lie outside. The key point is that for all *finite* values of *m*≥1, , thereby leaving vertical gaps or windows through which trajectories can potentially escape to infinity. However, while the gap between and closes for large *m*, the limit is forbidden, and so these windows can only be reduced to infinitesimally small holes that puncture a general upper bound. This result is consistent with that of Leray (1934) and Scheffer (1976). In terms of *Ω*_{m}, this punctured bound turns out to be
1.10
The length-scale equivalent to this upper bound is exceptionally small and is well below where the Navier–Stokes equations are valid. The conclusion is that, unless other unknown controlling mechanisms are shown to exist, the Navier–Stokes equations may formally possess solutions that either become singular or, if they continue to exist, may be unresolvable numerically.

### (b) Notation and functional setting

The setting is the incompressible (*div* ** u**=0), forced, three-dimensional Navier–Stokes equations for the velocity field

**(**

*u***,**

*x**t*) 1.11 with the equation for the vorticity expressed as 1.12 Various definitions are given in table 1. The domain is taken to be three-dimensional and periodic. The forcing function

**(**

*f***) is**

*x**L*

^{2}-bounded and the Grashof number

*Gr*is proportional to ∥

**∥**

*f*_{2}. The forcing is also endowed with the further property that it contains a smallest scale, such that, for

*n*≥1, 1.13 where ℓ is the smallest scale and a constant. The aspect ratio is

*a*

_{ℓ}=

*L*ℓ

^{−1}.

Now define
1.14
where the frequencies *Ω*_{m} are given by
1.15
The term *ϖ*_{0} in equation (1.14) provides a lower bound for *Ω*_{m}. Indeed, it is easy to prove that
1.16
The symbol 〈·〉_{T} denotes the time average up to time *T*,
1.17

## 2. Some properties of *Ω*_{m}(*t*)

This section firstly contains a result concerning the differential inequalities that govern the set of frequencies *Ω*_{m}(*t*). Secondly, it contains a result that is an estimate for an upper bound on a set of time averages over the interval [0, *T*]. Finally, it contains a result on the nature of exponential bounds on [0, *T*]. All of the proofs, which lie in appendices A–C, are based on the contradiction strategy explained in §1*a*. Firstly, we define
2.1

### Proposition 2.1.

*On [0,T], for* *,* *and Gr≥1, the D*_{m} *satisfy*
2.2
*where ρ*_{m}*=2m(4m+1)/3. For the unforced case, the last term on the right-hand side of equation (2.2) is proportional to c*_{3,m}*.*

### Remark 2.2.

Note the strict inequality ; the Riesz transform used in the proof in appendix A requires the introduction of higher derivatives when . Some multiplicative terms in the forcing aspect ratio *a*_{ℓ} have been absorbed into the constant *c*_{3,m} to save algebra.

### Theorem 2.3.

For and *Gr*≥1,
2.3
where *E*_{0}=*E*(0) is the initial value of the energy. For the unforced case, the estimate is
2.4

## 3. Trajectories on good, bad and neutral intervals

### (c) The ratio *D*_{m+1}/*D*_{m}

Given proposition 2.1, understanding the behaviour of the ratio *D*_{m}/*D*_{m+1} is an important step. The figures in this section are there to illustrate some of the estimates.

### Theorem 3.1.

For the parameters *μ*_{m}=*μ*_{m}(*T*, *p*, *Gr*) with values in the range 0<*μ*_{m}<1, the ratio *D*_{m}/*D*_{m+1} obeys the inequality
3.1

### Remark 3.2.

The proof lies in appendix C and is dependent on the result of theorem 2.3.

### Remark 3.3.

Theorem 3.1 implies that, while there must be intervals where the integrand is positive, there could also be intervals where it is negative. It tells us nothing about the interval size or distribution.

Formally, the theorem leads to the conclusion that there exists at least one good interval of time within [0, *T*] on which
3.2
while there potentially exist bad intervals of time on which
3.3
Neutral points or intervals represented^{4} by the zeros of the integrand in equation (3.1) lie at
3.4

In terms of *Ω*_{m+1} and *Ω*_{m}, equations (3.2) and (3.3) become
3.5
where and *γ*_{m} are defined by
3.6
3.7and
3.8

The positivity of *γ*_{m} requires that *μ*_{m} be bounded away from zero such that
3.9

Because *Ω*_{m+1}≥*Ω*_{m}, equation (3.5) shows that, on good and neutral intervals,
3.10

Now we turn to the bad intervals: consider equation (3.3) in (2.2), in which case (*Ω*_{n}≤*Ω*_{m}),
3.11
where *ρ*_{m}=2*m*(4*m*+1)/3, but is forbidden. The range of validity of *μ*_{m} in equation (3.9) can be re-written as *ρ*_{m}>*μ*_{m}*ρ*_{m}>2. Thus, if, at the time of entry *τ*_{i} into a bad interval,
3.12

Given that *ρ*_{m}*μ*_{m}>2 and *α*_{n}≥*α*_{m}, the first term on the right-hand side of equation (3.12) is dominant. Using the lower bound *D*_{m}≥1, it is found that
3.13
where
3.14and
3.15

### (b) How large are and ?

Figure 1 illustrates the case *m*=1, for which we have *b*_{1}/*a*_{1}=1 and *ρ*_{1}=10/3; the difference in the sizes of and lies in the upper bounds on *μ*_{1} and . The latter has been defined in equation (3.8),
3.16

From equations (3.6) and (3.10), we have 3.17 which, on minimization of the right-hand side, gives 3.18

The equivalent estimate for *D*_{1,bad} is
3.19

It is useful to re-work these estimates in terms of a point-wise inverse^{5} length scale, with a point-wise energy dissipation rate, . The result,
3.20
is shown in figure 2, where the constant on the bad estimate is slightly smaller.

