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I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations

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

$\displaystyle \nabla \cdot u = 0$

that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time ${T_*}$, then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:

• (Leray blowup criterion, 1934) If ${u}$ blows up at a finite time ${T_*}$, and ${3 < p \leq \infty}$, then ${\liminf_{t \rightarrow T_*} (T_* - t)^{\frac{1}{2}-\frac{3}{2p}} \| u(t) \|_{L^p_x({\bf R}^3)} \geq c}$ for an absolute constant ${c>0}$.
• (ProdiSerrinLadyzhenskaya blowup criterion, 1959-1967) If ${u}$ blows up at a finite time ${T_*}$, and ${3 < p \leq \infty}$, then ${\| u \|_{L^q_t L^p_x([0,T_*) \times {\bf R}^3)} =+\infty}$, where ${\frac{1}{q} := \frac{1}{2} - \frac{3}{2p}}$.
• (Beale-Kato-Majda blowup criterion, 1984) If ${u}$ blows up at a finite time ${T_*}$, then ${\| \omega \|_{L^1_t L^\infty_x([0,T_*) \times {\bf R}^3)} = +\infty}$, where ${\omega := \nabla \times u}$ is the vorticity.
• (Kato blowup criterion, 1984) If ${u}$ blows up at a finite time ${T_*}$, then ${\liminf_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} \geq c}$ for some abs}{olute constant ${c>0}$.
• (Escauriaza-Seregin-Sverak blowup criterion, 2003) If ${u}$ blows up at a finite time ${T_*}$, then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} = +\infty}$.
• (Seregin blowup criterion, 2012) If ${u}$ blows up at a finite time ${T_*}$, then ${\lim_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} = +\infty}$.
• (Phuc blowup criterion, 2015) If ${u}$ blows up at a finite time ${T_*}$, then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{L^{3,q}_x({\bf R}^3)} = +\infty}$ for any ${q < \infty}$.
• (Gallagher-Koch-Planchon blowup criterion, 2016) If ${u}$ blows up at a finite time ${T_*}$, then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{\dot B_{p,q}^{-1+3/p}({\bf R}^3)} = +\infty}$ for any ${3 < p, q < \infty}$.
• (Albritton blowup criterion, 2016) If ${u}$ blows up at a finite time ${T_*}$, then ${\lim_{t \rightarrow T_*} \|u(t) \|_{\dot B_{p,q}^{-1+3/p}({\bf R}^3)} = +\infty}$ for any ${3 < p, q < \infty}$.

My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely ${L^3_x({\bf R}^3)}$, had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by Gallagher-Koch-Planchon (building on ideas of Kenig-Koch), and by Dong-Du. However, all of these proofs rely on some form of a compactness argument: given a finite time blowup, one extracts some suitable family of rescaled solutions that converges in some weak sense to a limiting solution that has some additional good properties (such as almost periodicity modulo symmetries), which one can then rule out using additional qualitative tools, such as unique continuation and backwards uniqueness theorems for parabolic heat equations. In particular, all known proofs use some version of the backwards uniqueness theorem of Escauriaza, Seregin, and Sverak. Because of this reliance on compactness, the existing proofs of the Escauriaza-Seregin-Sverak blowup criterion are qualitative, in that they do not provide any quantitative information on how fast the ${\|u(t)\|_{L^3_x({\bf R}^3)}}$ norm will go to infinity (along a subsequence of times).

On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:

Theorem 1 If ${u}$ is a classical solution to Navier-Stokes on ${[0,T) \times {\bf R}^3}$ with

$\displaystyle \| u \|_{L^\infty_t L^3_x([0,T) \times {\bf R}^3)} \leq A \ \ \ \ \ (1)$

for some ${A \geq 2}$, then

$\displaystyle | \nabla^j u(t,x)| \leq \exp\exp\exp(A^{O(1)}) t^{-\frac{j+1}{2}}$

and

$\displaystyle | \nabla^j \omega(t,x)| \leq \exp\exp\exp(A^{O(1)}) t^{-\frac{j+2}{2}}$

for ${(t,x) \in [0,T) \times {\bf R}^3}$ and ${j=0,1}$.

As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to

$\displaystyle \limsup_{t \rightarrow T_*} \frac{\|u(t) \|_{L^3_x({\bf R}^3)}}{(\log\log\log \frac{1}{T_*-t})^c} = +\infty$

for some absolute constant ${c>0}$, which to my knowledge is the first (very slightly) supercritical blowup criterion for Navier-Stokes in the literature.

The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points ${(t_0,x_0)}$ where there is a frequency ${N_0}$ for which one has the bound

$\displaystyle |N_0^{-1} P_{N_0} u(t_0,x_0)| \geq A^{-C_0} \ \ \ \ \ (2)$

for a large absolute constant ${C_0}$, where ${P_{N_0}}$ is a Littlewood-Paley projection to frequencies ${\sim N_0}$. (This can be compared with the upper bound of ${O(A)}$ for the quantity on the left-hand side that follows from (1).) The factor of ${N_0^{-1}}$ normalises the left-hand side of (2) to be dimensionless (i.e., critical). The main task is to show that the dimensionless quantity ${t_0 N_0^2}$ cannot get too large; in particular, we end up establishing a bound of the form

$\displaystyle t_0 N_0^2 \lesssim \exp\exp\exp A^{O(C_0^6)}$

from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.

The strategy is to show that any concentration such as (2) when ${t_0 N_0^2}$ is large must force a significant component of the ${L^3_x}$ norm of ${u(t_0)}$ to also show up at many other locations than ${x_0}$, which eventually contradicts (1) if one can produce enough such regions of non-trivial ${L^3_x}$ norm. (This can be viewed as a quantitative variant of the “rigidity” theorems in some of the previous proofs of the Escauriaza-Seregin-Sverak theorem that rule out solutions that exhibit too much “compactness” or “almost periodicity” in the ${L^3_x}$ topology.) The chain of causality that leads from a concentration (2) at ${(t_0,x_0)}$ to significant ${L^3_x}$ norm at other regions of the time slice ${t_0 \times {\bf R}^3}$ is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):

1. Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with ${t_0 N_0^2}$ large) if there was also a preceding concentration

$\displaystyle |N_1^{-1} P_{N_1} u(t_1,x_1)| \geq A^{-C_0} \ \ \ \ \ (3)$

at some slightly previous point ${(t_1,x_1)}$ in spacetime, with ${N_1}$ also close to ${N_0}$ (more precisely, we have ${t_1 = t_0 - A^{-O(C_0^3)} N_0^{-2}}$, ${N_1 = A^{O(C_0^2)} N_0}$, and ${x_1 = x_0 + O( A^{O(C_0^4)} N_0^{-1})}$). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound ${A^{-C_0}}$ in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.

