This article describes some recent work on ‘direct air capture’ of carbon dioxide—essentially, sucking it out of the air:
• Jon Gerntner, The tiny Swiss company that thinks it can help stop climate change, New York Times Magazine, 12 February 2019.
There’s a Swiss company called Climeworks that’s built machines that do this—shown in the picture above. So far they are using these machines for purposes other than reducing atmospheric CO2 concentrations: namely, making carbonated water for soft drinks, and getting greenhouses to have lots of carbon dioxide in the air, for tastier vegetables. And they’re just experimental, not economically viable yet:
The company is not turning a profit. To build and install the 18 units at Hinwil, hand-assembled in a second-floor workshop in Zurich, cost between $3 million and $4 million, which is the primary reason it costs the firm between $500 and $600 to remove a metric ton of CO₂ from the air. Even as the company has attracted about $50 million in private investments and grants, it faces the same daunting task that confronted Carl Bosch a century ago: How much can it bring costs down? And how fast can it scale up?
If they ever make it in these markets, greenhouses and carbonation might want 6 megatonnes of CO₂ annually. This is nothing compared to the 37 gigatonnes of CO₂ that we put into the atmosphere in 2018. In principle the technology Climeworks is using could be massively scaled up. After all, Napoleon used aluminum silverware, back when aluminum was more precious than gold… and only later did the technology for making aluminum improve to the point where the metal gained a mass market.
But can Climeworks’ technology actually be scaled up? Some are dubious:
M.I.T.’s Howard Herzog, for instance, an engineer who has spent years looking at the potential for these machines, told me that he thinks the costs will remain between $600 and $1,000 per metric ton. Some of Herzog’s reasons for skepticism are highly technical and relate to the physics of separating gases. Some are more easily grasped. He points out that because direct-air-capture machines have to move tremendous amounts of air through a filter or solution to glean a ton of CO₂ — the gas, for all its global impact, makes up only about 0.04 percent of our atmosphere — the process necessitates large expenditures for energy and big equipment. What he has likewise observed, in analyzing similar industries that separate gases, suggests that translating spreadsheet projections for capturing CO₂ into real-world applications will reveal hidden costs. “I think there has been a lot of hype about this, and it’s not going to revolutionize anything,” he told me, adding that he thinks other negative-emissions technologies will prove cheaper. “At best it’s going to be a bit player.”
What actually is the technology Climeworks is using? And what other technologies are available for sucking carbon dioxide out of the air—or out of the exhaust from fossil-fuel-burning power plants, or out of water?
I’ll have a lot more to say about the latter question in future articles. As for Climeworks, they describe their technology rather briefly here:
• Climeworks, Our technology.
They write:
Our plants capture atmospheric carbon with a filter. Air is drawn into the plant and the CO2 within the air is chemically bound to the filter.
Once the filter is saturated with CO2 it is heated (using mainly low-grade heat as an energy source) to around 100 °C (212 °F). The CO2 is then released from the filter and collected as concentrated CO2 gas to supply to customers or for negative emissions technologies.
CO2-free air is released back into the atmosphere. This continuous cycle is then ready to start again. The filter is reused many times and lasts for several thousand cycles.
What is the filter material?
The filter material is made of porous granulates modified with amines, which bind the CO2 in conjunction with the moisture in the air. This bond is dissolved at temperatures of 100 °C.
So, it seems their technology is an example of ‘amine gas treating’:
• Wikipedia, Amine gas treating.
In future posts I’ll talk a bit more about amine gas treating, but also other methods for absorbing carbon dioxide from air or from solution in water. Maybe you can help me figure out what’s the best method!
, which we write in divergence form as 
is the (inverse) Euclidean metric. Here we use the summation conventions for indices such as 
(reserving the symbol 
for other purposes), and are retaining the convention from 
is the Euclidean metric. Indeed, if one assumes that 
are continuously differentiable in both space and time on ![{[0,T] \times \mathbf{T}}](https://s0.wp.com/latex.php?latex=%7B%5B0%2CT%5D+%5Ctimes+%5Cmathbf%7BT%7D%7D&bg=ffffff&fg=000000&s=0 "{[0,T] \times \mathbf{T}}")
, then one can multiply the equation 
and contract against 
to obtain 
![{[0,T] \times \mathbf{T}_E}](https://s0.wp.com/latex.php?latex=%7B%5B0%2CT%5D+%5Ctimes+%5Cmathbf%7BT%7D_E%7D&bg=ffffff&fg=000000&s=0 "{[0,T] \times \mathbf{T}_E}")
and uses Stokes’ theorem, one obtains the required energy conservation law 
is not a test function and so one cannot immediately integrate 
is low enough, energy can “escape to frequency infinity”, leading to failure of the energy conservation law, a phenomenon known in physics as anomalous energy dissipation.
for 
, defined as the space of continuous functions 
whose norm 
lies between the space 
of continuous functions and the space 
of continuously differentiable functions, and informally describes a space of functions that is “
times differentiable” in some sense. The above derivation of the energy conservation law involved the integral 
once 
. More precisely, one can make
, and let 
. ![{u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}](https://s0.wp.com/latex.php?latex=%7Bu+%5Cin+C%5E0_t+C%5E%7B0%2C%5Calpha%7D%28%5B0%2CT%5D+%5Ctimes+%5Cmathbf%7BT%7D+%5Crightarrow+%7B%5Cbf+R%7D%29%7D&bg=ffffff&fg=000000&s=0 "{u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}")
to the Euler equations (in the Leray form 
) obeys the energy conservation law 
, then there exist weak solutions 
(using 
-based Besov spaces instead of Hölder spaces, see Exercise 
in three and higher dimensions 
; the endpoint 
remains open. (In two dimensions it may be the case that the positive results extend to a larger range than Onsager’s conjecture predicts; see 
for 
close to 
, which is slightly easier to establish, though still rather intricate. Finally, we discuss the modifications of that construction needed to establish (ii), though we shall stop short of a full proof of that part of the conjecture.
