I have just uploaded to the arXiv my paper “A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation”, submitted to Dynamics of PDE. This is a short note on one formulation of the Clay Millennium prize problem, namely that there exists a global smooth solution to the Navier-Stokes equation on the torus ({\Bbb R}/{\Bbb Z})^3 given any smooth divergence-free data. (I should emphasise right off the bat that I am not claiming any major breakthrough on this problem, which remains extremely challenging in my opinion.)
This problem is formulated in a qualitative way: the conjecture asserts that the velocity field u stays smooth for all time, but does not ask for a quantitative bound on the smoothness of that field in terms of the smoothness of the initial data. Nevertheless, it turns out that the compactness properties of the periodic Navier-Stokes flow allow one to equate the qualitative claim with a more concrete quantitative one. More precisely, the paper shows that the following three statements are equivalent:

  1. (Qualitative regularity conjecture) Given any smooth divergence-free data u_0: ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3, there exists a global smooth solution u: [0,+\infty) \times ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3 to the Navier-Stokes equations.
  2. (Local-in-time quantitative regularity conjecture)
    Given any smooth solution u: [0,T] \times ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3 to the Navier-Stokes equations with 0 < T \leq 1, one has the a priori bound\| u(T) \|_{H^1(({\Bbb R}/{\Bbb Z})^3)} \leq F( \| u(0) \|_{H^1(({\Bbb R}/{\Bbb Z})^3)} ) for some non-decreasing function F:[0,+\infty) \to [0,+\infty).
  3. (Global-in-time quantitative regularity conjecture) This is the same conjecture as 2, but with the condition 0 < T \leq 1 replaced by 0 < T < \infty.

It is easy to see that Conjecture 3 implies Conjecture 2, which implies Conjecture 1. By using the compactness of the local periodic Navier-Stokes flow in H^1, one can show that Conjecture 1 implies Conjecture 2; and by using the energy identity (and in particular the fact that the energy dissipation is bounded) one can deduce Conjecture 3 from Conjecture 2. The argument uses only standard tools and is likely to generalise in a number of ways, which I discuss in the paper. (In particular one should be able to replace the H^1 norm here by any other subcritical norm.)

When I previously discussed the Navier-Stokes equations, I suggested that perhaps the best hope to attack this equation was by what I called “Strategy 1”: by obtaining a new a priori bound on solutions to this equation. What this result indicates (in the periodic case) is that this strategy is in fact essentially the only strategy for solving this equation, since the regularity problem is in fact equivalent to that of obtaining an a priori bound.

As the qualitative result is now logically equivalent to a quantitative one, it seems to me that purely “soft” approaches to the problem are now extremely unlikely to work, and that a substantial amount of “hard analysis” would have to go into any putative proof of this problem. In particular, it is clear that if one attempts to construct solutions by expressing them as the limit of some sort of regularised (or discretised) solutions, this can only work if one can obtain a priori bounds on the approximating solutions which are uniform in the approximation parameter, since all other bounds will be lost in the passage to the limit. One of course has the energy inequality for such approximate solutions (which is how Leray constructed global weak solutions to the Navier-Stokes equations), but to obtain smooth solutions one needs to control a norm such as H^1. [Incidentally, the energy inequality does show that \|u(T)\|_{H^1} is square-integrable in time, but this is quite far away from what we really need, which is that \|u(T)\|_{H^1} is bounded (or at least fourth power integrable). Actually, these two statements are radically different: the former is a supercritical control on u and the latter is subcritical (or at least critical) control.]

One amusing consequence of this equivalence is that the Navier-Stokes regularity conjecture is also equivalent to a non-perturbative global stability result for the Navier-Stokes flow. In particular, if the Navier-Stokes regularity conjecture is true, then the solution map maps any bounded set in H^1_x({\Bbb R}^3/{\Bbb Z}^3) to a bounded set in C^0_t  H^1_x([0,\infty) \times {\Bbb R}^3/{\Bbb Z}^3) in a Lipschitz manner, if one fixes the mean velocity \int_{{\Bbb R}^3/{\Bbb Z}^3} u_0 to avoid issues with drift. (The dissipation is what allows one to control things at very late times; this is one of the strange features of supercritical estimates such as the energy inequality, in that they offer very poor short-time control but give excellent long-time control.)