2. Iterating the previous step, one can find a sequence of concentration points

$\displaystyle |N_n^{-1} P_{N_n} u(t_n,x_n)| \geq A^{-C_0} \ \ \ \ \ (4)$

with the ${(t_n,x_n)}$ propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the ${t_0-t_n}$ increase geometrically with time, the ${N_n}$ are comparable (up to polynomial factors in ${A}$) to the natural frequency scale ${(t_0-t_n)^{-1/2}}$, and one has ${x_n = x_0 + O( |t_0-t_n|^{1/2} )}$. Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the ${t_n}$ slightly, one can also place the times ${t_n}$ in intervals ${I_n}$ (of length comparable to a small multiple of ${|t_0-t_n|}$) in which the solution is quite regular (in particular, ${u, \nabla u, \omega, \nabla \omega}$ enjoy good ${L^\infty_t L^\infty_x}$ bounds on ${I_n \times {\bf R}^3}$).

3. The concentration (4) can be used to establish a lower bound for the ${L^2_t L^2_x}$ norm of the vorticity ${\omega}$ near ${(t_n,x_n)}$. As is well known, the vorticity obeys the vorticity equation

$\displaystyle \partial_t \omega = \Delta \omega - (u \cdot \nabla) \omega + (\omega \cdot \nabla) u.$

In the epoch of regularity ${I_n \times {\bf R}^3}$, the coefficients ${u, \nabla u}$ of this equation obey good ${L^\infty_x}$ bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower ${L^2}$ bounds on the vorticity (and its first derivative) in annuli of the form ${\{ (t,x) \in I_n \times {\bf R}^3: R \leq |x-x_n| \leq R' \}}$ for various radii ${R,R'}$, although the lower bounds decay at a gaussian rate with ${R}$.

4. Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli ${\{ x \in {\bf R}^3: R \leq |x-x_n| \leq R'\}}$ where (a slightly normalised form of) the entrosphy is small at time ${t=t_n}$; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form ${\{ (t,x) \in [t_n,t_0] \times {\bf R}^3: R_n \leq |x-x_n| \leq R'_n\}}$ in which one has good estimates; in particular, one has ${L^\infty_x}$ type bounds on ${u, \nabla u, \omega, \nabla \omega}$ in this cylindrical annulus.
5. By intersecting the previous epoch of regularity ${I_n \times {\bf R}^3}$ with the above annulus of regularity, we have some lower bounds on the ${L^2}$ norm of the vorticity (and its first derivative) in the annulus of regularity. Using a Carleman estimate first introduced by theorem of Escauriaza, Seregin, and Sverak, as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time ${t=t_0}$, establishing a lower bound for the vorticity on the spatial annulus ${\{ (t_0,x): R_n \leq |x-x_n| \leq R'_n \}}$. By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the ${L^3}$ norm of the velocity ${u}$; crucially, this lower bound is uniform in ${n}$.
6. If ${t_0 N_0^2}$ is very large (triple exponential in ${A}$!), one can then find enough scales ${n}$ with disjoint ${\{ (t_0,x): R_n \leq |x-x_n| \leq R'_n \}}$ annuli that the total lower bound on the ${L^3_x}$ norm of ${u(t_0)}$ provided by the above arguments is inconsistent with (1), thus establishing the claim.

The chain of causality is summarised in the following image:

It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the ${L^3}$ upper bound costs the third.

Let ${\Omega}$ be some domain (such as the real numbers). For any natural number ${p}$, let ${L(\Omega^p)_{sym}}$ denote the space of symmetric real-valued functions ${F^{(p)}: \Omega^p \rightarrow {\bf R}}$ on ${p}$ variables ${x_1,\dots,x_p \in \Omega}$, thus

$\displaystyle F^{(p)}(x_{\sigma(1)},\dots,x_{\sigma(p)}) = F^{(p)}(x_1,\dots,x_p)$

for any permutation ${\sigma: \{1,\dots,p\} \rightarrow \{1,\dots,p\}}$. For instance, for any natural numbers ${k,p}$, the elementary symmetric polynomials

$\displaystyle e_k^{(p)}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} x_{i_1} \dots x_{i_k}$

will be an element of ${L({\bf R}^p)_{sym}}$. With the pointwise product operation, ${L(\Omega^p)_{sym}}$ becomes a commutative real algebra. We include the case ${p=0}$, in which case ${L(\Omega^0)_{sym}}$ consists solely of the real constants.

Given two natural numbers ${k,p}$, one can “lift” a symmetric function ${F^{(k)} \in L(\Omega^k)_{sym}}$ of ${k}$ variables to a symmetric function ${[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}$ of ${p}$ variables by the formula

$\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})$

$\displaystyle = \frac{1}{k!} \sum_\pi F^{(k)}( x_{\pi(1)}, \dots, x_{\pi(k)} )$

where ${\pi}$ ranges over all injections from ${\{1,\dots,k\}}$ to ${\{1,\dots,p\}}$ (the latter formula making it clearer that ${[F^{(k)}]_{k \rightarrow p}}$ is symmetric). Thus for instance

$\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)$

$\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)$

and

$\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.$

Also we have

$\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.$

With these conventions, we see that ${[F^{(k)}]_{k \rightarrow p}}$ vanishes for ${p=0,\dots,k-1}$, and is equal to ${F}$ if ${k=p}$. We also have the transitivity

$\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}$

if ${k \leq l \leq p}$.

