I’ve just uploaded to the arXiv the paper “Finite time blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the “supercriticality barrier” for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any “abstract” approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of the Navier-Stokes equation without the energy identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity (see also the work of Plechac and Sverak and of Hou and Lei that also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions), but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity. Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).

To state the results more precisely, recall that the Navier-Stokes equations can be written in the form

$\displaystyle \partial_t u + (u \cdot \nabla) u = \nu \Delta u + \nabla p$

for a divergence-free velocity field ${u}$ and a pressure field ${p}$, where ${\nu>0}$ is the viscosity, which we will normalise to be one. We will work in the non-periodic setting, so the spatial domain is ${{\bf R}^3}$, and for sake of exposition I will not discuss matters of regularity or decay of the solution (but we will always be working with strong notions of solution here rather than weak ones). Applying the Leray projection ${P}$ to divergence-free vector fields to this equation, we can eliminate the pressure, and obtain an evolution equation

$\displaystyle \partial_t u = \Delta u + B(u,u) \ \ \ \ \ (1)$

purely for the velocity field, where ${B}$ is a certain bilinear operator on divergence-free vector fields (specifically, ${B(u,v) = -\frac{1}{2} P( (u \cdot \nabla) v + (v \cdot \nabla) u)}$. The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation (1).

An important feature of the bilinear operator ${B}$ appearing in (1) is the cancellation law

$\displaystyle \langle B(u,u), u \rangle = 0$

(using the ${L^2}$ inner product on divergence-free vector fields), which leads in particular to the fundamental energy identity

$\displaystyle \frac{1}{2} \int_{{\bf R}^3} |u(T,x)|^2\ dx + \int_0^T \int_{{\bf R}^3} |\nabla u(t,x)|^2\ dx dt = \frac{1}{2} \int_{{\bf R}^3} |u(0,x)|^2\ dx.$

This identity (and its consequences) provide essentially the only known a priori bound on solutions to the Navier-Stokes equations from large data and arbitrary times. Unfortunately, as discussed in this previous post, the quantities controlled by the energy identity are supercritical with respect to scaling, which is the fundamental obstacle that has defeated all attempts to solve the global regularity problem for Navier-Stokes without any additional assumptions on the data or solution (e.g. perturbative hypotheses, or a priori control on a critical norm such as the ${L^\infty_t L^3_x}$ norm).

Our main result is then (slightly informally stated) as follows

Theorem 1 There exists an averaged version ${\tilde B}$ of the bilinear operator ${B}$, of the form

$\displaystyle \tilde B(u,v) := \int_\Omega m_{3,\omega}(D) Rot_{3,\omega}$

$\displaystyle B( m_{1,\omega}(D) Rot_{1,\omega} u, m_{2,\omega}(D) Rot_{2,\omega} v )\ d\mu(\omega)$

for some probability space ${(\Omega, \mu)}$, some spatial rotation operators ${Rot_{i,\omega}}$ for ${i=1,2,3}$, and some Fourier multipliers ${m_{i,\omega}}$ of order ${0}$, for which one still has the cancellation law

$\displaystyle \langle \tilde B(u,u), u \rangle = 0$

and for which the averaged Navier-Stokes equation

$\displaystyle \partial_t u = \Delta u + \tilde B(u,u) \ \ \ \ \ (2)$

admits solutions that blow up in finite time.

(There are some integrability conditions on the Fourier multipliers ${m_{i,\omega}}$ required in the above theorem in order for the conclusion to be non-trivial, but I am omitting them here for sake of exposition.)

Because spatial rotations and Fourier multipliers of order ${0}$ are bounded on most function spaces, ${\tilde B}$ automatically obeys almost all of the upper bound estimates that ${B}$ does. Thus, this theorem blocks any attempt to prove global regularity for the true Navier-Stokes equations which relies purely on the energy identity and on upper bound estimates for the nonlinearity; one must use some additional structure of the nonlinear operator ${B}$ which is not shared by an averaged version ${\tilde B}$. Such additional structure certainly exists – for instance, the Navier-Stokes equation has a vorticity formulation involving only differential operators rather than pseudodifferential ones, whereas a general equation of the form (2) does not. However, “abstract” approaches to global regularity generally do not exploit such structure, and thus cannot be used to affirmatively answer the Navier-Stokes problem.