for some 
. Establish the bounds 
![{u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T}_E \rightarrow {\bf R})}](https://s0.wp.com/latex.php?latex=%7Bu+%5Cin+C%5E0_t+C%5E%7B0%2C%5Calpha%7D%28%5B0%2CT%5D+%5Ctimes+%5Cmathbf%7BT%7D_E+%5Crightarrow+%7B%5Cbf+R%7D%29%7D&bg=ffffff&fg=000000&s=0 "{u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T}_E \rightarrow {\bf R})}")
be a weak solution to the Euler equations for some 
, thus 
. Applying 
to 
, and thus 
. Thus 
solves the PDE 
onto the divergence-free factor 
over there as well, we now reduce to showing that 
is a total derivative (as ![\displaystyle \int_0^T \int_{\mathbf{T}_E} \partial_j P_{\leq N} u \cdot [ P_{\leq N} (u^j u) - P_{\leq N} u^j P_{\leq N} u] dx dt = o(1).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_0%5ET+%5Cint_%7B%5Cmathbf%7BT%7D_E%7D+%5Cpartial_j+P_%7B%5Cleq+N%7D+u+%5Ccdot+%5B+P_%7B%5Cleq+N%7D+%28u%5Ej+u%29+-+P_%7B%5Cleq+N%7D+u%5Ej+P_%7B%5Cleq+N%7D+u%5D+dx+dt+%3D+o%281%29.&bg=ffffff&fg=000000&s=0 "\displaystyle \int_0^T \int_{\mathbf{T}_E} \partial_j P_{\leq N} u \cdot [ P_{\leq N} (u^j u) - P_{\leq N} u^j P_{\leq N} u] dx dt = o(1).")
that 
is finite, it thus suffices to establish the pointwise bound 
. 
, which is acceptable since 
. Observe that this commutator would vanish if 
, thus we may write this commutator as 
. If we write 
of total mass one, we have 
, we thus have the bound 
to be the space of functions 
such that the Besov space norm 
![{u \in L^3_t B^3_{\alpha,\infty}([0,T] \times \mathbf{T}_E \rightarrow {\bf R}^d)}](https://s0.wp.com/latex.php?latex=%7Bu+%5Cin+L%5E3_t+B%5E3_%7B%5Calpha%2C%5Cinfty%7D%28%5B0%2CT%5D+%5Ctimes+%5Cmathbf%7BT%7D_E+%5Crightarrow+%7B%5Cbf+R%7D%5Ed%29%7D&bg=ffffff&fg=000000&s=0 "{u \in L^3_t B^3_{\alpha,\infty}([0,T] \times \mathbf{T}_E \rightarrow {\bf R}^d)}")
is a weak solution to the Euler equations, the energy 
is conserved in time. 
of the above exercise is still open; however energy conservation in the slightly smaller space 
is known thanks to the 
is a 
.) 
, show that the energy 
function of time.
. Establish the energy identity 
, then from Exercise 
has amplitude 
at most 
; in particular, the amount of energy at frequencies 
is at most 
. On the other hand, by the heuristics in Remark 11 of 
needed for the portion of the solution at frequency 
is of order 
. Thus the rate of energy flux at frequency 
. For 
, the energy flux goes to zero as 
at a single Fourier mode 
evolves according to the equation 
for any 
, hence by Plancherel’s theorem 
to be of magnitude about 
. Onsager posited that for typically “turbulent” or “chaotic” flows, the main contributions to 
have magnitude roughly comparable to that of 
, one might expect about 
significant terms in the sum, which according to standard “square root cancellation heuristics” (cf. the central limit theorem) suggests that the sum is about as large as 
times the main term. Thus the total flux of energy in or out of a single mode 
, and so the total flux in or out of the frequency range 
(which consists of 
modes 
. As such, for 
threshold by a more physical space based approach, a rigorous version of which was 
of the origin 
in 
into Euclidean space 
.
of smooth maps 
for 
, such that for each point 
, the matrix 
is symmetric and positive definite. Any such metric 
gives 
. An isometric embedding of this manifold into a Euclidean space 
which is continuously differentiable, injective, and obeys the equation 
is the usual inner product (or dot product) on 
such that the Euclidean metric 
on 
.
, whether there is an isometric embedding from this manifold to a Euclidean space 
into 
of the origin. In particular for the local problem we do not need to worry about injectivity since 
is injective at the origin, and hence 
is large enough. For sake of discussion we just present the local version:
. Then for any smooth metric 
on some smaller neighbourhood 
of the origin. 
:
. Then there exists a smooth Riemannian metric 
has 
), but there are only 
on 
(here it is important to restrict to 
in order to avoid the symmetry constraint 
; also, the positive definiteness of the metric will be automatic as long as one restricts to sufficiently small perturbations). Comparing dimensions, we conclude that if every smooth metric 
and sending 
. Taking contrapositives, the claim follows. 