The lifting map ${[]_{k \rightarrow p}}$ is a linear map from ${L(\Omega^k)_{sym}}$ to ${L(\Omega^p)_{sym}}$, but it is not a ring homomorphism. For instance, when ${\Omega={\bf R}}$, one has

$\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)$

$\displaystyle = \sum_{i=1}^p x_i^2 + 2 \sum_{1 \leq i < j \leq p} x_i x_j$

$\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}$

$\displaystyle \neq [x_1^2]_{1 \rightarrow p}.$

In general, one has the identity

$\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)$

$\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}$

for all natural numbers ${k,l,p}$ and ${F^{(k)} \in L(\Omega^k)_{sym}}$, ${G^{(l)} \in L(\Omega^l)_{sym}}$, where ${\pi, \rho}$ range over all injections ${\pi: \{1,\dots,k\} \rightarrow \{1,\dots,m\}}$, ${\rho: \{1,\dots,l\} \rightarrow \{1,\dots,m\}}$ with ${\pi(\{1,\dots,k\}) \cup \rho(\{1,\dots,l\}) = \{1,\dots,m\}}$. Combinatorially, the identity (2) follows from the fact that given any injections ${\tilde \pi: \{1,\dots,k\} \rightarrow \{1,\dots,p\}}$ and ${\tilde \rho: \{1,\dots,l\} \rightarrow \{1,\dots,p\}}$ with total image ${\tilde \pi(\{1,\dots,k\}) \cup \tilde \rho(\{1,\dots,l\})}$ of cardinality ${m}$, one has ${k,l \leq m \leq k+l}$, and furthermore there exist precisely ${m!}$ triples ${(\pi, \rho, \sigma)}$ of injections ${\pi: \{1,\dots,k\} \rightarrow \{1,\dots,m\}}$, ${\rho: \{1,\dots,l\} \rightarrow \{1,\dots,m\}}$, ${\sigma: \{1,\dots,m\} \rightarrow \{1,\dots,p\}}$ such that ${\tilde \pi = \sigma \circ \pi}$ and ${\tilde \rho = \sigma \circ \rho}$.

Example 1 When ${\Omega = {\bf R}}$, one has

$\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}$

$\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}$

which is just a restatement of the identity

$\displaystyle (\sum_{i < j} x_i x_j) (\sum_k x_k) = \sum_{i

Note that the coefficients appearing in (2) do not depend on the final number of variables ${p}$. We may therefore abstract the role of ${p}$ from the law (2) by introducing the real algebra ${L(\Omega^*)_{sym}}$ of formal sums

$\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}$

where for each ${k}$, ${F^{(k)}}$ is an element of ${L(\Omega^k)_{sym}}$ (with only finitely many of the ${F^{(k)}}$ being non-zero), and with the formal symbol ${[]_{k \rightarrow *}}$ being formally linear, thus

$\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}$

and

$\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}$

for ${F^{(k)}, G^{(k)} \in L(\Omega^k)_{sym}}$ and scalars ${c \in {\bf R}}$, and with multiplication given by the analogue

$\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)$

$\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}$

of (2). Thus for instance, in this algebra ${L(\Omega^*)_{sym}}$ we have

$\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}$

and

$\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.$

Informally, ${L(\Omega^*)_{sym}}$ is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that ${L(\Omega^*)_{sym}}$ is indeed a commutative real algebra, with a unit ${[1]_{0 \rightarrow *}}$. (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )

For natural numbers ${p}$, there is an obvious specialisation map ${[]_{* \rightarrow p}}$ from ${L(\Omega^*)_{sym}}$ to ${L(\Omega^p)_{sym}}$, defined by the formula

$\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.$

Thus, for instance, ${[]_{* \rightarrow p}}$ maps ${[x_1]_{1 \rightarrow *}}$ to ${[x_1]_{1 \rightarrow p}}$ and ${[x_1 x_2]_{2 \rightarrow *}}$ to ${[x_1 x_2]_{2 \rightarrow p}}$. From (2) and (3) we see that this map ${[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}$ is an algebra homomorphism, even though the maps ${[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}$ and ${[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}$ are not homomorphisms. By inspecting the ${p^{th}}$ component of ${L(\Omega^*)_{sym}}$ we see that the homomorphism ${[]_{* \rightarrow p}}$ is in fact surjective.

Now suppose that we have a measure ${\mu}$ on the space ${\Omega}$, which then induces a product measure ${\mu^p}$ on every product space ${\Omega^p}$. To avoid degeneracies we will assume that the integral ${\int_\Omega \mu}$ is strictly positive. Assuming suitable measurability and integrability hypotheses, a function ${F \in L(\Omega^p)_{sym}}$ can then be integrated against this product measure to produce a number

$\displaystyle \int_{\Omega^p} F\ d\mu^p.$

In the event that ${F}$ arises as a lift ${[F^{(k)}]_{k \rightarrow p}}$ of another function ${F^{(k)} \in L(\Omega^k)_{sym}}$, then from Fubini’s theorem we obtain the formula

$\displaystyle \int_{\Omega^p} F\ d\mu^p = \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}.$

Thus for instance, if ${\Omega={\bf R}}$,

$\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)$

and

$\displaystyle \int_{{\bf R}^p} [x_1 x_2]_{2 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)$

On summing, we see that if

$\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}$

is an element of the formal algebra ${L(\Omega^*)_{sym}}$, then

$\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)$

Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.

Now for a key observation: whereas the left-hand side of (6) only makes sense when ${p}$ is a natural number, the right-hand side is meaningful when ${p}$ takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient ${\binom{p}{k}}$ as a polynomial ${\frac{p(p-1) \dots (p-k+1)}{k!}}$ in ${p}$. As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space ${\Omega^p}$ for such values of ${p}$, and even to integrate such functions against product measures ${\mu^p}$, even if the fractional power ${\Omega^p}$ does not exist in the usual set-theoretic sense (and ${\mu^p}$ similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex ${p}$, we now define ${L(\Omega^p)_{sym}}$ to be the space of abstract objects

$\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}$

with ${F^{(*)} \in L(\Omega^*)_{sym}}$ and ${[]_{* \rightarrow p}}$ (and ${[]_{k \rightarrow p}}$ now interpreted as formal symbols, with the structure of a commutative real algebra inherited from ${L(\Omega^*)_{sym}}$, thus