### (c) How large are and for large m?

From the definitions of equations (3.6) and (3.14) and the fact that , it is clear that , keeping in mind that the limit is forbidden. Firstly, the *c*_{i,m} are polynomial in *m* and *ρ*_{m}∼*O*(*m*^{2}) for large *m*. Therefore,
3.21
Hence, for large *m*,
3.22

Specifically for *D*_{m}, for very large *m*, the upper bounds on *μ*_{m} and can now be taken arbitrarily close to unity, provided that *μ*_{m}<1 and . From equation (3.6), minimization of the right-hand side gives
3.23
The equivalent estimate for *D*_{1,bad} is
3.24

Figure 3 illustrates how a small gap remains owing to the forbidden limit .

## 4. Conclusion: what are the length scales corresponding to the upper bounds?

The key feature of this paper is the closure of the gaps between the good/bad intervals as , but with the actual limit forbidden. The origin of this lies in proposition 2.1 in the use of the inequality (),
4.1
whereas, when ,
4.2
Equation (4.1) has its origin in a double Riesz transform, while equation (4.2) arises from the work of Beale *et al.* (1984) on the three-dimensional Euler equations—see also Kato and Ponce (1986). The term in equation (4.2) prevents the closure of the set of inequalities for *D*_{m}. While the limit is valid for good intervals, it is not valid for the bad because of the necessary use of proposition 2.1. Thus, it is not possible to completely close the gaps between the two sets of intervals, although they can become arbitrarily small. This allows for the possibility of the escape of trajectories. The *m*-dependence of the *τ*_{i} means that the junction points can, in principle, lie at different places on the time axis as *m* varies. If the gaps fall randomly with respect to *m*, then a trajectory would have to thread its way through these to escape to infinity. However, a subtle alignment of the gaps cannot be ruled out.

The closeness of the upper bounds on both the time average and on point-wise values of *D*_{m} (*m*≫1) away from the gaps poses the question whether there exists dynamics that naturally lie either close to these bounds or even fulfill them. The point-wise energy dissipation rate per unit volume is
4.3
Defining a *local* Kolmogorov length as *λ*_{k,loc}=(*ε*/*ν*^{3})^{1/4}, we obtain
4.4
which is consistent with the estimate in equation (1.5) for large *m*. If the solution survives for large enough *T* to make sense of a Reynolds number based on , then the Doering–Foias result for Navier–Stokes solutions (Doering & Foias 2002), *Gr*≤*c* *Re*^{2}, can be invoked to give an estimate for a local Kolmorgorov scale^{6}
4.5
This length scale is immensely small, probably below molecular scales. Because the bounds on the good and bad intervals are very close to the time average, solutions could, in principle, spend long periods of time close to this bound and yet remain regular. Such a scale is not only unreachable computationally but would be outside the validity of the Navier–Stokes equations. Thus, a singularity is not necessary to produce unresolvable solutions.

## Acknowledgements

I would like to express very warm thanks to Claude Bardos, Matania Benartzi, Toti Daskalopoulos, Darryl Holm, Roger Lewandowski, Gustavo Ponce, James Robinson and Edriss Titi for discussions on this topic.

## Appendix A. Proof of proposition 2.1

### Proof.

Consider the time derivative of *J*_{m} defined in equation (1.14),
A1

Bounds on each of the three constituent parts of equation (A1) are dealt with in turn, culminating in a differential inequality for *J*_{m}. In what follows, *c*_{m} is a generic *m*-dependent constant.

*The Laplacian term.* Let *ϕ*=*ω*^{2}=** ω**·

**. Then, A2 Using the fact that Δ(**

*ω**ϕ*

^{m})=

*m*{(

*m*−1)

*ϕ*

^{m−2}|∇

*ϕ*|

^{2}+

*ϕ*

^{m−1}Δ

*ϕ*}, we obtain A3 Thus, we have A4 where A5 where there is equality for

*m*=1. The negativity of the right-hand side of equation (A4) is important. Both ∥∇

*A*

_{m}∥

_{2}and ∥

*A*

_{m}∥

_{2}will appear later in the proof.

*The nonlinear term in equation (A 1 ).* The second term in equation (A1) is
A6
where the inequality ∥∇** u**∥

_{p}≤

*c*

_{p}∥

**∥**

*ω*_{p}for has been used.

