I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that talk are here.)
One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity ![{\omega = \nabla \times u}]()
, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become
![\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u + \nu \Delta \omega]()
![\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).]()
(Throughout this post we will be working in three spatial dimensions ![{{\bf R}^3}]()
.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.
Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case ![{\nu=0}]()
of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity (which, as explained in the above slides, heuristically means that the effects of viscosity are negligible when compared against the nonlinear components of the equation). In vorticity-stream formulation, the Euler equations can be written as
![\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u]()
![\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).]()
As discussed in this previous blog post, a natural generalisation of this system of equations is the system
![\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (1)]()
![\displaystyle u = T (-\Delta)^{-1} (\nabla \times \omega).]()
where ![{T}]()
is a linear operator on divergence-free vector fields that is “zeroth order” in some sense; ideally it should also be invertible, self-adjoint, and positive definite (in order to have a Hamiltonian that is comparable to the kinetic energy ![{\frac{1}{2} \int_{{\bf R}^3} |u|^2}]()
). (In the previous blog post, it was observed that the surface quasi-geostrophic (SQG) equation could be embedded in a system of the form (1).) The system (1) has many features in common with the Euler equations; for instance vortex lines are transported by the velocity field ![{u}]()
, and Kelvin’s circulation theorem is still valid.
So far, I have not been able to fully achieve this goal. However, I have the following partial result, stated somewhat informally:
Theorem 1 There is a “zeroth order” linear operator ![{T}]()
(which, unfortunately, is not invertible, self-adjoint, or positive definite) for which the system (1) exhibits smooth solutions that blowup in finite time.
The operator ![{T}]()
constructed is not quite a zeroth-order pseudodifferential operator; it is instead merely in the “forbidden” symbol class ![{S^0_{1,1}}]()
, and more precisely it takes the form
![\displaystyle T v = \sum_{j \in {\bf Z}} 2^{3j} \langle v, \phi_j \rangle \psi_j \ \ \ \ \ (2)]()
for some compactly supported divergence-free ![{\phi,\psi}]()
of mean zero with
![\displaystyle \phi_j(x) := \phi(2^j x); \quad \psi_j(x) := \psi(2^j x)]()
being ![{L^2}]()
rescalings of ![{\phi,\psi}]()
. This operator is still bounded on all ![{L^p({\bf R}^3)}]()
spaces ![{1 < p < \infty}]()
, and so is arguably still a zeroth order operator, though not as convincingly as I would like. Another, less significant, issue with the result is that the solution constructed does not have good spatial decay properties, but this is mostly for convenience and it is likely that the construction can be localised to give solutions that have reasonable decay in space. But the biggest drawback of this theorem is the fact that ![{T}]()
is not invertible, self-adjoint, or positive definite, so in particular there is no non-negative Hamiltonian for this equation. It may be that some modification of the arguments below can fix these issues, but I have so far been unable to do so. Still, the construction does show that the circulation theorem is insufficient by itself to prevent blowup.
We sketch the proof of the above theorem as follows. We use the barrier method, introducing the time-varying hyperboloid domains
![\displaystyle \Omega(t) := \{ (r,\theta,z): r^2 \leq 1-t + z^2 \}]()
for ![{t>0}]()
(expressed in cylindrical coordinates ![{(r,\theta,z)}]()
). We will select initial data ![{\omega(0)}]()
to be ![{\omega(0,r,\theta,z) = (0,0,\eta(r))}]()
for some non-negative even bump function ![{\eta}]()
supported on ![{[-1,1]}]()
, normalised so that
![\displaystyle \int\int \eta(r)\ r dr d\theta = 1;]()
in particular ![{\omega(0)}]()
is divergence-free supported in ![{\Omega(0)}]()
, with vortex lines connecting ![{z=-\infty}]()
to ![{z=+\infty}]()
. Suppose for contradiction that we have a smooth solution ![{\omega}]()
to (1) with this initial data; to simplify the discussion we assume that the solution behaves well at spatial infinity (this can be justified with the choice (2) of vorticity-stream operator, but we will not do so here). Since the domains ![{\Omega(t)}]()
disconnect ![{z=-\infty}]()
from ![{z=+\infty}]()
at time ![{t=1}]()
, there must exist a time ![{0 < T_* < 1}]()
which is the first time where the support of ![{\omega(T_*)}]()
touches the boundary of ![{\Omega(T_*)}]()
, with ![{\omega(t)}]()
supported in ![{\Omega(t)}]()
.
From (1) we see that the support of ![{\omega(t)}]()
is transported by the velocity field ![{u(t)}]()
. Thus, at the point of contact of the support of ![{\omega(T_*)}]()
with the boundary of ![{\Omega(T_*)}]()
, the inward component of the velocity field ![{u(T_*)}]()
cannot exceed the inward velocity of ![{\Omega(T_*)}]()
. We will construct the functions ![{\phi,\psi}]()
so that this is not the case, leading to the desired contradiction. (Geometrically, what is going on here is that the operator ![{T}]()
is pinching the flow to pass through the narrow cylinder ![{\{ z, r = O( \sqrt{1-t} )\}}]()
, leading to a singularity by time ![{t=1}]()
at the latest.)