; this is a 
), whereas the definition of an isometric embedding only required the regularity of 
. Then for any smooth metric 
; Kuiper then obtained the optimal condition 
fails due to curvature obstructions; for instance, if the Riemannian metric 
will have (Riemannian) volume slightly less than their Euclidean counterparts, whereas any 
(with the usual metric) into Euclidean space 
. It is a classical result (see e.g., 
isometric embeddings of 
with an isometry of the Euclidean space; however, the Nash-Kuiper construction allows one to create an 
is a short isometric embedding on 
solve the equation 
symmetric and positive definite for all 
. With the additional unknown field 
it is much easier to solve the short isometric problem than the true problem. For instance:
, and let 
. 
and 
for a sufficiently small 
, where 
is the standard embedding, and 
of short embeddings with 
converging to zero in a suitable sense, and then pass to a limit. The key observation is then that by using the fact that the positive matrices lie in the convex hull of the rank one matrices, one can add a high frequency perturbation to the first component 
, replacing it instead with a much higher frequency error.
be a closed ball in 
, one can find a short isometric embedding 
to 
on a ball 
on 
. From this we see that 
converges uniformly to zero, while 
converges in 
be a smooth immersion, and let 
be a smooth Riemannian metric on 
of smooth immersions for 
obeying the bounds 
denotes a quantity that goes to zero as 
). 
(which will be positive definite for 
small enough), we obtain a sequence 
solving 
of unit vectors 
in 
, such that 
. Furthermore, for each 
of the 
are non-zero. 
should belong to the dual space 
of 
appear as subscripts instead of superscripts. A similar consideration occurs for the frequency vectors 
from Remark 
in 
is symmetric and positive definite; one can thus write 
, where 
is an orthonormal basis of (column) eigenvectors of 
are the eigenvalues (we suppress for now the dependence of these objects on 
, where 
unit vectors of the form 
for 
, enumerated in an arbitrary order. From the further parallelogram identity 
also has a representation of the form 
that depend smoothly on the perturbation. As 
sufficiently close to 
varying smoothly with 
centred at 
(say), covering these balls by balls 
of a fixed radius 
smaller than all the 
in the finite cover, in such a way that any point lies in at most 
, constructing a smooth partition of unity 
adapted to the 
previously obtained on 
(which each lie in one of the 
, and summing to obtain the required decomposition on 
-principle
be a unit vector, and let 
be smooth. Then there exists a sequence 
is contained in the support of 
. 
times (and diagonalising the sequences as necessary), we obtain sequences 
of smooth immersions for 
such that 
and one has 
is contained in that of 
be an immersion will be automatic from 
positive definite uniformly for 
.
, it will suffice to find a sequence of smooth maps 
supported in the support of 
, we use the method of slow and fast variables. First we observe by applying a rotation that we may assume without loss of generality that 
is the unit vector 
, thus 
. We then use the ansatz 
is a smooth function independent of 
taking values in the“fast torus” 
. (We adopt the convention here of using boldface symbols to denote functions of both the fast and slow variables. The fast torus is isomorphic to the Eulerian torus 
from the introduction, but we denote them by slightly different notation as they play different roles.) Thus 
that are independent of the fast variable 
as being of “low frequency”, and conversely functions that have mean zero in the fast variable 
for all 
can be uniquely decomposed into a “low frequency” component and a “high frequency” component, with the two components orthogonal to each other. In later sections we will start inverting “fast derivatives” 
on “high frequency” functions, which will effectively gain important factors of 
in the analysis. See also the table below for the dictionary between ordinary physical coordinates and fast-slow coordinates. 
and 
to denote partial differentiation in the coordinates of the slow and fast variables respectively, and noting that all terms with at least one power of 
can be absorbed into the 
to obey the bounds 
are viewed as functions of the slow variable 
. Taking derivatives in 
one would conclude that 
were unit vectors varying smoothly with respect to the slow variable; this obeys 
and 
were both always orthogonal to the entire gradient of 
(as 
linearly independent vectors), but there is no obstruction for 
are unit vectors orthogonal to each other and to 
for 
, depending smoothly on 
from the Gram-Schmidt process. Now we extend in the 
. To do this we evolve the fields 
by the 
by using the previous extension to the segment as initial data and solving the parallel transport ODE 
by repeating the above arguments. By Gram-Schmidt, we can then locate a smooth unit vector field 
which is orthogonal to 
for 
, but for which the quantity 
is positive. If we use the “Kuiper corrugation” ansatz 
, one is reduced to locating such functions 
that obey the bounds 
and any 
, show that there exist smooth functions 
solving the ODE 
(even at the endpoint 
), and obey the bounds 
. The problem is basically to locate a periodic function 
mapping 
to the circle 
of mean zero and Lipschitz norm 
that varies smoothly with 
for some smooth and small 
that is even and compactly supported in 
with mean zero on each interval, and then choose 
to be odd.) 
. For the local isometric embedding problem, we have seen that we have rigidity at regularities at or above 
level for 
, while 
. For recent results in higher dimensions, see 
denote the periodic spacetime 
denote the space of smooth periodic functions 
that have mean zero and are divergence-free at every time 
, thus 
as an abbreviation for 
for various vector spaces 
(the choice of which will be clear from context).
(for now, our discussion will apply both to the Navier-Stokes equations 
and the Euler equations 
). Smooth solutions to Navier-Stokes equations then take the form 
and smooth 
. Here of course 
denotes the spatial Laplacian.