It turns out that the particular averaged bilinear operator ${B}$ that we will use will be a finite linear combination of local cascade operators, which take the form

$\displaystyle C(u,v) := \sum_{n \in {\bf Z}} (1+\epsilon_0)^{5n/2} \langle u, \psi_{1,n} \rangle \langle v, \psi_{2,n} \rangle \psi_{3,n}$

where ${\epsilon_0>0}$ is a small parameter, ${\psi_1,\psi_2,\psi_3}$ are Schwartz vector fields whose Fourier transform is supported on an annulus, and ${\psi_{i,n}(x) := (1+\epsilon_0)^{3n/2} \psi_i( (1+\epsilon_0)^n x)}$ is an ${L^2}$-rescaled version of ${\psi_i}$ (basically a “wavelet” of wavelength about ${(1+\epsilon_0)^{-n}}$ centred at the origin). Such operators were essentially introduced by Katz and Pavlovic as dyadic models for ${B}$; they have the essentially the same scaling property as ${B}$ (except that one can only scale along powers of ${1+\epsilon_0}$, rather than over all positive reals), and in fact they can be expressed as an average of ${B}$ in the sense of the above theorem, as can be shown after a somewhat tedious amount of Fourier-analytic symbol manipulations.

If we consider nonlinearities ${\tilde B}$ which are a finite linear combination of local cascade operators, then the equation (2) more or less collapses to a system of ODE in certain “wavelet coefficients” of ${u}$. The precise ODE that shows up depends on what precise combination of local cascade operators one is using. Katz and Pavlovic essentially considered a single cascade operator together with its “adjoint” (needed to preserve the energy identity), and arrived (more or less) at the system of ODE

$\displaystyle \partial_t X_n = - (1+\epsilon_0)^{2n} X_n + (1+\epsilon_0)^{\frac{5}{2}(n-1)} X_{n-1}^2 - (1+\epsilon_0)^{\frac{5}{2} n} X_n X_{n+1} \ \ \ \ \ (3)$

where ${X_n: [0,T] \rightarrow {\bf R}}$ are scalar fields for each integer ${n}$. (Actually, Katz-Pavlovic worked with a technical variant of this particular equation, but the differences are not so important for this current discussion.) Note that the quadratic terms on the RHS carry a higher exponent of ${1+\epsilon_0}$ than the dissipation term; this reflects the supercritical nature of this evolution (the energy ${\frac{1}{2} \sum_n X_n^2}$ is monotone decreasing in this flow, so the natural size of ${X_n}$ given the control on the energy is ${O(1)}$). There is a slight technical issue with the dissipation if one wishes to embed (3) into an equation of the form (2), but it is minor and I will not discuss it further here.

In principle, if the ${X_n}$ mode has size comparable to ${1}$ at some time ${t_n}$, then energy should flow from ${X_n}$ to ${X_{n+1}}$ at a rate comparable to ${(1+\epsilon_0)^{\frac{5}{2} n}}$, so that by time ${t_{n+1} \approx t_n + (1+\epsilon_0)^{-\frac{5}{2} n}}$ or so, most of the energy of ${X_n}$ should have drained into the ${X_{n+1}}$ mode (with hardly any energy dissipated). Since the series ${\sum_{n \geq 1} (1+\epsilon_0)^{-\frac{5}{2} n}}$ is summable, this suggests finite time blowup for this ODE as the energy races ever more quickly to higher and higher modes. Such a scenario was indeed established by Katz and Pavlovic (and refined by Cheskidov) if the dissipation strength ${(1+\epsilon)^{2n}}$ was weakened somewhat (the exponent ${2}$ has to be lowered to be less than ${\frac{5}{3}}$). As mentioned above, this is enough to give a version of Theorem 1 in five and higher dimensions.

On the other hand, it was shown a few years ago by Barbato, Morandin, and Romito that (3) in fact admits global smooth solutions (at least in the dyadic case ${\epsilon_0=1}$, and assuming non-negative initial data). Roughly speaking, the problem is that as energy is being transferred from ${X_n}$ to ${X_{n+1}}$, energy is also simultaneously being transferred from ${X_{n+1}}$ to ${X_{n+2}}$, and as such the solution races off to higher modes a bit too prematurely, without absorbing all of the energy from lower modes. This weakens the strength of the blowup to the point where the moderately strong dissipation in (3) is enough to kill the high frequency cascade before a true singularity occurs. Because of this, the original Katz-Pavlovic model cannot quite be used to establish Theorem 1 in three dimensions. (Actually, the original Katz-Pavlovic model had some additional dispersive features which allowed for another proof of global smooth solutions, which is an unpublished result of Nazarov.)