$\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}$

$\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}$

$\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.$

In particular, the multiplication law (2) continues to hold for such values of ${p}$, thanks to (3). Given any measure ${\mu}$ on ${\Omega}$, we formally define a measure ${\mu^p}$ on ${\Omega^p}$ with regards to which we can integrate elements ${F^{(p)}}$ of ${L(\Omega^p)_{sym}}$ by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of ${p}$, even though the formal space ${{\bf R}^p}$ no longer makes sense as a set, and the formal measure ${\mu^p}$ no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)

Example 2 Suppose ${\mu}$ is a probability measure on ${\Omega}$, and ${X: \Omega \rightarrow {\bf R}}$ is a random variable; on any power ${\Omega^k}$, we let ${X_1,\dots,X_k: \Omega^k \rightarrow {\bf R}}$ be the usual independent copies of ${X}$ on ${\Omega^k}$, thus ${X_j(\omega_1,\dots,\omega_k) := X(\omega_j)}$ for ${(\omega_1,\dots,\omega_k) \in \Omega^k}$. Then for any real or complex ${p}$, the formal integral

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p$

can be evaluated by first using the identity

$\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}$

(cf. (1)) and then using (6) and the probability measure hypothesis ${\int_\Omega\ d\mu = 1}$ to conclude that

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2$

$\displaystyle = p (\int_\Omega X^2\ d\mu - (\int_\Omega X\ d\mu)^2) + p^2 (\int_\Omega X\ d\mu)^2$

or in probabilistic notation

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)$

For ${p}$ a natural number, this identity has the probabilistic interpretation

$\displaystyle \mathbf{E}( X_1 + \dots + X_p)^2 = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2 \ \ \ \ \ (8)$

whenever ${X_1,\dots,X_p}$ are jointly independent copies of ${X}$, which reflects the well known fact that the sum ${X_1 + \dots + X_p}$ has expectation ${p \mathbf{E} X}$ and variance ${p \mathbf{Var}(X)}$. One can thus view (7) as an abstract generalisation of (8) to the case when ${p}$ is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about ${p}$ independent copies ${X_1,\dots,X_p}$ of ${X}$ in the standard framework of probability theory.

In this particular case, the quantity (7) is non-negative for every nonnegative ${p}$, which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if ${X}$ has mean zero, one can check that

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2$

and the right-hand side can become negative for ${p < 2/3}$. This is a shame, because otherwise one could hope to start endowing ${L(X^p)_{sym}}$ with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the ${p^{th}}$ power, need not be a characteristic function of any random variable once ${p}$ is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if ${F: \Omega \rightarrow {\bf R}}$ is non-negative, then so is

$\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.$

One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents ${p}$ and appealing to a general extrapolation theorem to then obtain various identities in the fractional ${p}$ case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity

$\displaystyle (\int_\Omega\ d\mu)^p = \int_{\Omega^p}\ d\mu^p$

(where the right-hand side should be viewed as the fractional dimensional integral of the unit ${[1]_{0 \rightarrow p}}$ against ${\mu^p}$). As such, one can manipulate ${p^{th}}$ powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is

Lemma 3 (Differentiation formula) Suppose that a positive measure ${\mu = \mu(t)}$ on ${\Omega}$ depends on some parameter ${t}$ and varies by the formula

$\displaystyle \frac{d}{dt} \mu(t) = a(t) \mu(t) \ \ \ \ \ (9)$

for some function ${a(t): \Omega \rightarrow {\bf R}}$. Let ${p}$ be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have

$\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)$

for all ${F^{(p)} \in L(\Omega^p)_{sym}}$ that are independent of ${t}$. If we allow ${F^{(p)}(t)}$ to now depend on ${t}$ also, then we have the more general total derivative formula

$\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}(t)\ d\mu(t)^p \ \ \ \ \ (11)$

$\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,$

again assuming sufficient amounts of smoothness and regularity.

Proof: We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case ${F^{(p)} = [F^{(k)}]_{k \rightarrow p}}$ for some symmetric function ${F^{(k)} \in L(\Omega^k)_{sym}}$ for a natural number ${k}$. By (6), the left-hand side of (10) is then

$\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)$

Differentiating under the integral sign using (9) we have

$\displaystyle \frac{d}{dt} \int_\Omega\ d\mu(t) = \int_\Omega\ a(t)\ d\mu(t)$

and similarly

$\displaystyle \frac{d}{dt} \int_{\Omega^k} F^{(k)}\ d\mu(t)^k = \int_{\Omega^k} F^{(k)}(a_1+\dots+a_k)\ d\mu(t)^k$

where ${a_1,\dots,a_k}$ are the standard ${k}$ copies of ${a = a(t)}$ on ${\Omega^k}$:

$\displaystyle a_j(\omega_1,\dots,\omega_k) := a(\omega_j).$

By the product rule, we can thus expand (12) as

$\displaystyle \binom{p}{k} (\int_{\Omega^k} F^{(k)}(a_1+\dots+a_k)\ d\mu^k ) (\int_\Omega\ d\mu)^{p-k}$

$\displaystyle + \binom{p}{k} (p-k) (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ a\ d\mu) (\int_\Omega\ d\mu)^{p-k-1}$

where we have suppressed the dependence on ${t}$ for brevity. Since ${\binom{p}{k} (p-k) = \binom{p}{k+1} (k+1)}$, we can write this expression using (6) as

$\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p$

where ${F^{(k)} \ast a \in L(\Omega^{k+1})_{sym}}$ is the symmetric function

$\displaystyle F^{(k)} \ast a(\omega_1,\dots,\omega_{k+1}) := \sum_{j=1}^{k+1} F^{(k)}(\omega_1,\dots,\omega_{j-1},\omega_{j+1} \dots \omega_{k+1}) a(\omega_j).$

But from (2) one has

$\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}$

and the claim follows. $\Box$

Remark 4 It is also instructive to prove this lemma in the special case when ${p}$ is a natural number, in which case the fractional dimensional integral ${\int_{\Omega^p} F^{(p)}\ d\mu(t)^p}$ can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude that

$\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.$

One could in fact derive (10) for arbitrary real or complex ${p}$ from the case when ${p}$ is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.

