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\displaystyle \nabla \cdot u = 0



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

\displaystyle  \partial_t u + u \cdot \nabla u = \nu \Delta u + \nabla p + f

\displaystyle  \nabla \cdot u = 0

where {u: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3} is the velocity field, {f: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3} is a forcing term, {p: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}} is a pressure field, and {\nu > 0} is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

\displaystyle  \hat u(t,k) := \int_{{\bf R}^3/{\bf Z}^3} u(t,x) e^{-2\pi i k \cdot x}

so that the system becomes

\displaystyle  \partial_t \hat u(t,k) + 2\pi \sum_{k = k_1 + k_2} (\hat u(t,k_1) \cdot ik_2) \hat u(t,k_2) =

\displaystyle  - 4\pi^2 \nu |k|^2 \hat u(t,k) + 2\pi ik \hat p(t,k) + \hat f(t,k) \ \ \ \ \ (1)

\displaystyle  k \cdot \hat u(t,k) = 0.

We may normalise {u} (and {f}) to have mean zero, so that {\hat u(t,0)=0}. Then we introduce the dyadic energies

\displaystyle  E_N(t) := \sum_{|k| \sim N} |\hat u(t,k)|^2

where {N \geq 1} ranges over the powers of two, and {|k| \sim N} is shorthand for {N \leq |k| < 2N}. Taking the inner product of (1) with {\hat u(t,k)}, we obtain the energy flow equation

\displaystyle  \partial_t E_N = \sum_{N_1,N_2} \Pi_{N,N_1,N_2} - D_N + F_N \ \ \ \ \ (2)

where {N_1,N_2} range over powers of two, {\Pi_{N,N_1,N_2}} is the energy flow rate

\displaystyle  \Pi_{N,N_1,N_2} := -2\pi \sum_{k=k_1+k_2: |k| \sim N, |k_1| \sim N_1, |k_2| \sim N_2}

\displaystyle  (\hat u(t,k_1) \cdot ik_2) (\hat u(t,k) \cdot \hat u(t,k_2)),

{D_N} is the energy dissipation rate

\displaystyle  D_N := 4\pi^2 \nu \sum_{|k| \sim N} |k|^2 |\hat u(t,k)|^2

and {F_N} is the energy injection rate

\displaystyle  F_N := \sum_{|k| \sim N} \hat u(t,k) \cdot \hat f(t,k).

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 {E_N} 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 {N} into three regimes:

  • The injection regime in which the energy injection rate {F_N} dominates the right-hand side of (2);
  • The energy flow regime in which the flow rates {\Pi_{N,N_1,N_2}} dominate the right-hand side of (2); and
  • The dissipation regime in which the dissipation {D_N} dominates the right-hand side of (2).

If we assume a fairly steady and smooth forcing term {f}, then {\hat f} will be supported on the low frequency modes {k=O(1)}, and so we heuristically expect the injection regime to consist of the low scales {N=O(1)}. Conversely, if we take the viscosity {\nu} to be small, we expect the dissipation regime to only occur for very large frequencies {N}, 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 {\Pi_{N,N_1,N_2}}, it turns out in practice that the terms in which {N_1,N_2 = N+O(1)} (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 {u} into Littlewood-Paley components

\displaystyle  u_N(t,x) := \sum_{|k| \sim N} \hat u(t,k) e^{2\pi i k \cdot x}

of the velocity field {u(t,x)} at frequency {N}. By Plancherel’s theorem, this field will have an {L^2} norm of {E_N(t)^{1/2}}, 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

\displaystyle  |u_N(t,x)| = O( E_N(t)^{1/2} ),

and a similar heuristic applied to {\nabla u_N} gives

\displaystyle  |\nabla u_N(t,x)| = O( N E_N(t)^{1/2} ).

(One can consider modifications of the Kolmogorov model in which {u_N} 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

\displaystyle  \Pi_{N,N_1,N_2} = - \int_{{\bf R}^3/{\bf Z}^3} u_N \cdot ( (u_{N_1} \cdot \nabla) u_{N_2} )\ dx

we thus arrive at the heuristic

\displaystyle  \Pi_{N,N_1,N_2} = O( N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} ).