, known as the Reynolds stress (cf. the 
, 
a Navier-Stokes-Reynolds flow (or Euler-Reynolds flow, if 
then we recover a solution to the true Navier-Stokes equations. Thus, heuristically, the smaller 
should become to a solution to the true Navier-Stokes equations. (The Reynolds stress tensor 
here is a rank 
tensor, as opposed to the rank 
tensor 
, 
, 
are smooth functions, then 
will also be a Navier-Stokes-Reynolds flow if and only if 
obeys the difference equation 
small, as one could hopefully iterate this procedure and take a limit to construct weak solutions to the true Euler equations. The main strategy here will be to choose a highly oscillating (and divergence-free) correction velocity field 
such that 
approximates 
up to an error which is also highly oscillating (and somewhat divergence-free). The effect of this error can then eventually be absorbed efficiently into the new Reynolds stress tensor 
. Of course, one also has to manage the other terms 
, 
, 
, 
appearing in 
norm, and can thus be easily disposed of. In three dimensions the situation is considerably more delicate, particularly with regards to the 
is an approximate difference Navier-Stokes-Reynolds flow at 
, 
, 
, as well as the mean of 
with only small adjustments:
is the mean of 
has mean zero in space at every time; this can be achieved by integrating 
) are sufficiently small.
for some compact time interval 
, then 
can be chosen to also be supported in this region. 
is an approximate difference Navier-Stokes-Reynolds flow at 
, where 
can be approximated (in an 
sense) as the limit of Navier-Stokes-Reynolds flows 
, with the Reynolds stress 
be a Navier-Stokes-Reynolds flow. Then for sufficiently large 
obeying the estimates 
, and such that 
for some interval 
, then one can arrange for 
to be supported on 
for any interval 
containing 
). 
is oscillating at frequencies 
, at least in an 
sense. In the next section, we obtain a stronger version of this proposition with more quantitative estimates that can be iterated to obtain higher regularity weak solutions.
of differential operators, we use 
to denote the 
-tuple of differential operators 
with 
. We use 
to denote the 
-tuple formed by concatenating 
for 
. Thus for instance the estimate 
. Informally, one should read the above estimate as asserting that 
, and oscillates with frequency 
in time and 
and a spatial wavelength of 
).
. We may assume that 
.
at 
and obeying the bounds 
at 
obeying the desired estimates (using in particular the fact that 
is bounded on 
is very small).
, and introduce the fast-slow spacetime 
, which we coordinatise as 
; we use 
, and 
. We also use 
as shorthand for 
. Define an approximate fast-slow solution to the difference Navier-Stokes-Reynolds equation at 
) to be a tuple 
of smooth functions 
, 
, 
that obey the system of equations 
denote the tuple 
. Suppose that for any sufficiently large 
that obeys the bounds 
.) From 
, we see from the chain rule that 
is an approximate difference Navier-Stokes-Reynolds flow at 
, we have 
outside of an exceptional set of measure (say) 
. Similarly for the other equations above. Applying the union bound, we can then find a 
obeying the simplified equations 
will be an approximate fast-slow solution supported on 
and 
, and also such that 
has mean zero in the fast variable 
and 
, but we will then apply a “divergence corrector” to almost completely eliminate 
will be a “Mikado flow”, consisting of flows along narrow tubes. (Earlier literature used other flows, such as 
matrices 
. Show that there exist non-zero lattice vectors 
and smooth functions 
for some 
such that 
. (This decomposition is essentially 
so that this matrix is positive definite. There are many choices available for this pressure; we will take 
is the Frobenius norm of 
. Then 
, we conclude that there exist non-zero lattice vectors 
for some 
. As 
depend only on 
, and 
; they are independent of 
on 
are supported in 
, the closed subgroup 
is a one-dimensional subset of 
, so the 
-neighbourhood of this subgroup has measure 
; crucially, this will be a large negative power of 
be a translate of this 
are disjoint; this is easily accomplished by the probabilistic method for 
by an independent random shift and noting that the probability of a collision goes to zero as 
be a large integer (depending on 
be a scalar test function supported on 
direction, thus 
of 
is also not identically zero (thanks to the unique continuation property of harmonic functions). We can normalise so that 
and the applying an affine transformation to map onto 
denote the function 
ensuring that this function is extremely well balanced (for instance, it will have mean zero, and thus “high frequency” in the sense of Remark 
is divergence free, and one also has the steady-state Euler equation 
. If we then set 
is a simplified fast-slow solution supported in 
, which has measure 
, which for 
. Thus by Cauchy-Schwarz one has 
are not yet strong enough. We now apply a “divergence corrector” to make 
where 
is supported on 
and obeys the estimates 
as 
. Iterating the above identity 
is supported in 
is a simplified fast-slow solution supported in 
continues to be supported on the thin set 
also has mean zero. We can thus iterate and obtain 
is a smooth function whose exact form is explicit but irrelevant for our argument. We then see that 
is a simplified fast-slow solution supported in 
. Since 
is bounded in 
, we see from 
, let 
be the Banach space of periodic functions 
which are divergence free, and of mean zero at every time. For 
solution 
that solves the equation 
maps test functions to bounded functions.)