To get around this, I had to “engineer” an ODE system with similar features to (3) (namely, a quadratic nonlinearity, a monotone total energy, and the indicated exponents of ${(1+\epsilon_0)}$ for both the dissipation term and the quadratic terms), but for which the cascade of energy from scale ${n}$ to scale ${n+1}$ was not interrupted by the cascade of energy from scale ${n+1}$ to scale ${n+2}$. To do this, I needed to insert a delay in the cascade process (so that after energy was dumped into scale ${n}$, it would take some time before the energy would start to transfer to scale ${n+1}$), but the process also needed to be abrupt (once the process of energy transfer started, it needed to conclude very quickly, before the delayed transfer for the next scale kicked in). It turned out that one could build a “quadratic circuit” out of some basic “quadratic gates” (analogous to how an electrical circuit could be built out of basic gates such as amplifiers or resistors) that achieved this task, leading to an ODE system essentially of the form

$\displaystyle \partial_t X_{1,n} = - (1+\epsilon_0)^{2n} X_{1,n}$

$\displaystyle + (1+\epsilon_0)^{5n/2} (- \epsilon^{-2} X_{3,n} X_{4,n} - \epsilon X_{1,n} X_{2,n} - \epsilon^2 \exp(-K^{10}) X_{1,n} X_{3,n}$

$\displaystyle + K X_{4,n-1}^2)$

$\displaystyle \partial_t X_{2,n} = - (1+\epsilon_0)^{2n} X_{2,n} + (1+\epsilon_0)^{5n/2} (\epsilon X_{1,n}^2 - \epsilon^{-1} K^{10} X_{3,n}^2)$

$\displaystyle \partial_t X_{3,n} = - (1+\epsilon_0)^{2n} X_{3,n} + (1+\epsilon_0)^{5n/2} (\epsilon^2 \exp(-K^{10}) X_{1,n}^2$

$\displaystyle + \epsilon^{-1} K^{10} X_{2,n} X_{3,n} )$

$\displaystyle \partial_t X_{4,n} =- (1+\epsilon_0)^{2n} X_{4,n} + (1+\epsilon_0)^{5n/2} (\epsilon^{-2} X_{3,n} X_{1,n}$

$\displaystyle - (1+\epsilon_0)^{5/2} K X_{4,n} X_{1,n+1})$

where ${K \geq 1}$ is a suitable large parameter and ${\epsilon > 0}$ is a suitable small parameter (much smaller than ${1/K}$). To visualise the dynamics of such a system, I found it useful to describe this system graphically by a “circuit diagram” that is analogous (but not identical) to the circuit diagrams arising in electrical engineering:

The coupling constants here range widely from being very large to very small; in practice, this makes the ${X_{2,n}}$ and ${X_{3,n}}$ modes absorb very little energy, but exert a sizeable influence on the remaining modes. If a lot of energy is suddenly dumped into ${X_{1,n}}$, what happens next is roughly as follows: for a moderate period of time, nothing much happens other than a trickle of energy into ${X_{2,n}}$, which in turn causes a rapid exponential growth of ${X_{3,n}}$ (from a very low base). After this delay, ${X_{3,n}}$ suddenly crosses a certain threshold, at which point it causes ${X_{1,n}}$ and ${X_{4,n}}$ to exchange energy back and forth with extreme speed. The energy from ${X_{4,n}}$ then rapidly drains into ${X_{1,n+1}}$, and the process begins again (with a slight loss in energy due to the dissipation). If one plots the total energy ${E_n := \frac{1}{2} ( X_{1,n}^2 + X_{2,n}^2 + X_{3,n}^2 + X_{4,n}^2 )}$ as a function of time, it looks schematically like this:

As in the previous heuristic discussion, the time between cascades from one frequency scale to the next decay exponentially, leading to blowup at some finite time ${T}$. (One could describe the dynamics here as being similar to the famous “lighting the beacons” scene in the Lord of the Rings movies, except that (a) as each beacon gets ignited, the previous one is extinguished, as per the energy identity; (b) the time between beacon lightings decrease exponentially; and (c) there is no soundtrack.)

There is a real (but remote) possibility that this sort of construction can be adapted to the true Navier-Stokes equations. The basic blowup mechanism in the averaged equation is that of a von Neumann machine, or more precisely a construct (built within the laws of the inviscid evolution ${\partial_t u = \tilde B(u,u)}$) that, after some time delay, manages to suddenly create a replica of itself at a finer scale (and to largely erase its original instantiation in the process). In principle, such a von Neumann machine could also be built out of the laws of the inviscid form of the Navier-Stokes equations (i.e. the Euler equations). In physical terms, one would have to build the machine purely out of an ideal fluid (i.e. an inviscid incompressible fluid). If one could somehow create enough “logic gates” out of ideal fluid, one could presumably build a sort of “fluid computer”, at which point the task of building a von Neumann machine appears to reduce to a software engineering exercise rather than a PDE problem (providing that the gates are suitably stable with respect to perturbations, but (as with actual computers) this can presumably be done by converting the analog signals of fluid mechanics into a more error-resistant digital form). The key thing missing in this program (in both senses of the word) to establish blowup for Navier-Stokes is to construct the logic gates within the laws of ideal fluids. (Compare with the situation for cellular automata such as Conway’s “Game of Life“, in which Turing complete computers, universal constructors, and replicators have all been built within the laws of that game.)