Of course, there is the possibility that due to significant cancellation, the energy flow is significantly less than {O( N E_N(t)^{3/2} )}, but we will assume that cancellation effects are not that significant, so that we typically have

\displaystyle  \Pi_{N,N_1,N_2} \sim N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} \ \ \ \ \ (3)

or (assuming that {E_N} does not oscillate too much in {N}, and {N_1,N_2} are close to {N})

\displaystyle  \Pi_{N,N_1,N_2} \sim N E_N^{3/2}.

On the other hand, we clearly have

\displaystyle  D_N \sim \nu N^2 E_N.

We thus expect to be in the dissipation regime when

\displaystyle  N \gtrsim \nu^{-1} E_N^{1/2} \ \ \ \ \ (4)

and in the energy flow regime when

\displaystyle  1 \lesssim N \lesssim \nu^{-1} E_N^{1/2}. \ \ \ \ \ (5)

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

\displaystyle  E_N \sim A N^{-\alpha} \ \ \ \ \ (6)

for some {A,\alpha > 0}. From (3), we then expect an average asymptotic of the form

\displaystyle  \Pi_{N,N_1,N_2} \approx A^{3/2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \ \ \ \ \ (7)

for some structure constants {c_{N,N_1,N_2} \sim 1} that depend on the exact nature of the turbulence; here we have replaced the factor {N_2} by the comparable term {(N N_1 N_2)^{1/3}} 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:

\displaystyle  \sum_{N_1,N_2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \approx 0. \ \ \ \ \ (8)

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

\displaystyle  c_{\lambda N, \lambda N_1, \lambda N_2} = c_{N,N_1,N_2} \ \ \ \ \ (9)

for dyadic {\lambda > 0}. Also, since the Euler equations conserve energy, the energy flows {\Pi_{N,N_1,N_2}} symmetrise to zero,

\displaystyle  \Pi_{N,N_1,N_2} + \Pi_{N,N_2,N_1} + \Pi_{N_1,N,N_2} + \Pi_{N_1,N_2,N} + \Pi_{N_2,N,N_1} + \Pi_{N_2,N_1,N} = 0,

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

\displaystyle  c_{N,N_1,N_2} + c_{N,N_2,N_1} + c_{N_1,N,N_2} + c_{N_1,N_2,N} + c_{N_2,N,N_1} + c_{N_2,N_1,N} \approx 0.

Combining this with the scale-invariance (9), we see that for fixed {N}, we may organise the structure constants {c_{N,N_1,N_2}} for dyadic {N_1,N_2} 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

\displaystyle  \frac{1}{3} - \frac{\alpha}{2} = 0

or in other words

\displaystyle  \alpha = \frac{2}{3};

for any other value of {\alpha}, 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 {E_N} is the {2/3} law

\displaystyle  E_N \sim A N^{-2/3} \ \ \ \ \ (10)

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

\displaystyle  \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim A k_0^{-5/3}.

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 {\alpha > 2/3} are likely to lose energy in the near-term, while scales at which a law (6) hold for some {\alpha< 2/3} are conversely expected to gain energy, thus nudging the exponent of power law towards {2/3}.

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

\displaystyle  N_* \sim \nu^{-1} E_{N_*}

and hence by (10)

\displaystyle  N_* \sim \nu^{-1} A N_*^{-2/3}.

On the other hand, if we let {\epsilon := D_{N_*}} be the energy dissipation at this scale {N_*} (which we expect to be the dominant scale of energy dissipation), we have

\displaystyle  \epsilon \sim \nu N_*^2 E_N \sim \nu N_*^2 A N_*^{-2/3}.

Some simple algebra then lets us solve for {A} and {N_*} as

\displaystyle  N_* \sim (\frac{\epsilon}{\nu^3})^{1/4}


\displaystyle  A \sim \epsilon^{2/3}.

Thus, we have the Kolmogorov prediction

\displaystyle  \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim \epsilon^{2/3} k_0^{-5/3}


\displaystyle  1 \lesssim k_0 \lesssim (\frac{\epsilon}{\nu^3})^{1/4}

with energy dissipation occuring at the high end {k_0 \sim (\frac{\epsilon}{\nu^3})^{1/4}} of this scale, which is counterbalanced by the energy injection at the low end {k_0 \sim 1} of the scale.