solution 
, but is not identically zero. In particular, periodic weak 
be an element of 
that is supported on ![{[0.4, 0.6] \times \mathbf{T}_E}](https://s0.wp.com/latex.php?latex=%7B%5B0.4%2C+0.6%5D+%5Ctimes+%5Cmathbf%7BT%7D_E%7D&bg=ffffff&fg=000000&s=0 "{[0.4, 0.6] \times \mathbf{T}_E}")
and is not identically zero (it is easy to show that such an element exists). By Proposition 
also supported on 
be sufficiently large. By applying Proposition 
) and with a sufficiently rapidly increasing sequence 
, we can find a sequence 
of Navier-Stokes-Reynolds flows supported on (say) ![{[0.3 + 2^{-n}/100, 0.7 - 2^{-n}/100] \times \mathbf{T}_E}](https://s0.wp.com/latex.php?latex=%7B%5B0.3+%2B+2%5E%7B-n%7D%2F100%2C+0.7+-+2%5E%7B-n%7D%2F100%5D+%5Ctimes+%5Cmathbf%7BT%7D_E%7D&bg=ffffff&fg=000000&s=0 "{[0.3 + 2^{-n}/100, 0.7 - 2^{-n}/100] \times \mathbf{T}_E}")
obeying the bounds 
. For 
sufficiently rapidly growing, this implies that 
converges strongly in 
converges strongly in 
supported in ![{[0.3, 0.7] \times \mathbf{T}_E}](https://s0.wp.com/latex.php?latex=%7B%5B0.3%2C+0.7%5D+%5Ctimes+%5Cmathbf%7BT%7D_E%7D&bg=ffffff&fg=000000&s=0 "{[0.3, 0.7] \times \mathbf{T}_E}")
. From the triangle inequality we have 
is the vector field with components 
for 
, we conclude that 
, and assume that the dimension 
) and a higher dimensional result of Luo (taking 
, and restricting attention to time-independent solutions). In high dimensions one can create fairly regular functions which are large in 
in (variants of) the Navier-Stokes equation, as well as interaction terms between low and high frequencies. As such, the analysis here is simpler than that required to establish the Onsager conjecture. The construction used to prove this theorem shows in fact that periodic weak 
for a small 
, assumed to be sufficiently large depending on 
; and 
.
. To simplify the notation we allow all implied constants to depend on 
unless otherwise specified. We recall from the previous section the notion of a Navier-Stokes-Reynolds flow 
for the spatial gradient to distinguish it from the time derivative 
.
oscillating at spatial scales up to some small wavelength 
, and modifies it to a Navier-Stokes-Reynolds flow 
oscillating at a slightly smaller wavelength 
, with a smaller Reynolds stress. It can be viewed as a more quantitative version of Proposition 
. Suppose that one has a Navier-Stokes-Reynolds flow 
obeying the estimates 
. Set 
. Then there exists a Navier-Stokes-Reynolds flow 
obeying the estimates 
is supported in 
-neighbourhood of 
be chosen to be supported on (say) ![{[0.4,0.6] \times \mathbf{T}_E}](https://s0.wp.com/latex.php?latex=%7B%5B0.4%2C0.6%5D+%5Ctimes+%5Cmathbf%7BT%7D_E%7D&bg=ffffff&fg=000000&s=0 "{[0.4,0.6] \times \mathbf{T}_E}")
and not be identically zero. By Proposition 
supported on 
be a sufficiently large parameter, and set 
, then the hypotheses 
for all 
. By iteratively applying Theorem 
of Navier-Stokes-Reynolds flows, all supported on (say) 
. In particular, the 
converge weakly to zero on 
converges strongly in 
(and in particular also in 
for some 
) to some limit 
; as the 
. Next, to cancel the Reynolds stress 
up to a smaller error 
, we will perturb 
), so that 
(minus a pressure term) after averaging at spatial scales 
. Given the size bound 
. By exploiting the small gap between 
and 
), which in high dimension allows us to make linear terms such as 
and 
(as well as the correlation terms 
and 
) negligible in size (as measured using 
(cf. the proof of Proposition 
will then oscillate at frequency 
in 
or below). Using the ellipticity of the Laplacian, we can then express this defect as 
where the 
is of order 
, this is slightly less than 
, allowing one to close the argument.
obeying the estimates 
is supported in 
, where 
is the 
-neighbourhood of 
be a bump function (depending only on 
of total mass 
on smooth functions 
by the formula 
will behave like low and high frequency Littlewood-Paley projections. (We cannot directly use these projections here because their convolution kernels are not localised in time.)
are convolution operators and thus commute with each other and with the partial derivatives 
. If we apply the operator 
to 
, 
is a linear combination of the operators 
. In particular, we see that 
. 
. For any 
, thanks to Young’s inequality. A similar application of Young’s inequality gives 
as linear combinations of 
, we have 
and noting that 
. For 
) this is bounded by 
. For 
, we rewrite the expression 
) by 
. Let 
be the Navier-Stokes-Reynolds flow constructed by Proposition 
at 
to the difference equation at 
denote the tuple 
, in the slow variable 
, and in the fast variable 
.
rather than 
and 
, and also such that 
is “ high frequency” (mean zero in 
is positive definite. In the previous section we essentially took 
, but now we will need good control on the derivatives of 
, where 
, 
is not identically zero, since otherwise the claim is trivial. For brevity we write 
. From 
, and let 
denote the maximal function 
, so we apply a smoothing operator. Let 
denote the function 
, then by Fubini’s theorem (or Young’s inequality) 
is smooth and strictly positive everywhere, and from differentation under the integral sign and integration by parts we have 
. From many applications of the chain rule (or the 
, 
is a linear combination of 
terms of the form 
sum up to 
is a linear combination of expressions of the above form in which one works with individual components of each factor 
rather than the full tuple 
, then by the product rule 
is not supported in 
for a sufficiently large constant 
depending only on 
, then 
takes values in a compact subset of positive definite 
for some 
, we thus have 
be the disjoint tubes in 
rather than 
has measure (say) 
. Thus by Cauchy-Schwarz (in just the 
supported in 
in the third estimate, with the first factor 
coming from essentially inverting a “fast” derivative 
.
supported in 
coming from applying a slow derivative and inverting a fast one.