Let us give a simple PDE application of this lemma as illustration:

Proposition 5 (Heat flow monotonicity) Let ${u: [0,+\infty) \times {\bf R}^d \rightarrow {\bf R}}$ be a solution to the heat equation ${u_t = \Delta u}$ with initial data ${\mu_0}$ a rapidly decreasing finite non-negative Radon measure, or more explicitly

$\displaystyle u(t,x) = \frac{1}{(4\pi t)^{d/2}} \int_{{\bf R}^d} e^{-|x-y|^2/4t}\ d\mu_0(y)$

for al ${t>0}$. Then for any ${p>0}$, the quantity

$\displaystyle Q_p(t) := t^{\frac{d}{2} (p-1)} \int_{{\bf R}^d} u(t,x)^p\ dx$

is monotone non-decreasing in ${t \in (0,+\infty)}$ for ${1 < p < \infty}$, constant for ${p=1}$, and monotone non-increasing for ${0 < p < 1}$.

Proof: By a limiting argument we may assume that ${d\mu_0}$ is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.

For any ${(t,x) \in (0,+\infty) \times {\bf R}^d}$, let ${\mu(t,x)}$ denote the Radon measure

$\displaystyle d\mu(t,x)(y) := \frac{1}{(4\pi)^{d/2}} e^{-|x-y|^2/4t}\ d\mu_0(y).$

Then the quantity ${Q_p(t)}$ can be written as a fractional dimensional integral

$\displaystyle Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx.$

Observe that

$\displaystyle \frac{\partial}{\partial t} d\mu(t,x) = \frac{|x-y|^2}{4t^2} d\mu(t,x)$

and thus by Lemma 3 and the product rule

$\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)$

where we use ${y}$ for the variable of integration in the factor space ${{\bf R}^d}$ of ${({\bf R}^d)^p}$.

To simplify this expression we will take advantage of integration by parts in the ${x}$ variable. Specifically, in any direction ${x_j}$, we have

$\displaystyle \frac{\partial}{\partial x_j} d\mu(t,x) = -\frac{x_j-y_j}{2t} d\mu(t,x)$

and hence by Lemma 3

$\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

Multiplying by ${x_j}$ and integrating by parts, we see that

$\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

$\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

where we use the Einstein summation convention in ${j}$. Similarly, if ${F_j(y)}$ is any reasonable function depending only on ${y}$, we have

$\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

$\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

and hence on integration by parts

$\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

We conclude that

$\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx$

and thus by (13)

$\displaystyle \frac{d}{dt} Q_p(t) = \frac{1}{4t^{\frac{d}{2}+2}} \int_{{\bf R}^d} \int_{({\bf R}^d)^p}$

$\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

The choice of ${F_j}$ that then achieves the most cancellation turns out to be ${F_j(y) = \frac{1}{p} y_j}$ (this cancels the terms that are linear or quadratic in the ${x_j}$), so that ${x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}$. Repeating the calculations establishing (7), one has

$\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}$

and

$\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p$

$\displaystyle = (p \mathbf{Var}(x-Y) + p^2 |\mathop{\bf E} x-Y|^2) (\int_{{\bf R}^d}\ d\mu)^{p}$

where ${Y}$ is the random variable drawn from ${{\bf R}^d}$ with the normalised probability measure ${\mu / \int_{{\bf R}^d}\ d\mu}$. Since ${\mathop{\bf E} |x-Y|^2 = \mathbf{Var}(x-Y) + |\mathop{\bf E} x-Y|^2}$, one thus has

$\displaystyle \frac{d}{dt} Q_p(t) = (p-1) \frac{1}{4t^{\frac{d}{2}+2}} \int_{{\bf R}^d} \mathbf{Var}(x-Y) (\int_{{\bf R}^d}\ d\mu)^{p}\ dx. \ \ \ \ \ (14)$

This expression is clearly non-negative for ${p>1}$, equal to zero for ${p=1}$, and positive for ${0 < p < 1}$, giving the claim. (One could simplify ${\mathbf{Var}(x-Y)}$ here as ${\mathbf{Var}(Y)}$ if desired, though it is not strictly necessary to do so for the proof.) $\Box$

Remark 6 As with Remark 4, one can also establish the identity (14) first for natural numbers ${p}$ by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of ${p}$. This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of ${p}$.

A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ${R}$); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number ${p}$ first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)

I was recently asked to contribute a short comment to Nature Reviews Physics, as part of a series of articles on fluid dynamics on the occasion of the 200th anniversary (this August) of the birthday of George Stokes.  My contribution is now online as “Searching for singularities in the Navier–Stokes equations“, where I discuss the global regularity problem for Navier-Stokes and my thoughts on how one could try to construct a solution that blows up in finite time via an approximately discretely self-similar “fluid computer”.  (The rest of the series does not currently seem to be available online, but I expect they will become so shortly.)

I was pleased to learn this week that the 2019 Abel Prize was awarded to Karen Uhlenbeck. Uhlenbeck laid much of the foundations of modern geometric PDE. One of the few papers I have in this area is in fact a joint paper with Gang Tian extending a famous singularity removal theorem of Uhlenbeck for four-dimensional Yang-Mills connections to higher dimensions. In both these papers, it is crucial to be able to construct “Coulomb gauges” for various connections, and there is a clever trick of Uhlenbeck for doing so, introduced in another important paper of hers, which is absolutely critical in my own paper with Tian. Nowadays it would be considered a standard technique, but it was definitely not so at the time that Uhlenbeck introduced it.

Suppose one has a smooth connection ${A}$ on a (closed) unit ball ${B(0,1)}$ in ${{\bf R}^n}$ for some ${n \geq 1}$, taking values in some Lie algebra ${{\mathfrak g}}$ associated to a compact Lie group ${G}$. This connection then has a curvature ${F(A)}$, defined in coordinates by the usual formula

$\displaystyle F(A)_{\alpha \beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha + [A_\alpha,A_\beta]. \ \ \ \ \ (1)$

It is natural to place the curvature in a scale-invariant space such as ${L^{n/2}(B(0,1))}$, and then the natural space for the connection would be the Sobolev space ${W^{n/2,1}(B(0,1))}$. It is easy to see from (1) and Sobolev embedding that if ${A}$ is bounded in ${W^{n/2,1}(B(0,1))}$, then ${F(A)}$ will be bounded in ${L^{n/2}(B(0,1))}$. One can then ask the converse question: if ${F(A)}$ is bounded in ${L^{n/2}(B(0,1))}$, is ${A}$ bounded in ${W^{n/2,1}(B(0,1))}$? This can be viewed as asking whether the curvature equation (1) enjoys “elliptic regularity”.