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space {{\cal L} = ({\bf R}^n, \hbox{vol})} and Eulerian space {{\cal E} = ({\bf R}^n, \eta, \hbox{vol})}, we let {M} be the space of volume-preserving, orientation-preserving maps {\Phi: {\cal L} \rightarrow {\cal E}} from Lagrangian space to Eulerian space. Given a curve {\Phi: {\bf R} \rightarrow M}, we can define the Lagrangian velocity field {\dot \Phi: {\bf R} \times {\cal L} \rightarrow T{\cal E}} as the time derivative of {\Phi}, and the Eulerian velocity field {u := \dot \Phi \circ \Phi^{-1}: {\bf R} \times {\cal E} \rightarrow T{\cal E}}. The volume-preserving nature of {\Phi} ensures that {u} is a divergence-free vector field:

\displaystyle  \nabla \cdot u = 0. \ \ \ \ \ (1)

If we formally define the functional

\displaystyle  J[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} |u(t,x)|^2\ dx dt = \frac{1}{2} \int_R \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx dt

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

\displaystyle  [\partial_t + u \cdot \nabla] u = - \nabla p

\displaystyle  \nabla \cdot u = 0

for some pressure field {p: {\bf R} \times {\cal E} \rightarrow {\bf R}}. As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

\displaystyle  \frac{1}{2} \int_{{\cal E}} |u(t,x)|^2\ dx = \frac{1}{2} \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx;

the rigid motion symmetries of Eulerian space give conservation of the total momentum

\displaystyle  \int_{{\cal E}} u(t,x)\ dx

and total angular momentum

\displaystyle  \int_{{\cal E}} x \wedge u(t,x)\ dx;

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

\displaystyle  \int_{\Phi(\gamma)} u^*

for any closed loop {\gamma} in {{\cal L}}, or equivalently pointwise conservation of the Lagrangian vorticity {\Phi^* \omega = \Phi^* du^*}, where {u^*} is the {1}-form associated with the vector field {u} using the Euclidean metric {\eta} on {{\cal E}}, with {\Phi^*} denoting pullback by {\Phi}.

It turns out that one can generalise the above calculations. Given any self-adjoint operator {A} on divergence-free vector fields {u: {\cal E} \rightarrow {\bf R}}, we can define the functional

\displaystyle  J_A[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx dt;

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

\displaystyle  [\partial_t + u \cdot \nabla] Au + (\nabla u) \cdot Au= - \nabla \tilde p \ \ \ \ \ (2)

\displaystyle  \nabla \cdot u = 0

for some pressure field {\tilde p: {\bf R} \times {\cal E} \rightarrow {\bf R}}, where {(\nabla u) \cdot Au} in coordinates is {\partial_i u_j Au_j} with the usual summation conventions. (When {A=1}, {(\nabla u) \cdot Au = \nabla(\frac{1}{2} |u|^2)}, and this term can be absorbed into the pressure {\tilde p}, and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian


If the operator {A} commutes with rigid motions on {{\cal E}}, then we have conservation of total momentum

\displaystyle  \int_{{\cal E}} Au(t,x)\ dx

and total angular momentum

\displaystyle  \int_{{\cal E}} x \wedge Au(t,x)\ dx,

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

\displaystyle  \int_{\Phi(\gamma)} (Au)^*

or pointwise conservation of the Lagrangian vorticity {\Phi^* \theta := \Phi^* d(Au)^*}. These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions {n=2}, when {A} is the inverse derivative {A = |\nabla|^{-1} = (-\Delta)^{-1/2}}. The vorticity {\theta = d(Au)^*} is a {2}-form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write {u = (u_1,u_2)}, then

\displaystyle  \theta = \partial_{x_1} |\nabla|^{-1} u_2 - \partial_{x_2} |\nabla|^{-1} u_1.

Since {u} is also divergence-free, we may therefore write

\displaystyle  u = (- \partial_{x_2} \psi, \partial_{x_1} \psi )

where the stream function {\psi} is given by the formula

\displaystyle  \psi = |\nabla|^{-1} \theta.