^{7}This can be proved in the following way: write

**=curl(−Δ)**

*u*^{−1}

**, therefore,**

*ω**u*

_{i,j}=

*R*

_{j}

*R*

_{i}

*ω*

_{i}, where

*R*

_{i}is a Riesz transform.

Together with equation (A2), this makes equation (A1) into A7

*The forcing term in equation (A 1 ).* Now we use the smallest scale in the forcing ℓ defined in equation (1.13) to estimate the last term in equation (A7)
A8
However, by going up to at least *n*≥3 derivatives in a Sobolev inequality and using equation (1.13), it can be easily shown that ∥∇** f**∥

_{2m}≤

*c*∥

**∥**

*f*_{2}ℓ

^{(3−5m)/2m}. Equation (A8) becomes A9

*A differential inequality for J_{m}.* Recalling that

*A*

_{m}=

*ω*

^{m}, A10

A Gagliardo–Nirenberg inequality yields
A11
which means that
A12
With *β*_{m}=*m*(*m*+1), equation (A12) can be used to form *Ω*_{m+1}
A13
which converts to
A14

This motivates us to re-write equation (A7) as A15

Converting the *J*_{m} into *Ω*_{m} and using *Gr*≥1 (the *a*_{ℓ}-term has been absorbed in *c*_{6,m})
A16

Using a Hölder inequality on the central term on the right-hand side, equation (A16) finally becomes
A17
With no forcing, the final term in equation (A17) is proportional to . Converting to the dimensionless quantity already defined in equation (2.1), with *α*_{m}=2*m*/(4*m*−3), finally gives
A18
with . ■

## Appendix B. Proof of theorem 2.3

### Proof.

A result of Foias *et al.* (1981), which uses higher derivatives: define *H*_{n} for *n*≥1
B1
together with an integration of Leray’s energy inequality
B2

Then, the result of Foias *et al.* (1981) for *n*≥3 is
B3
where *E*_{0}=*E*(0)=*H*_{0}(0) is the initial energy. In the unforced case,
B4
A Sobolev inequality gives
B5
where *a*=3(*m*−1)/4*m* for *m*≥1. Moreover, the constant *c* can be taken as finite for each finite *m* because in the case it is bounded. Thus, taking *n*=3 in equation (B3), which fixes the constant *c*_{n}, we have
B6

Using equations (B2) and (B4), this gives
B7
and thus the final result with an *m*-independent constant. In the unforced case,
B8

There is also a way of reproducing the *Gr*^{2} estimate from proposition 2.1 but with worse constants. Based on for *n*=1/2(*m*+1), the relation in terms of *D*_{n} and *D*_{m} is
B9

Inequality (A18) is now divided by , where *δ*≥(1/(2*m*−1)). Noting that *D*_{m}≥1, the term is handled as follows:
B10

It follows that
B11
where the coefficients from the Hölder inequality have been absorbed into the constants. Define Δ_{m}=2*m*(4*m*+1)/3, and consider
B12
where a Hölder inequality has been used at the last step. The end result is
B13

Because *n*=(1/2)(*m*+1), when *m*=1, then *n*=1. Moreover, only when *δ*=1 does an estimate exist for 〈*D*_{1}〉 through equation (B2), in which case equation (B13) is a generating inequality and gives the *Gr*^{2} estimate but with worse constants. ■

## Appendix C. Proof of theorem 3.1

### Proof.

With 0<*μ*_{m}<1, we write
C1
which becomes
C2

The estimate for the time average of 〈*D*_{m+1}〉_{T} from equation (2.3) and the lower bound *D*_{m}≥1 are now used to give
C3
■

## Footnotes

↵1 Cao & Titi (2007) have recently proved the regularity of the primitive equations of the atmosphere and oceans, even though these have been considered by many to be a problem harder than the Navier–Stokes equations. Unfortunately, the methods used do not appear to successfully transfer to the Navier–Stokes equations.

↵2 See also papers on geometric control of the direction of vorticity (Constantin & Fefferman 1993; Constantin

*et al.*1996).↵3 Doering & Foias (2002) have shown that, for Navier–Stokes solutions,

*Gr*≤*c**Re*^{2}, which would be valid if solutions were assumed to exist for large enough values of*T*. In this case, the*Gr*^{2}term on the right-hand side of equation (1.4) would be replaced by*Re*^{3}, in which case the right-hand side of equation (1.6) would be*Re*^{3/2αm}. Thus, is the Kolmogorov estimate. For large*m*, this becomes significantly larger, running to .↵4 There is no information on the nature of the set {

*τ*_{i}} and how the*τ*_{i}are distributed; this could be an important issue that requires more investigation.↵5 The context of this is the estimate for the inverse length of §1.

↵6 The correspondence is that

*Gr*^{2}is replaced by*Re*^{3}.↵7 I am grateful to G. Ponce for pointing out this result to me. Note that the case is forbidden because an extra term is needed (Beale

*et al.*1984; Kato & Ponce 1986). It is this forbidden limit that ultimately prevents the closure of the gaps in the figures in §3, which allows trajectories to escape.- Received December 5, 2009.
- Accepted February 15, 2010.

- © 2010 The Royal Society