First we observe from conservation of circulation, and from the fact that ![{\omega(t)}]()
is supported in ![{\Omega(t)}]()
, that the integrals
![\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta]()
are constant in both space and time for ![{0 \leq t \leq T_*}]()
. From the choice of initial data we thus have
![\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta = 1]()
for all ![{t \leq T_*}]()
and all ![{z}]()
. On the other hand, if ![{T}]()
is of the form (2) with ![{\phi = \nabla \times \eta}]()
for some bump function ![{\eta = (0,0,\eta_z)}]()
that only has ![{z}]()
-components, then ![{\phi}]()
is divergence-free with mean zero, and
![\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = 2^{-j} \langle (-\Delta) (\nabla \times \omega), \nabla \times \eta_j \rangle]()
![\displaystyle = 2^{-j} \langle \omega, \eta_j \rangle]()
![\displaystyle = 2^{-j} \int\int\int \omega_z(t,r,\theta,z) \eta_z(2^j r, \theta, 2^j z)\ r dr d\theta dz,]()
where ![{\eta_j(x) := \eta(2^j x)}]()
. We choose ![{\eta_z}]()
to be supported in the slab ![{\{ C \leq z \leq 2C\}}]()
for some large constant ![{C}]()
, and to equal a function ![{f(z)}]()
depending only on ![{z}]()
on the cylinder ![{\{ C \leq z \leq 2C; r \leq 10C \}}]()
, normalised so that ![{\int f(z)\ dz = 1}]()
. If ![{C/2^j \geq (1-t)^{1/2}}]()
, then ![{\Omega(t)}]()
passes through this cylinder, and we conclude that
![\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = -2^{-j} \int f(2^j z)\ dz]()
![\displaystyle = 2^{-2j}.]()
Inserting ths into (2), (1) we conclude that
![\displaystyle u = \sum_{j: C/2^j \geq (1-t)^{1/2}} 2^j \psi_j + \sum_{j: C/2^j < (1-t)^{1/2}} c_j(t) \psi_j]()
for some coefficients ![{c_j(t)}]()
. We will not be able to control these coefficients ![{c_j(t)}]()
, but fortunately we only need to understand ![{u}]()
on the boundary ![{\partial \Omega(t)}]()
, for which ![{r+|z| \gg (1-t)^{1/2}}]()
. So, if ![{\psi}]()
happens to be supported on an annulus ![{1 \ll r+|z| \ll 1}]()
, then ![{\psi_j}]()
vanishes on ![{\partial \Omega(t)}]()
if ![{C}]()
is large enough. We then have
![\displaystyle u = \sum_j 2^j \psi_j]()
on the boundary of ![{\partial \Omega(t)}]()
.
Let ![{\Phi(r,\theta,z)}]()
be a function of the form
![\displaystyle \Phi(r,\theta,z) = C z \varphi(z/r)]()
where ![{\varphi}]()
is a bump function supported on ![{[-2,2]}]()
that equals ![{1}]()
on ![{[-1,1]}]()
. We can perform a dyadic decomposition ![{\Phi = \sum_j \Psi_j}]()
where
![\displaystyle \Psi_j(r,\theta,z) = \Phi(r,\theta,z) a(2^j r)]()
where ![{a}]()
is a bump function supported on ![{[1/2,2]}]()
with ![{\sum_j a(2^j r) = 1}]()
. If we then set
![\displaystyle \psi_j = \frac{2^{-j}}{r} (-\partial_z \Psi_j, 0, \partial_r \Psi_j)]()
then one can check that ![{\psi_j(x) = \psi(2^j x)}]()
for a function ![{\psi}]()
that is divergence-free and mean zero, and supported on the annulus ![{1 \ll r+|z| \ll 1}]()
, and
![\displaystyle \sum_j 2^j \psi_j = \frac{1}{r} (-\partial_z \Phi, 0, \partial_r \Phi)]()
so on ![{\partial \Omega(t)}]()
(where ![{|z| \leq r}]()
) we have
![\displaystyle u = (-\frac{C}{r}, 0, 0 ).]()
One can manually check that the inward velocity of this vector on ![{\partial \Omega(t)}]()
exceeds the inward velocity of ![{\Omega(t)}]()
if ![{C}]()
is large enough, and the claim follows.
Remark 2 The type of blowup suggested by this construction, where a unit amount of circulation is squeezed into a narrow cylinder, is of “Type II” with respect to the Navier-Stokes scaling, because Navier-Stokes-critical norms such ![{L^3({\bf R}^3)}]()
(or at least ![{L^{3,\infty}({\bf R}^3)}]()
) look like they stay bounded during this squeezing procedure (the velocity field is of size about ![{2^j}]()
in cylinders of radius and length about ![{2^j}]()
). So even if the various issues with ![{T}]()
are repaired, it does not seem likely that this construction can be directly adapted to obtain a corresponding blowup for a Navier-Stokes type equation. To get a “Type I” blowup that is consistent with Kelvin’s circulation theorem, it seems that one needs to coil the vortex lines around a loop multiple times in order to get increased circulation in a small space. This seems possible to pull off to me – there don’t appear to be any unavoidable obstructions coming from topology, scaling, or conservation laws – but would require a more complicated construction than the one given above.
![{[-1,1]}](http://s0.wp.com/latex.php?latex=%7B%5B-1%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{[-2,2]}](http://s0.wp.com/latex.php?latex=%7B%5B-2%2C2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{[1/2,2]}](http://s0.wp.com/latex.php?latex=%7B%5B1%2F2%2C2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
(or at least 
) look like they stay bounded during this squeezing procedure (the velocity field is of size about 
in cylinders of radius and length about