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

\displaystyle  [\partial_t + u \cdot \nabla] \theta = 0 \ \ \ \ \ (3)

\displaystyle  u = (-\partial_{x_2} |\nabla|^{-1} \theta, \partial_{x_1} |\nabla|^{-1} \theta).

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

\displaystyle  \frac{1}{2} \int_{{\bf R}^2} u\cdot |\nabla|^{-1} u\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta \psi\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta |\nabla|^{-1} \theta\ dx

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As {A} commutes with rigid motions, we also have (formally, at least) conservation of momentum

\displaystyle  \int_{{\bf R}^2} Au\ dx

(which up to trivial transformations is also expressible in impulse form as {\int_{{\bf R}^2} \theta x\ dx}, after integration by parts), and conservation of angular momentum

\displaystyle  \int_{{\bf R}^2} x \wedge Au\ dx

(which up to trivial transformations is {\int_{{\bf R}^2} \theta |x|^2\ dx}). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity {\Phi^* \theta}, thus {\theta} is transported by the flow (which is also evident from (3). In particular, all integrals of the form {\int F(\theta)\ dx} for a fixed function {F} are conserved by the flow.

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  1. The two postulates of special relativity;



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A few days ago, I released a preprint entitled “Localisation and compactness properties of the Navier-Stokes global regularity problem“, discussed in this previous blog post.  As it turns out, I was somewhat impatient to finalise the paper and move on to other things, and the original preprint was still somewhat rough in places (contradicting my own advice on this matter), with a number of typos of minor to moderate severity.  But a bit more seriously, I discovered on a further proofreading that there was a subtle error in a component of the argument that I had believed to be routine – namely the persistence of higher regularity for mild solutions.   As a consequence, some of the implications stated in the first version were not exactly correct as stated; but they can be repaired by replacing a “bad” notion of global regularity for a certain class of data with a “good” notion.   I have completed (and proofread) an updated version of the ms, which should appear at the arXiv link of the paper in a day or two (and which I have also placed at this link).  (In the meantime, it is probably best not to read the original ms too carefully, as this could lead to some confusion.)   I’ve also added a new section that shows that, due to this technicality, one can exhibit smooth H^1 initial data to the Navier-Stokes equation for which there are no smooth solutions, which superficially sounds very close to a negative solution to the global regularity problem, but is actually nothing of the sort.

Let me now describe the issue in more detail (and also to explain why I missed it previously).  A standard principle in the theory of evolutionary partial differentiation equations is that regularity in space can be used to imply regularity in time.  To illustrate this, consider a solution u to the supercritical nonlinear wave equation

-\partial_{tt} u + \Delta u = u^7  (1)

for some field u: {\bf R} \times {\bf R}^3 \to {\bf R}.   Suppose one already knew that u had some regularity in space, and in particular the C^0_t C^2_x \cap C^1_t C^1_x norm of u was bounded (thus u and up to two spatial derivatives of u were bounded).  Then, by (1), we see that two time derivatives of u were also bounded, and one then gets the additional regularity of C^2_t C^0_x.

In a similar vein, suppose one initially knew that u had the regularity C^0_t C^3_x \cap C^1_t C^2_x.  Then (1) soon tells us that u also has the regularity C^2_t C^1_x; then, if one differentiates (1) in time to obtain

-\partial_{ttt} u + \Delta \partial_t u = 7 u^6 \partial_t u

one can conclude that u also has the regularity of C^3_t C^0_x.  One can continue this process indefinitely; in particular, if one knew that u \in C^0_t C^\infty_x \cap C^1_t C^\infty_x, then these sorts of manipulations show that u is infinitely smooth in both space and time.

The issue that caught me by surprise is that for the Navier-Stokes equations

\partial_t u + (u \cdot \nabla) u =\Delta u -\nabla p  (2)

\nabla \cdot u = 0

(setting the forcing term f equal to zero for simplicity), infinite regularity in space does not automatically imply infinite regularity in time, even if one assumes the initial data lies in a standard function space such as the Sobolev space H^1_x({\bf R}^3).  The problem lies with the pressure term p, which is recovered from the velocity via the elliptic equation

\Delta p = -\nabla^2 \cdot (u \otimes u) (3)

that can be obtained by taking the divergence of (2).   This equation is solved by a non-local integral operator:

\displaystyle p(t,x) = \int_{{\bf R}^3} \frac{\nabla^2 \cdot (u \otimes u)(t,y)}{4\pi |x-y|}\ dy.