, we see (for 
(although all of the arguments here also apply without much modification to 
as well). In this section we establish an analogue of Corollary 
in the difference Euler-Reynolds equations is no longer negligible, and needs to be treated by working with an ansatz in Lagrangian coordinates (or equivalently, an ansatz transported by the flow). (This use of Lagrangian coordinates is implicit in 
. Let 
are smooth functions obeying the bounds 
is mostly “high frequency” in nature, in that its low frequency components are small. Together with 
norm. Unlike the previous estimates, it will be important for our arguments that 
topology – but one which is still sufficent to ensure that the limiting solution constructed is not identically zero.) 
, and for which we have a decomposition 
obeying 
obeying 
has to obey the system of equations 
is the material Lie derivative of 
in 
. We will therefore invoke Lagrangian coordinates in order to convert the material Lie derivative 
into the more tractable time derivative 
that is an isomorphic copy of the Eulerian torus 
; as in the previous section, we paramterise this torus by 
, and adopt the usual summation conventions for the indices 
. Let 
be a trajectory map for 
, the map 
is a diffeomorphism and one obeys the ODE 
. The existence of such a trajectory map is guaranteed by the Picard existence theorem (it is important here that 
on 
can be pulled back to a Lagrangian scalar field 
on 
by the formula
on 
is the inverse of the matrix 
, defined by 
on 
tensor 
on 
(also known as pushforward by 
for 
with the following property: for any smooth Eulerian rank 
, one has
can be given explicitly as 
, where 
is the 
of the Euclidean metric – but we will not need this precise formula. The right-hand side may also be written (in 
, where 
is the 
to the system 
denotes the rank 
supported on 
obeying the equations 
denotes the supremum on the Lagrangian spacetime 
depend on 
rather than the Eulerian torus 
(thus we think of 
as “low frequency” functions of time 
). Set 
to the above system supported on 
obeying the estimates 
.
to 
now cancels off 
. Construct a smooth vector field 
not depending on 
have mean zero in 
, we conclude that the matrix with entries 
; taking traces one concludes 
, and obtains a new Euler-Reynolds flow 
that we omit here. In contrast to previous arguments, it is useful for technical reasons to not impose time regularity in the estimates. Once this claim is formalised and proved, conclusions such as Onsager’s conjecture follow from the usual iteration arguments.
appearing in 
for 
. Suppose for the time being that we could magically replace the material Lie derivative 
, oscillating at frequency 
is high frequency (mean zero in the fast variable) and has amplitude about 
. We can also arrange matters (using something like 
vanishes, leaving only a slower derivative of size about 
and oscillating at frequency about 
. Such a term would be acceptable (smaller than 
) for 
, which (in contrast to the high-dimensional Sobolev scale setting) cannot be ignored in this low-dimensional Hölder scale problem. The natural time scale of oscillation here is 
, coming from the usual heuristics concerning the Euler equation (see Remark 11 from 
and 
, these expressions would be expected to have amplitude 
. They still oscillate at the high frequency 
. This however remains acceptable for 
, then it turns out that one only gets good control on the flow map and its derivatives for times 
within the natural time scale 
of that initial time 
of length about 
of the velocity perturbation 
will overlap, and their corresponding local corrections 
will also overlap. This leads to some highly undesirable interaction terms between 
(such as the fast derivative of 
) which are very difficult to make small (for instance, one cannot simply ensure that 
in which the new Reynolds stress 
-separated sequence of times 
, and for each such time 
at 
on a time interval centred at 
, that agree with 
in time. This gives fields 
that solve the Euler equation near each time 
for the intervals at which the Euler equation is not solved exactly. Isett’s original construction of this stress was rather intricate; see Sections 7-10 of 
th row consists of all 1’s. If you look at the triangle consisting of the first 
you get a fractal called the 
are prime. But Euler crushed this dream by showing that the next Fermat number, 
is not prime.
They’ve even checked a few bigger ones, the largest being
options, if we include the regular 1-gon.
of real 
matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the 
of two generators 
and their inverses (i.e., an element of the 
), one can define the word map 
simply by substituting a pair of matrices in 
, then the corresponding word map 
. Because 
of non-trivial word maps 
that converge uniformly to the identity map?
, does there exist a non-trivial word 
for all 
denotes (say) the operator norm, and 
denotes the identity matrix in 
(or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” 
(or “labels space”) that we will introduce shortly (but the reader is free to also ignore the 
or 
subscripts if he or she wishes). Elements of Eulerian space 
to denote Lebesgue measure on 
for the 
to index these coordinates (with the usual summation conventions), for instance 
denotes partial differentiation along the 
coordinate. (We use superscripts for coordinates 
to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)
is the velocity field and 
is the pressure field. These are functions of time 
and on the spatial location variable 
. We will refer to the coordinates 
as 
, which can be thought of as a single function 
from the coordinates 
(where 
and the velocity field was given by the informal relationship 
to the Euler equations on a spacetime slab 
in Eulerian coordinates; assume furthermore that the velocity field 
to denote Lebesgue measure on 
to refer to the 
denotes partial differentiation along the 
coordinate. We will use summation conventions for both the Eulerian coordinates 
that obeys the ODE 
assigned to each label 
, so that we label each particle by its initial location at time 
is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map 
is also a trajectory map, albeit one with different initial conditions 
is an important feature of Eulerian space 
is a smooth map, show that there exists a unique smooth trajectory map 
for all 
is a diffeomorphism and 
is also a diffeomorphism.
. 