There is a basic obstruction provided by gauge invariance. For any smooth gauge ${U: B(0,1) \rightarrow G}$ taking values in the Lie group, one can gauge transform ${A}$ to

$\displaystyle A^U_\alpha := U^{-1} \partial_\alpha U + U^{-1} A_\alpha U$

and then a brief calculation shows that the curvature is conjugated to

$\displaystyle F(A^U)_{\alpha \beta} = U^{-1} F_{\alpha \beta} U.$

This gauge symmetry does not affect the ${L^{n/2}(B(0,1))}$ norm of the curvature tensor ${F(A)}$, but can make the connection ${A}$ extremely large in ${W^{n/2,1}(B(0,1))}$, since there is no control on how wildly ${U}$ can oscillate in space.

However, one can hope to overcome this problem by gauge fixing: perhaps if ${F(A)}$ is bounded in ${L^{n/2}(B(0,1))}$, then one can make ${A}$ bounded in ${W^{n/2,1}(B(0,1))}$ after applying a gauge transformation. The basic and useful result of Uhlenbeck is that this can be done if the ${L^{n/2}}$ norm of ${F(A)}$ is sufficiently small (and then the conclusion is that ${A}$ is small in ${W^{n/2,1}}$). (For large connections there is a serious issue related to the Gribov ambiguity.) In my (much) later paper with Tian, we adapted this argument, replacing Lebesgue spaces by Morrey space counterparts. (This result was also independently obtained at about the same time by Meyer and Riviére.)

To make the problem elliptic, one can try to impose the Coulomb gauge condition

$\displaystyle \partial^\alpha A_\alpha = 0 \ \ \ \ \ (2)$

(also known as the Lorenz gauge or Hodge gauge in various papers), together with a natural boundary condition on ${\partial B(0,1)}$ that will not be discussed further here. This turns (1), (2) into a divergence-curl system that is elliptic at the linear level at least. Indeed if one takes the divergence of (1) using (2) one sees that

$\displaystyle \partial^\alpha F(A)_{\alpha \beta} = \Delta A_\beta + \partial^\alpha [A_\alpha,A_\beta] \ \ \ \ \ (3)$

and if one could somehow ignore the nonlinear term ${\partial^\alpha [A_\alpha,A_\beta]}$ then we would get the required regularity on ${A}$ by standard elliptic regularity estimates.

The problem is then how to handle the nonlinear term. If we already knew that ${A}$ was small in the right norm ${W^{n/2,1}(B(0,1))}$ then one can use Sobolev embedding, Hölder’s inequality, and elliptic regularity to show that the second term in (3) is small compared to the first term, and so one could then hope to eliminate it by perturbative analysis. However, proving that ${A}$ is small in this norm is exactly what we are trying to prove! So this approach seems circular.

Uhlenbeck’s clever way out of this circularity is a textbook example of what is now known as a “continuity” argument. Instead of trying to work just with the original connection ${A}$, one works with the rescaled connections ${A^{(t)}_\alpha(x) := t A_\alpha(tx)}$ for ${0 \leq t \leq 1}$, with associated rescaled curvatures ${F(A^{(t)})_\alpha = t^2 F(A)_{\alpha \beta}(tx)}$. If the original curvature ${F(A)}$ is small in ${L^{n/2}}$ norm (e.g. bounded by some small ${\varepsilon>0}$), then so are all the rescaled curvatures ${F(A^{(t)})}$. We want to obtain a Coulomb gauge at time ${t=1}$; this is difficult to do directly, but it is trivial to obtain a Coulomb gauge at time ${t=0}$, because the connection vanishes at this time. On the other hand, once one has successfully obtained a Coulomb gauge at some time ${t \in [0,1]}$ with ${A^{(t)}}$ small in the natural norm ${W^{n/2,1}}$ (say bounded by ${C \varepsilon}$ for some constant ${C}$ which is large in absolute terms, but not so large compared with say ${1/\varepsilon}$), the perturbative argument mentioned earlier (combined with the qualitative hypothesis that ${A}$ is smooth) actually works to show that a Coulomb gauge can also be constructed and be small for all sufficiently close nearby times ${t' \in [0,1]}$ to ${t}$; furthermore, the perturbative analysis actually shows that the nearby gauges enjoy a slightly better bound on the ${W^{n/2,1}}$ norm, say ${C\varepsilon/2}$ rather than ${C\varepsilon}$. As a consequence of this, the set of times ${t}$ for which one has a good Coulomb gauge obeying the claimed estimates is both open and closed in ${[0,1]}$, and also contains ${t=0}$. Since the unit interval ${[0,1]}$ is connected, it must then also contain ${t=1}$. This concludes the proof.

One of the lessons I drew from this example is to not be deterred (especially in PDE) by an argument seeming to be circular; if the argument is still sufficiently “nontrivial” in nature, it can often be modified into a usefully non-circular argument that achieves what one wants (possibly under an additional qualitative hypothesis, such as a continuity or smoothness hypothesis).