If, say, u lies in H^1_x({\bf R}^3), then there is no difficulty establishing a bound on p in terms of u (for instance, one can use singular integral theory and Sobolev embedding to place p in L^3_x({\bf R}^3).  However, one runs into difficulty when trying to compute time derivatives of p.  Differentiating (3) once, one gets

\Delta \partial_t p = -2\nabla^2 \cdot (u \otimes \partial_t u).

At the regularity of H^1, one can still (barely) control this quantity by using (2) to expand out \partial_t u and using some integration by parts.  But when one wishes to compute a second time derivative of the pressure, one obtains (after integration by parts) an expansion of the form

\Delta \partial_{tt} p = -4\nabla^2 \cdot (\Delta u \otimes \Delta u) + \ldots

and now there is not enough regularity on u available to get any control on \partial_{tt} p, even if one assumes that u is smooth.   Indeed, following this observation, I was able to show that given generic smooth H^1 data, the pressure p will instantaneously fail to be C^2 in time, and thence (by (2)) the velocity will instantaneously fail to be C^3 in time.  (Switching to the vorticity formulation buys one further degree of time differentiability, but does not fully eliminate the problem; the vorticity \omega will fail to be C^4 in time.  Switching to material coordinates seems to makes things very slightly better, but I believe there is still a breakdown of time regularity in these coordinates also.)

For later times t>0 (and assuming homogeneous data f=0 for simplicity), this issue no longer arises, because of the instantaneous smoothing effect of the Navier-Stokes flow, which for instance will upgrade H^1_x regularity to H^\infty_x regularity instantaneously.  It is only the initial time at which some time irregularity can occur.

This breakdown of regularity does not actually impact the original formulation of the Clay Millennium Prize problem, though, because in that problem the initial velocity is required to be Schwartz class (so all derivatives are rapidly decreasing).  In this class, the regularity theory works as expected; if one has a solution which already has some reasonable regularity (e.g. a mild H^1 solution) and the data is Schwartz, then the solution will be smooth in spacetime.   (Another class where things work as expected is when the vorticity is Schwartz; in such cases, the solution remains smooth in both space and time (for short times, at least), and the Schwartz nature of the vorticity is preserved (because the vorticity is subject to fewer non-local effects than the velocity, as it is not directly affected by the pressure).)

This issue means that one of the implications in the original paper (roughly speaking, that global regularity for Schwartz data implies global regularity for smooth H^1 data) is not correct as stated.  But this can be fixed by weakening the notion of global regularity in the latter setting, by limiting the amount of time differentiability available at the initial time.  More precisely, call a solution u: [0,T] \times {\bf R}^3 \to {\bf R}^3 and p: [0,T] \times {\bf R}^3 \to {\bf R} almost smooth if

  • u and p are smooth on the half-open slab (0,T] \times {\bf R}^3; and
  • For every k \geq 0, \nabla^k_x u, \nabla^k_x p, \nabla^x_u \partial_t u exist and are continuous on the full slab [0,T] \times {\bf R}^3.

Thus, an almost smooth solution is the same concept as a smooth solution, except that at time zero, the velocity field is only C^1_t C^\infty_x, and the pressure field is only C^0_t C^\infty_x.  This is still enough regularity to interpret the Navier-Stokes equation (2) in a classical manner, but falls slightly short of full smoothness.

(I had already introduced this notion of almost smoothness in the more general setting of smooth finite energy solutions in the first draft of this paper, but had failed to realise that it was also necessary in the smooth H^1 setting also.)

One can now “fix” the global regularity conjectures for Navier-Stokes in the smooth H^1 or smooth finite energy setting by requiring the solutions to merely be almost smooth instead of smooth.  Once one does so, the results in my paper then work as before: roughly speaking, if one knows that Schwartz data produces smooth solutions, one can conclude that smooth H^1 or smooth finite energy data produces almost smooth solutions (and the paper now contains counterexamples to show that one does not always have smooth solutions in this category).

