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Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes. A typical example arises from the three-dimensional periodic Navier-Stokes equations

where is the velocity field, is a forcing term, is a pressure field, and is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

so that the system becomes

We may normalise (and ) to have mean zero, so that . Then we introduce the dyadic energies

where ranges over the powers of two, and is shorthand for . Taking the inner product of (1) with , we obtain the energy flow equation

where range over powers of two, is the energy flow rate

is the energy dissipation rate

and is the energy injection rate

The Navier-Stokes equations are notoriously difficult to solve in general. Despite this, Kolmogorov in 1941 was able to give a convincing heuristic argument for what the distribution of the dyadic energies should become over long times, assuming that some sort of distributional steady state is reached. It is common to present this argument in the form of dimensional analysis, but one can also give a more “first principles” form Kolmogorov’s argument, which I will do here. Heuristically, one can divide the frequency scales into three regimes:

- The
*injection regime*in which the energy injection rate dominates the right-hand side of (2); - The
*energy flow regime*in which the flow rates dominate the right-hand side of (2); and - The
*dissipation regime*in which the dissipation dominates the right-hand side of (2).

If we assume a fairly steady and smooth forcing term , then will be supported on the low frequency modes , and so we heuristically expect the injection regime to consist of the low scales . Conversely, if we take the viscosity to be small, we expect the dissipation regime to only occur for very large frequencies , with the energy flow regime occupying the intermediate frequencies.

We can heuristically predict the dividing line between the energy flow regime. Of all the flow rates , it turns out in practice that the terms in which (i.e., interactions between comparable scales, rather than widely separated scales) will dominate the other flow rates, so we will focus just on these terms. It is convenient to return back to physical space, decomposing the velocity field into Littlewood-Paley components

of the velocity field at frequency . By Plancherel’s theorem, this field will have an norm of , and as a naive model of turbulence we expect this field to be spread out more or less uniformly on the torus, so we have the heuristic

and a similar heuristic applied to gives

(One can consider modifications of the Kolmogorov model in which is concentrated on a lower-dimensional subset of the three-dimensional torus, leading to some changes in the numerology below, but we will not consider such variants here.) Since

we thus arrive at the heuristic

Of course, there is the possibility that due to significant cancellation, the energy flow is significantly less than , but we will assume that cancellation effects are not that significant, so that we typically have

or (assuming that does not oscillate too much in , and are close to )

On the other hand, we clearly have

We thus expect to be in the dissipation regime when

and in the energy flow regime when

Now we study the energy flow regime further. We assume a “statistically scale-invariant” dynamics in this regime, in particular assuming a power law

for some . From (3), we then expect an average asymptotic of the form

for some structure constants that depend on the exact nature of the turbulence; here we have replaced the factor by the comparable term to make things more symmetric. In order to attain a steady state in the energy flow regime, we thus need a cancellation in the structure constants:

On the other hand, if one is assuming statistical scale invariance, we expect the structure constants to be scale-invariant (in the energy flow regime), in that

for dyadic . Also, since the Euler equations conserve energy, the energy flows symmetrise to zero,

which from (7) suggests a similar cancellation among the structure constants

Combining this with the scale-invariance (9), we see that for fixed , we may organise the structure constants for dyadic into sextuples which sum to zero (including some degenerate tuples of order less than six). This will *automatically* guarantee the cancellation (8) required for a steady state energy distribution, provided that

or in other words

for any other value of , there is no particular reason to expect this cancellation (8) to hold. Thus we are led to the heuristic conclusion that the most stable power law distribution for the energies is the law

or in terms of shell energies, we have the famous Kolmogorov 5/3 law

Given that frequency interactions tend to cascade from low frequencies to high (if only because there are so many more high frequencies than low ones), the above analysis predicts a stablising effect around this power law: scales at which a law (6) holds for some are likely to lose energy in the near-term, while scales at which a law (6) hold for some are conversely expected to gain energy, thus nudging the exponent of power law towards .

We can solve for in terms of energy dissipation as follows. If we let be the frequency scale demarcating the transition from the energy flow regime (5) to the dissipation regime (4), we have

and hence by (10)

On the other hand, if we let be the energy dissipation at this scale (which we expect to be the dominant scale of energy dissipation), we have

Some simple algebra then lets us solve for and as

and

Thus, we have the Kolmogorov prediction

for

with energy dissipation occuring at the high end of this scale, which is counterbalanced by the energy injection at the low end of the scale.

It is always dangerous to venture an opinion as to why a problem is hard (cf. Clarke’s first law), but I’m going to stick my neck out on this one, because (a) it seems that there has been a lot of effort expended on this problem recently, sometimes perhaps without full awareness of the main difficulties, and (b) I would love to be proved wrong on this opinion :-) .

The global regularity problem for Navier-Stokes is of course a Clay Millennium Prize problem and it would be redundant to describe it again here. I will note, however, that it asks for existence of global smooth solutions to a Cauchy problem for a nonlinear PDE. There are countless other global regularity results of this type for many (but certainly not all) other nonlinear PDE; for instance, global regularity is known for Navier-Stokes in two spatial dimensions rather than three (this result essentially dates all the way back to Leray’s thesis in 1933!). Why is the three-dimensional Navier-Stokes global regularity problem considered so hard, when global regularity for so many other equations is easy, or at least achievable?

(For this post, I am only considering the global regularity problem for Navier-Stokes, from a purely mathematical viewpoint, and in the precise formulation given by the Clay Institute; I will not discuss at all the question as to what implications a rigorous solution (either positive or negative) to this problem would have for physics, computational fluid dynamics, or other disciplines, as these are beyond my area of expertise. But if anyone qualified in these fields wants to make a comment along these lines, by all means do so.)

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