I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds ${(M,g)}$, as I conjecture that the freedom to set the metric ${g}$ ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

$\displaystyle \partial_t u^k + u^j \nabla_j u^k = - \nabla^k p \ \ \ \ \ (1)$

$\displaystyle \nabla_k u^k = 0$

where ${p: [0,T] \rightarrow C^\infty(M)}$ is the pressure field and ${u: [0,T] \rightarrow \Gamma(TM)}$ is the velocity field, and ${\nabla}$ denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where ${u,p}$ are smooth and ${M}$ is compact, smooth, orientable, connected, and without boundary. Let’s call ${u}$ an Euler flow on ${M}$ (for the time interval ${[0,T]}$) if it solves the above system of equations for some pressure ${p}$, and an incompressible flow if it just obeys the divergence-free relation ${\nabla_k u^k=0}$. Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be extended to an Euler flow by adding some additional dimensions to ${M}$. In my paper, I formalise this by considering warped products ${\tilde M}$ of ${M}$ which (as a smooth manifold) are products ${\tilde M = M \times ({\bf R}/{\bf Z})^m}$ of ${M}$ with a torus, with a metric ${\tilde g}$ given by

$\displaystyle d \tilde g^2 = g_{ij}(x) dx^i dx^j + \sum_{s=1}^m \tilde g_{ss}(x) (d\theta^s)^2$

for ${(x,\theta) \in \tilde M}$, where ${\theta^1,\dots,\theta^m}$ are the coordinates of the torus ${({\bf R}/{\bf Z})^m}$, and ${\tilde g_{ss}: M \rightarrow {\bf R}^+}$ are smooth positive coefficients for ${s=1,\dots,m}$; in order to preserve the incompressibility condition, we also require the volume preservation property

$\displaystyle \prod_{s=1}^m \tilde g_{ss}(x) = 1 \ \ \ \ \ (2)$

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus ${({\bf R}/{\bf Z})^m}$. We say that an incompressible flow ${u}$ is extendible to an Euler flow if there exists a warped product ${\tilde M}$ extending ${M}$, and an Euler flow ${\tilde u}$ on ${\tilde M}$ of the form

$\displaystyle \tilde u(t,(x,\theta)) = u^i(t,x) \frac{d}{dx^i} + \sum_{s=1}^m \tilde u^s(t,x) \frac{d}{d\theta^s}$

for some “swirl” fields ${\tilde u^s: [0,T] \times M \rightarrow {\bf R}}$. The situation here is motivated by the familiar situation of studying axisymmetric Euler flows ${\tilde u}$ on ${{\bf R}^3}$, which in cylindrical coordinates take the form

$\displaystyle \tilde u(t,(r,z,\theta)) = u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz} + \tilde u^\theta(t,r,z) \frac{d}{d\theta}.$

The base component

$\displaystyle u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz}$

of this flow is then a flow on the two-dimensional ${r,z}$ plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field ${\rho}$ that is basically the square of the swirl ${\tilde u^\rho}$) that is reminiscent of the Boussinesq equations.

On a fixed ${d}$-dimensional manifold ${(M,g)}$, let ${{\mathcal F}}$ denote the space of incompressible flows ${u: [0,T] \rightarrow \Gamma(TM)}$, equipped with the smooth topology (in spacetime), and let ${{\mathcal E} \subset {\mathcal F}}$ denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

• (i) (Generic inextendibility) Assume ${d \geq 3}$. Then ${{\mathcal E}}$ is of the first category in ${{\mathcal F}}$ (the countable union of nowhere dense sets in ${{\mathcal F}}$).
• (ii) (Non-rigidity) Assume ${M = ({\bf R}/{\bf Z})^d}$ (with an arbitrary metric ${g}$). Then ${{\mathcal E}}$ is somewhere dense in ${{\mathcal F}}$ (that is, the closure of ${{\mathcal E}}$ has non-empty interior).

More informally, starting with an incompressible flow ${u}$, one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be perturbed to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “${h}$-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow ${u}$ is essentially determined by ${d-1}$ independent functions of space and time, whereas the warping factors ${\tilde g_{ss}}$ are functions of space only, the pressure field is one function of space and time, and the swirl fields ${u^s}$ are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When ${d>2}$, this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when ${d=2}$, but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates ${x^i}$ (coming from the warping factors) and the Lagrangian spatial coordinates ${a^\beta}$ (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of ${{\mathcal F}}$, the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that ${M}$ be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the ${M = ({\bf R}/{\bf Z})^d}$ case at least, that ${{\mathcal E}}$ is dense in all of ${{\mathcal F}}$, not just in a non-empty open subset).

We consider the incompressible Euler equations on the (Eulerian) torus ${\mathbf{T}_E := ({\bf R}/{\bf Z})^d}$, which we write in divergence form as

$\displaystyle \partial_t u^i + \partial_j(u^j u^i) = - \eta^{ij} \partial_j p \ \ \ \ \ (1)$

$\displaystyle \partial_i u^i = 0, \ \ \ \ \ (2)$

where ${\eta^{ij}}$ is the (inverse) Euclidean metric. Here we use the summation conventions for indices such as ${i,j,l}$ (reserving the symbol ${k}$ for other purposes), and are retaining the convention from Notes 1 of denoting vector fields using superscripted indices rather than subscripted indices, as we will eventually need to change variables to Lagrangian coordinates at some point. In principle, much of the discussion in this set of notes (particularly regarding the positive direction of Onsager’s conjecture) could also be modified to also treat non-periodic solutions that decay at infinity if desired, but some non-trivial technical issues do arise non-periodic settings for the negative direction.

As noted previously, the kinetic energy

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} |u(t,x)|^2\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(t,x) u^j(t,x)\ dx$

is formally conserved by the flow, where ${\eta_{ij}}$ is the Euclidean metric. Indeed, if one assumes that ${u,p}$ are continuously differentiable in both space and time on ${[0,T] \times \mathbf{T}}$, then one can multiply the equation (1) by ${u^l}$ and contract against ${\eta_{il}}$ to obtain

$\displaystyle \eta_{il} u^l \partial_t u^i + \eta_{il} u^l \partial_j (u^j u^i) = - \eta_{il} u^l \eta^{ij} \partial_j p = 0$

which rearranges using (2) and the product rule to

$\displaystyle \partial_t (\frac{1}{2} \eta_{ij} u^i u^j) + \partial_j( \frac{1}{2} \eta_{il} u^i u^j u^l ) + \partial_j (u^j p)$

and then if one integrates this identity on ${[0,T] \times \mathbf{T}_E}$ and uses Stokes’ theorem, one obtains the required energy conservation law

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(T,x) u^j(T,x)\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(0,x) u^j(0,x)\ dx. \ \ \ \ \ (3)$

It is then natural to ask whether the energy conservation law continues to hold for lower regularity solutions, in particular weak solutions that only obey (1), (2) in a distributional sense. The above argument no longer works as stated, because ${u^i}$ is not a test function and so one cannot immediately integrate (1) against ${u^i}$. And indeed, as we shall soon see, it is now known that once the regularity of ${u}$ 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.