The diagram of implications between conjectures has been adjusted to reflect this issue, and now reads as follows:

I’ve just uploaded to the arXiv my paper “Localisation and compactness properties of the Navier-Stokes global regularity problem“, submitted to Analysis and PDE. This paper concerns the global regularity problem for the Navier-Stokes system of equations

\displaystyle  \partial_t u + (u \cdot \nabla) u = \Delta u - \nabla p + f \ \ \ \ \ (1)

\displaystyle  \nabla \cdot u = 0 \ \ \ \ \ (2)

\displaystyle  u(0,\cdot) = u_0 \ \ \ \ \ (3)

in three dimensions. Thus, we specify initial data {(u_0,f,T)}, where {0 < T < \infty} is a time, {u_0: {\bf R}^3 \rightarrow {\bf R}^3} is the initial velocity field (which, in order to be compatible with (2), (3), is required to be divergence-free), {f: [0,T] \times {\bf R}^3 \rightarrow {\bf R}^3} is the forcing term, and then seek to extend this initial data to a solution {(u,p,u_0,f,T)} with this data, where the velocity field {u: [0,T] \times {\bf R}^3 \rightarrow {\bf R}^3} and pressure term {p: [0,T] \times {\bf R}^3 \rightarrow {\bf R}} are the unknown fields.

Roughly speaking, the global regularity problem asserts that given every smooth set of initial data {(u_0,f,T)}, there exists a smooth solution {(u,p,u_0,f,T)} to the Navier-Stokes equation with this data. However, this is not a good formulation of the problem because it does not exclude the possibility that one or more of the fields {u_0, f, u, p} grows too fast at spatial infinity. This problem is evident even for the much simpler heat equation

\displaystyle  \partial_t u = \Delta u

\displaystyle  u(0,\cdot) = u_0.

As long as one has some mild conditions at infinity on the smooth initial data {u_0: {\bf R}^3 \rightarrow {\bf R}} (e.g. polynomial growth at spatial infinity), then one can solve this equation using the fundamental solution of the heat equation:

\displaystyle  u(t,x) = \frac{1}{(4\pi t)^{3/2}} \int_{{\bf R}^3} u_0(y) e^{-|x-y|^2/4t}\ dy.

If furthermore {u} is a tempered distribution, one can use Fourier-analytic methods to show that this is the unique solution to the heat equation with this data. But once one allows sufficiently rapid growth at spatial infinity, existence and uniqueness can break down. Consider for instance the backwards heat kernel

\displaystyle  u(t,x) = \frac{1}{(4\pi(T-t))^{3/2}} e^{|x|^2/(T-t)}

for some {T>0}, which is smooth (albeit exponentially growing) at time zero, and is a smooth solution to the heat equation for {0 \leq t < T}, but develops a dramatic singularity at time {t=T}. A famous example of Tychonoff from 1935, based on a power series construction, also shows that uniqueness for the heat equation can also fail once growth conditions are removed. An explicit example of non-uniqueness for the heat equation is given by the contour integral

\displaystyle  u(t,x_1,x_2,x_3) = \int_\gamma \exp(e^{\pi i/4} x_1 z + e^{5\pi i/8} z^{3/2} - itz^2)\ dz

where {\gamma} is the {L}-shaped contour consisting of the positive real axis and the upper imaginary axis, with {z^{3/2}} being interpreted with the standard branch (with cut on the negative axis). One can show by contour integration that this function solves the heat equation and is smooth (but rapidly growing at infinity), and vanishes for {t<0}, but is not identically zero for {t>0}.

Thus, in order to obtain a meaningful (and physically realistic) problem, one needs to impose some decay (or at least limited growth) hypotheses on the data {u_0,f} and solution {u,p} in addition to smoothness. For the data, one can impose a variety of such hypotheses, including the following:

  • (Finite energy data) One has {\|u_0\|_{L^2_x({\bf R}^3)} < \infty} and {\| f \|_{L^\infty_t L^2_x([0,T] \times {\bf R}^3)} < \infty}.
  • ({H^1} data) One has {\|u_0\|_{H^1_x({\bf R}^3)} < \infty} and {\| f \|_{L^\infty_t H^1_x([0,T] \times {\bf R}^3)} < \infty}.
  • (Schwartz data) One has {\sup_{x \in {\bf R}^3} ||x|^m \nabla_x^k u_0(x)| < \infty} and {\sup_{(t,x) \in [0,T] \times {\bf R}^3} ||x|^m \nabla_x^k \partial_t^l f(t,x)| < \infty} for all {m,k,l \geq 0}.
  • (Periodic data) There is some {0 < L < \infty} such that {u_0(x+Lk) = u_0(x)} and {f(t,x+Lk) = f(t,x)} for all {(t,x) \in [0,T] \times {\bf R}^3} and {k \in {\bf Z}^3}.
  • (Homogeneous data) {f=0}.

