I’ve just uploaded to the arXiv my paper “Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation“, submitted to Analysis & PDE.  It is a famous problem to establish the existence of global smooth solutions to the three-dimensional Navier-Stokes system of equations

$\partial_t u + (u \cdot \nabla) u = \Delta u - \nabla p$
$\nabla \cdot u = 0$
$u(0,x) = u_0(x)$

given smooth, compactly supported, divergence-free initial data $u_0: {\Bbb R}^3 \to {\Bbb R}^3$.

I do not claim to have any substantial progress on this problem here.  Instead, the paper makes a small observation about the hyper-dissipative version of the Navier-Stokes equations, namely

$\partial_t u + (u \cdot \nabla) u = - |\nabla|^{2\alpha} u - \nabla p$
$\nabla \cdot u = 0$
$u(0,x) = u_0(x)$

for some $\alpha > 1$.  It is a folklore result that global regularity for this equation holds for $\alpha \geq 5/4$; the significance of the exponent $5/4$ is that it is energy-critical, in the sense that the scaling which preserves this particular hyper-dissipative Navier-Stokes equation, also preserves the energy.

Values of $\alpha$ below $5/4$ (including, unfortunately, the case $\alpha=1$, which is the original Navier-Stokes equation) are supercritical and thus establishing global regularity beyond the reach of most known methods (see my earlier blog post for more discussion).

A few years ago, I observed (in the case of the spherically symmetric wave equation) that this “criticality barrier” had a very small amount of flexibility to it, in that one could push a critical argument to a slightly supercritical one by exploiting spacetime integral estimates a little bit more.  I realised recently that the same principle applied to hyperdissipative Navier-Stokes; here, the relevant spacetime integral estimate is the energy dissipation inequality

$\int_0^T \int_{{\Bbb R}^d} | |\nabla|^\alpha u(t,x)|^2\ dx dt \leq \frac{1}{2} \int_{{\Bbb R}^d} |u_0(x)|^2\ dx$

which ensures that the energy dissipation $a(t) := \int_{{\Bbb R}^d} | |\nabla|^\alpha u(t,x)|^2\ dx$ is locally integrable (and in fact globally integrable) in time.

In this paper I push the global regularity results by a fraction of a logarithm from $\alpha=5/4$ towards $\alpha=1$.  For instance, the argument shows that the logarithmically supercritical equation

$\partial_t u + (u \cdot \nabla) u = - \frac{|\nabla|^{5/2}}{\log^{1/2}(2-\Delta)} u - \nabla p$ (0)
$\nabla \cdot u = 0$
$u(0,x) = u_0(x)$

The argument is in fact quite simple (the paper is seven pages in length), and relies on known technology; one just applies the energy method and a logarithmically modified Sobolev inequality in the spirit of a well-known inequality of Brezis and Wainger.  It looks like it will take quite a bit of effort though to improve the logarithmic factor much further.

One way to explain the tiny bit of wiggle room beyond the critical case is as follows.  The standard energy method approach to the critical Navier-Stokes equation relies at one stage on Gronwall’s inequality, which among other things asserts that if a time-dependent non-negative quantity E(t) obeys the differential inequality

$\partial_t E(t) \leq a(t) E(t)$ (1)

and $a(t)$ was locally integrable, then E does not blow up in time; in fact, one has the inequality

$E(t) \leq E(0) \exp( \int_0^t a(s)\ ds )$.

A slight modification of the argument shows that one can replace the linear inequality with a slightly superlinear inequality.  For instance, the differential inequality

$\partial_t E(t) \leq a(t) E(t) \log E(t)$ (2)

also does not blow up in time; indeed, a separation of variables argument gives the explicit double-exponential bound

$E(t) \leq \exp(\exp( \int_0^t a(s)\ ds + \log \log E(0) ))$

(let’s take $E(0) > 1$ and all functions smooth, to avoid technicalities).  It is this ability to go beyond Gronwall’s inequality by a little bit which is really at the heart of the logarithmically supercritical phenomenon.  In the paper, I establish an inequality basically of the shape (2), where $E(t)$ is a suitably high-regularity Sobolev norm of $u(t)$, and $a(t)$ is basically the energy dissipation mentioned earlier.  The point is that the logarithmic loss of $\log(1 - \Delta)^{1/4}$ in the dissipation can eventually be converted (by a Brezis-Wainger type argument) to a logarithmic loss in the high-regularity energy, as this energy can serve as a proxy for the frequency $|\xi|$, which in turn serves as a proxy for the Laplacian $-\Delta$.

To put it another way, with a linear exponential growth model, such as $\partial_t E(t) = C E(t)$, it takes a constant amount of time for E to double, and so E never becomes infinite in finite time.  With an equation such as $\partial_t E(t) = C E(t) \log E(t)$, the time taken for E to double from (say) $2^n$ to $2^{n+1}$ now shrinks to zero, but only as quickly as the harmonic series $1/n$, so it still takes an infinite amount of time for E to blow up.  But because the divergence of $\sum_n 1/n$ is logarithmically slow, the growth of E is now a double exponential rather than a single one.  So there is a little bit of room to exploit between exponential growth and blowup.

Interestingly, there is a heuristic argument that suggests that the half-logarithmic loss in (0) can be widened to a full logarithmic loss, which I give below the fold.

Suppose the solution to (0) (with a full power of the logarithm) blows up at some finite time $T_*$.  We make the (somewhat improbable) ansatz that all the energy concentrates to a point (e.g. the origin) at this blowup time, thus for each $0 < t < T_*$ we assume that u is concentrated in a ball of radius $1/N(t)$, where $N(t)$ is a frequency scale that goes to infinity as $t \to T_*$.  (This type of concentration is, heuristically, the “worst case” for any argument involving Sobolev embedding, which the arguments in my paper certainly rely on.)  As the total energy is bounded, and the ball has volume $O( 1/N(t)^3)$, this suggests that u(t,x) should have magnitude $O(N(t)^{3/2})$ at time t.  In particular, from the convection term $(u \cdot \nabla) u$ in (0) this suggests that u propagates at speed $O(N(t)^{3/2})$.  In particular, the radius $1/N(t)$ should obey the ODE

$\frac{d}{dt} \frac{1}{N(t)} = O( N(t)^{3/2} )$

which after solving this ODE suggests that N(t) needs to blow up at the rate $(T_*-t)^{-2/5}$ or faster: $N(t) \gg (T_*-t)^{-2/5}$.  On the other hand, the energy identity for (0) (with a full power of the logarithm) implies that

$\int_0^{T_*} \int_{{\Bbb R}^3} \frac{|\nabla|^{5/2}}{\log(2-\Delta)} u \cdot u\ dx dt < \infty$;

heuristically substituting in our ansatz, this suggests

$\int_0^{T_*} \frac{N(t)^{5/2}}{\log(2 +N(t))} dt < \infty$

but this is incompatible with the blowup rate $N(t) \gg (T_*-t)^{-2/5}$ because $\int_0^\infty \frac{ds}{s \log s}$ is (barely) divergent at infinity.  Unfortunately, I do not know how to make this non-rigorous argument precise without taking on some unwanted logarithmic losses, but it may well be feasible to do so; readers are welcome to try, of course :-).