(oriented) volume preserving if one has the equation 
is the 
for 
and 
are the components of 
. (If one wishes, one can also view 
of Lagrangian space at 
of Eulerian space at 
.) Equivalently, 
, which is in turn equivalent to 
having the same Lebesgue measure as 
. 
then can be nicely expressed in terms of volume-preserving properties of the trajectory maps 
is volume-preserving for all 
we have the conservation law 
, defined by setting 
to be the inverse of the diffeomorphism 
. By the implicit function theorem, 
is the usual 
. In particular, for any 
, we can calculate 
, thus 
is volume-preserving, and hence 
be a continuously differentiable map from the time interval 
to the general linear group 
of invertible 
of Eulerian spacetime, one has 
of Eulerian spacetime is equivalent to the ODE 
are the smooth functions of Lagrangian spacetime defined by 
appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here. 
where 
, and similarly a subset 
of physical space can be pulled back to a subset 
of label space. A scalar field 
can be pulled back to a scalar field 
, defined by pre-composition: 
, and 
then 
if and only if 
. 
if and only if 
. 
is an isomorphism of 
-algebras. 
is an algebra isomorphism.
on 
for 
which is totally antisymmetric with respect to permutations of the indices 
, thus if one interchanges 
and 
for any 
, then 
flips to 
. Thus for instance 
; 
of 
of 
scalar functions with 
(so in particular 
); 
-form, when viewed in coordinates, is a collection 
of 
scalar functions with 
, 
, and 
.
, then the only 
, defined by setting 
for any permutation 
(with 
being the 
equal to 
when 
, 
when 
, and 
to denote the space of 
, we can define the product 
by pointwise multiplication of components: 
, we define the 
to be the 
-form given by the formula 
is the symmetric group of permutations on 
. For instance, for a scalar field 
), 
. Similarly, if 
and 
, we have the pointwise identities 
to 
that obeys the 
. (In other words, the space of formal linear combinations of forms, 
in coordinates as 
-form. Thus for instance: 
. 
is a continuously differentiable 
. 
.
. In particular, the exterior derivatives 
, 
is continuously differentiable, then 
is given in coordinates as 
by the formula 
is given by the same formula 
as before. 
is given by the formula 
. 
is given by the formula 
is a linear map from 
. It also preserves the exterior algebra and exterior derivative:
. Show that 
has a parameterisation 
, where 
and 
is an injective immersion. Then any continuous compactly supported 
into parameterised manifolds, and define the total integral to be the sum of the components; again, one can show (after some tedious calculation) that this is independent of the choice of parameterisation. If 
, then we have the identity 
, and 
here can often be relaxed by the usual limiting arguments; for the purposes of this set of notes, we shall proceed formally and assume that identities such as 
on 
, 
instead of subscripts 
to denote the components. On the other hand, we will transform vector fields under pullback in a different manner from 
, defined by setting 
to equal 
and 
of a 
. One can of course define vector fields on 
of 
such that 
). Equivalently, if 
is the inverse matrix to the total differential 
), so that 
.
are vector fields, one can form the scalar field 
by the formula 
are vector fields, then 
is a totally antisymmetric form in the 
. If 
and 
is a vector field, we define the 
in coordinates by the formula 
is the 
is just the directional derivative of 
is also denoted 
in the literature. If one contracts a vector field 
-form which we will call (by slight abuse of notation) the 
of 
: 
-forms for any 
, but we will not do so here as we will not have much use for the notion of a 
.
such that for each point 
, which is a collection of scalar functions 
such that 
is similarly equal to 
by 
. 
by the formula 
is a diffeomorphism, show that 
. 
is a vector field, the associated 
(also denoted 
or simply 
) is defined in coordinates as 
(also denoted 
or 
) is defined in coordinates as 
and 
. Note that 
can still be defined if 
can be written as 
can be written in differential geometry notation as 
of a rank 
tensor field 
for 
) that generalises the pullback of differential forms, vector fields, and Riemannian metrics. Argue why your definition is the natural one. 
is a continuously differentiable 
of 
vanishes if 
is just the directional derivative of 
. 
is the 
is the 
can be viewed as a smooth vector field on 
is a smooth 
be the trajectory map associated to a time-independent vector field 
and 
, then the above exercise shows that 
for any differential form 
(and has the advantage of readily extending to other tensors than differential forms, for which the Cartan formula is not available). 
also be continuously differentiable. Establish the Leibniz rule 
is the divergence of 
. 
of a continuously differentiable vector field 
is defined in coordinates as 
of a continuously differentiable rank 
are continuously differentiable, establish the Leibniz rule
, establish the variant Leibniz rule 
are continuously differentiable, establish the Leibniz rule
or a rank 
is a continuously differentiable differential form, vector field, or metric tensor. 
are smooth, define the Lie bracket ![{[Z,W] \in \Gamma(T {\bf R}^d_E)}](https://s0.wp.com/latex.php?latex=%7B%5BZ%2CW%5D+%5Cin+%5CGamma%28T+%7B%5Cbf+R%7D%5Ed_E%29%7D&bg=ffffff&fg=000000&s=0 "{[Z,W] \in \Gamma(T {\bf R}^d_E)}")
by the formula ![\displaystyle [Z,W] := {\mathcal L}_Z W.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BZ%2CW%5D+%3A%3D+%7B%5Cmathcal+L%7D_Z+W.&bg=ffffff&fg=000000&s=0 "\displaystyle [Z,W] := {\mathcal L}_Z W.")