Note that smoothness alone does not necessarily imply finite energy, {H^1}, or the Schwartz property. For instance, the (scalar) function {u(x) = \exp( i |x|^{10} ) (1+|x|)^{-2}} is smooth and finite energy, but not in {H^1} or Schwartz. Periodicity is of course incompatible with finite energy, {H^1}, or the Schwartz property, except in the trivial case when the data is identically zero.

Similarly, one can impose conditions at spatial infinity on the solution, such as the following:

  • (Finite energy solution) One has {\| u \|_{L^\infty_t L^2_x([0,T] \times {\bf R}^3)} < \infty}.
  • ({H^1} solution) One has {\| u \|_{L^\infty_t H^1_x([0,T] \times {\bf R}^3)} < \infty} and {\| u \|_{L^2_t H^2_x([0,T] \times {\bf R}^3)} < \infty}.
  • (Partially periodic solution) There is some {0 < L < \infty} such that {u(t,x+Lk) = u(t,x)} for all {(t,x) \in [0,T] \times {\bf R}^3} and {k \in {\bf Z}^3}.
  • (Fully periodic solution) There is some {0 < L < \infty} such that {u(t,x+Lk) = u(t,x)} and {p(t,x+Lk) = p(t,x)} for all {(t,x) \in [0,T] \times {\bf R}^3} and {k \in {\bf Z}^3}.

(The {L^2_t H^2_x} component of the {H^1} solution is for technical reasons, and should not be paid too much attention for this discussion.) Note that we do not consider the notion of a Schwartz solution; as we shall see shortly, this is too restrictive a concept of solution to the Navier-Stokes equation.

Finally, one can downgrade the regularity of the solution down from smoothness. There are many ways to do so; two such examples include

  • ({H^1} mild solutions) The solution is not smooth, but is {H^1} (in the preceding sense) and solves the equation (1) in the sense that the Duhamel formula

    \displaystyle  u(t) = e^{t\Delta} u_0 + \int_0^t e^{(t-t')\Delta} (-(u\cdot\nabla) u-\nabla p+f)(t')\ dt'


  • (Leray-Hopf weak solution) The solution {u} is not smooth, but lies in {L^\infty_t L^2_x \cap L^2_t H^1_x}, solves (1) in the sense of distributions (after rewriting the system in divergence form), and obeys an energy inequality.

Finally, one can ask for two types of global regularity results on the Navier-Stokes problem: a qualitative regularity result, in which one merely provides existence of a smooth solution without any explicit bounds on that solution, and a quantitative regularity result, which provides bounds on the solution in terms of the initial data, e.g. a bound of the form

\displaystyle  \| u \|_{L^\infty_t H^1_x([0,T] \times {\bf R}^3)} \leq F( \|u_0\|_{H^1_x({\bf R}^3)} + \|f\|_{L^\infty_t H^1_x([0,T] \times {\bf R}^3)}, T )

for some function {F: {\bf R}^+ \times {\bf R}^+ \rightarrow {\bf R}^+}. One can make a further distinction between local quantitative results, in which {F} is allowed to depend on {T}, and global quantitative results, in which there is no dependence on {T} (the latter is only reasonable though in the homogeneous case, or if {f} has some decay in time).

By combining these various hypotheses and conclusions, we see that one can write down quite a large number of slightly different variants of the global regularity problem. In the official formulation of the regularity problem for the Clay Millennium prize, a positive correct solution to either of the following two problems would be accepted for the prize:

  • Conjecture 1.4 (Qualitative regularity for homogeneous periodic data) If {(u_0,0,T)} is periodic, smooth, and homogeneous, then there exists a smooth partially periodic solution {(u,p,u_0,0,T)} with this data.
  • Conjecture 1.3 (Qualitative regularity for homogeneous Schwartz data) If {(u_0,0,T)} is Schwartz and homogeneous, then there exists a smooth finite energy solution {(u,p,u_0,0,T)} with this data.

(The numbering here corresponds to the numbering in the paper.)

Furthermore, a negative correct solution to either of the following two problems would also be accepted for the prize:

  • Conjecture 1.6 (Qualitative regularity for periodic data) If {(u_0,f,T)} is periodic and smooth, then there exists a smooth partially periodic solution {(u,p,u_0,f,T)} with this data.
  • Conjecture 1.5 (Qualitative regularity for Schwartz data) If {(u_0,f,T)} is Schwartz, then there exists a smooth finite energy solution {(u,p,u_0,f,T)} with this data.

I am not announcing any major progress on these conjectures here. What my paper does study, though, is the question of whether the answer to these conjectures is somehow sensitive to the choice of formulation. For instance:

  1. Note in the periodic formulations of the Clay prize problem that the solution is only required to be partially periodic, rather than fully periodic; thus the pressure has no periodicity hypothesis. One can ask the extent to which the above problems change if one also requires pressure periodicity.
  2. In another direction, one can ask the extent to which quantitative formulations of the Navier-Stokes problem are stronger than their qualitative counterparts; in particular, whether it is possible that each choice of initial data in a certain class leads to a smooth solution, but with no uniform bound on that solution in terms of various natural norms of the data.
  3. Finally, one can ask the extent to which the conjecture depends on the category of data. For instance, could it be that global regularity is true for smooth periodic data but false for Schwartz data? True for Schwartz data but false for smooth {H^1} data? And so forth.

One motivation for the final question (which was posed to me by my colleague, Andrea Bertozzi) is that the Schwartz property on the initial data {u_0} tends to be instantly destroyed by the Navier-Stokes flow. This can be seen by introducing the vorticity {\omega := \nabla \times u}. If {u(t)} is Schwartz, then from Stokes’ theorem we necessarily have vanishing of certain moments of the vorticity, for instance:

\displaystyle  \int_{{\bf R}^3} \omega_1 (x_2^2-x_3^2)\ dx = 0.

On the other hand, some integration by parts using (1) reveals that such moments are usually not preserved by the flow; for instance, one has the law

\displaystyle \partial_t \int_{{\bf R}^3} \omega_1(t,x) (x_2^2-x_3^2)\ dx = 4\int_{{\bf R}^3} u_2(t,x) u_3(t,x)\ dx,

and one can easily concoct examples for which the right-hand side is non-zero at time zero. This suggests that the Schwartz class may be unnecessarily restrictive for Conjecture 1.3 or Conjecture 1.5.

My paper arose out of an attempt to address these three questions, and ended up obtaining partial results in all three directions. Roughly speaking, the results that address these three questions are as follows:

  1. (Homogenisation) If one only assumes partial periodicity instead of full periodicity, then the forcing term {f} becomes irrelevant. In particular, Conjecture 1.4 and Conjecture 1.6 are equivalent.
  2. (Concentration compactness) In the {H^1} category (both periodic and nonperiodic, homogeneous or nonhomogeneous), the qualitative and quantitative formulations of the Navier-Stokes global regularity problem are essentially equivalent.
  3. (Localisation) The (inhomogeneous) Navier-Stokes problems in the Schwartz, smooth {H^1}, and finite energy categories are essentially equivalent to each other, and are also implied by the (fully) periodic version of these problems.

The first two of these families of results are relatively routine, drawing on existing methods in the literature; the localisation results though are somewhat more novel, and introduce some new local energy and local enstrophy estimates which may be of independent interest.

Broadly speaking, the moral to draw from these results is that the precise formulation of the Navier-Stokes equation global regularity problem is only of secondary importance; modulo a number of caveats and technicalities, the various formulations are close to being equivalent, and a breakthrough on any one of the formulations is likely to lead (either directly or indirectly) to a comparable breakthrough on any of the others.

This is only a caricature of the actual implications, though. Below is the diagram from the paper indicating the various formulations of the Navier-Stokes equations, and the known implications between them:

The above three streams of results are discussed in more detail below the fold.

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