![{[Z,W] = -[W,Z]}](https://s0.wp.com/latex.php?latex=%7B%5BZ%2CW%5D+%3D+-%5BW%2CZ%5D%7D&bg=ffffff&fg=000000&s=0 "{[Z,W] = -[W,Z]}")
(so in particular ![{[Z,Z]=0}](https://s0.wp.com/latex.php?latex=%7B%5BZ%2CZ%5D%3D0%7D&bg=ffffff&fg=000000&s=0 "{[Z,Z]=0}")
) and the ![\displaystyle [[Z_1,Z_2],Z_3] + [[Z_2,Z_3],Z_1] + [[Z_3,Z_1],Z_2] = 0,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5B%5BZ_1%2CZ_2%5D%2CZ_3%5D+%2B+%5B%5BZ_2%2CZ_3%5D%2CZ_1%5D+%2B+%5B%5BZ_3%2CZ_1%5D%2CZ_2%5D+%3D+0%2C&bg=ffffff&fg=000000&s=0 "\displaystyle [[Z_1,Z_2],Z_3] + [[Z_2,Z_3],Z_1] + [[Z_3,Z_1],Z_2] = 0,")
![\displaystyle {\mathcal L}_{[Z,W]} \phi = {\mathcal L}_Z {\mathcal L}_W - {\mathcal L}_W {\mathcal L}_Z](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cmathcal+L%7D_%7B%5BZ%2CW%5D%7D+%5Cphi+%3D+%7B%5Cmathcal+L%7D_Z+%7B%5Cmathcal+L%7D_W+-+%7B%5Cmathcal+L%7D_W+%7B%5Cmathcal+L%7D_Z&bg=ffffff&fg=000000&s=0 "\displaystyle {\mathcal L}_{[Z,W]} \phi = {\mathcal L}_Z {\mathcal L}_W - {\mathcal L}_W {\mathcal L}_Z")
are smooth, and 
of a (continuously differentiable) rank 
, which can be viewed as an element of 
, and a time-dependent velocity field 
) 
, formed by applying the Euclidean musical isomorphism to 
, thus 
for 
of 
, we can express the right-hand side as a total derivative 
, where 
is the modified pressure 
is another modification of the pressure: 
, thus pulling back under the time-independent map 
defined by 
, but we ignore this issue for now by only working away from the 
axis 
pulls back under this change of coordinates 
as 
is diagonal in 
coordinates with entries 
similarly pulls back to the familiar cylindrical coordinate volume form 
as 
, and the components of 
as 
, then the second equation 
to the Euler equations 
, and transforming to cylindrical coordinates 
, establish the chain rule formulae 
are a variant of cylindrical coordinates 
, defined by the formulae 
to Turkington coordinates 
is volume preserving. Show that in these coordinates, the Euler equations become 
do not depend on the 
variable, so that all the terms involving 
vanish; one can specialise further to the case of solutions that are “axisymmetric without swirl”, in which case 
also vanishes.) 
is symmetric in 
) we have 
; from Exercise 
, and hence by the Leibniz rule (Exercise 
as a total derivative: 
.
is a total derivative 
, and hence this contribution to the integral formally vanishes as 
can be written using the Leibniz rule as the difference of the total derivative 
and the quantity 
. The former quantity also gives no contribution to the integral as 
. Since 
is symmetric in 
as a total derivative 
. If we set the Killing vector field 
) (which one can easily verify to be Killing using 
. Unfortunately, this essentially exhausts the supply of Killing vector fields:
is defined as the exterior derivative of the covelocity: 
of the velocity 
, which upon integration on 
using Exercise 
norms of 
, and hence also for 
by a limiting argument, recovering Proposition 24 from 
of the covelocity and the vorticity is a 
. The 
. From 
, hence this expression is also a total derivative 
. From Stokes’ theorem 
; this was first observed 
(formed by taking the exterior product of 
copies of 
is conserved by the flow. (This observation is 
be a smooth solution to the Euler equations in three dimensions 
be an arbitrary smooth scalar field. Establish Ertel’s theorem 
takes the form 
. 
are smooth scalar fields obeying the transport equations 
initially 
was observed 
are annihilated by this vector field: 
lie in the joint level sets of 
and 
(and indeed, if 
are transverse to each other, then the vortex lines are locally the intersection of the two level sets, away from critical points at least).
, a Lagrangian covelocity field 
, a Lagrangian modified pressure field 
, and a Lagrangian vorticity field 
by the formulae 
is a pointwise conserved quantity: 
in terms of the trajectory map. Indeed, from 
of the linear transformation 
, we thus obtain the Cauchy vorticity formula 
, and we thus have 
.
and using Stokes’ theorem 
is known as the circulation of the fluid along the loop; the Kelvin circulation theorem then asserts that this circulation remains constant over time as long as the loop evolves along the flow.
in terms of the Euclidean metric 
is defined as 
of the Lagrangian vorticity, thus the conservation of the Cauchy invariants is equivalent to the Cauchy vorticity formula.
of vorticity is a vector field. A smooth curve 
at that point. Suppose that the trajectory map is normalised using 
is a vortex line at any other time 
in which, at every point 
, the vorticity vector field 
is tangent to 
is a vortex tube at time 
is a vortex tube at time 
on the vortex tube is formally conserved in time. 
, show that the local helicity 
for an arbitrary covelocity 
equal to the coordinate functions 
.)
of differentiation along the (Hodge dual of the) vorticity. 
are the partial derivatives raised by the Euclidean metric. For 
, we can use the fundamental solution 
of the Laplacian (see 
; see Chapter 4 of 
are real-analytic in 
and 
be smooth functions. 
defined by