But what is the precise level of regularity needed in order to for this anomalous energy dissipation to occur? To make this question precise, we need a quantitative notion of regularity. One such measure is given by the Hölder space ${C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})}$ for ${0 < \alpha < 1}$, defined as the space of continuous functions ${f: \mathbf{T}_E \rightarrow {\bf R}}$ whose norm

$\displaystyle \| f \|_{C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})} := \sup_{x \in \mathbf{T}_E} |f(x)| + \sup_{x,y \in \mathbf{T}_E: x \neq y} \frac{|f(x)-f(y)|}{|x-y|^\alpha}$

is finite. The space ${C^{0,\alpha}}$ lies between the space ${C^0}$ of continuous functions and the space ${C^1}$ of continuously differentiable functions, and informally describes a space of functions that is “${\alpha}$ times differentiable” in some sense. The above derivation of the energy conservation law involved the integral

$\displaystyle \int_{\mathbf{T}_E} \eta_{ik} u^k \partial_j (u^j u^i)\ dx$

that roughly speaking measures the fluctuation in energy. Informally, if we could take the derivative in this integrand and somehow “integrate by parts” to split the derivative “equally” amongst the three factors, one would morally arrive at an expression that resembles

$\displaystyle \int_{\mathbf{T}} \nabla^{1/3} u \nabla^{1/3} u \nabla^{1/3} u\ dx$

which suggests that the integral can be made sense of for ${u \in C^0_t C^{0,\alpha}_x}$ once ${\alpha > 1/3}$. More precisely, one can make

Conjecture 1 (Onsager’s conjecture) Let ${0 < \alpha < 1}$ and ${d \geq 2}$, and let ${0 < T < \infty}$.

• (i) If ${\alpha > 1/3}$, then any weak solution ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in the Leray form ${\partial_t u + \partial_j {\mathbb P} (u^j u) = u_0(x) \delta_0(t)}$) obeys the energy conservation law (3).
• (ii) If ${\alpha \leq 1/3}$, then there exist weak solutions ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in Leray form) which do not obey energy conservation.

This conjecture was originally arrived at by Onsager by a somewhat different heuristic derivation; see Remark 7. The numerology is also compatible with that arising from the Kolmogorov theory of turbulence (discussed in this previous post), but we will not discuss this interesting connection further here.

The positive part (i) of Onsager conjecture was established by Constantin, E, and Titi, building upon earlier partial results by Eyink; the proof is a relatively straightforward application of Littlewood-Paley theory, and they were also able to work in larger function spaces than ${C^0_t C^{0,\alpha}_x}$ (using ${L^3_x}$-based Besov spaces instead of Hölder spaces, see Exercise 3 below). The negative part (ii) is harder. Discontinuous weak solutions to the Euler equations that did not conserve energy were first constructed by Sheffer, with an alternate construction later given by Shnirelman. De Lellis and Szekelyhidi noticed the resemblance of this problem to that of the Nash-Kuiper theorem in the isometric embedding problem, and began adapting the convex integration technique used in that theorem to construct weak solutions of the Euler equations. This began a long series of papers in which increasingly regular weak solutions that failed to conserve energy were constructed, culminating in a recent paper of Isett establishing part (ii) of the Onsager conjecture in the non-endpoint case ${\alpha < 1/3}$ in three and higher dimensions ${d \geq 3}$; the endpoint ${\alpha = 1/3}$ 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 this paper of Cheskidov, Lopes Filho, Nussenzveig Lopes, and Shvydkoy for more discussion.) Further work continues into several variations of the Onsager conjecture, in which one looks at other differential equations, other function spaces, or other criteria for bad behavior than breakdown of energy conservation. See this recent survey of de Lellis and Szekelyhidi for more discussion.

In these notes we will first establish (i), then discuss the convex integration method in the original context of the Nash-Kuiper embedding theorem. Before tackling the Onsager conjecture (ii) directly, we discuss a related construction of high-dimensional weak solutions in the Sobolev space ${L^2_t H^s_x}$ for ${s}$ close to ${1/2}$, 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.

We thank Phil Isett for some comments and corrections.

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.

We consider the Euler equations for incompressible fluid flow on a Euclidean space ${{\bf R}^d}$; we will label ${{\bf R}^d}$ as the “Eulerian space” ${{\bf R}^d_E}$ (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” ${{\bf R}^d_L}$ (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the ${E}$ or ${L}$ subscripts if he or she wishes). Elements of Eulerian space ${{\bf R}^d_E}$ will be referred to by symbols such as ${x}$, we use ${dx}$ to denote Lebesgue measure on ${{\bf R}^d_E}$ and we will use ${x^1,\dots,x^d}$ for the ${d}$ coordinates of ${x}$, and use indices such as ${i,j,k}$ to index these coordinates (with the usual summation conventions), for instance ${\partial_i}$ denotes partial differentiation along the ${x^i}$ coordinate. (We use superscripts for coordinates ${x^i}$ instead of subscripts ${x_i}$ 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.)

In Eulerian coordinates, the Euler equations read

$\displaystyle \partial_t u + u \cdot \nabla u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ is the velocity field and ${p: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ is the pressure field. These are functions of time ${t \in [0,T)}$ and on the spatial location variable ${x \in {\bf R}^d_E}$. We will refer to the coordinates ${(t,x) = (t,x^1,\dots,x^d)}$ as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field ${u}$ or the pressure field ${p}$, but rather the trajectories ${(x^{(a)}(t))_{a \in A}}$, which can be thought of as a single function ${x: [0,T) \times A \rightarrow {\bf R}^d_E}$ from the coordinates ${(t,a)}$ (where ${t}$ is a time and ${a}$ is an element of the label set ${A}$) to ${{\bf R}^d}$. The relationship between the trajectories ${x^{(a)}(t) = x(t,a)}$ and the velocity field was given by the informal relationship

$\displaystyle \partial_t x(t,a) \approx u( t, x(t,a) ). \ \ \ \ \ (2)$

We will refer to the coordinates ${(t,a)}$ as (discrete) Lagrangian coordinates for describing the fluid.

In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version ${(t,a)}$ of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution ${u, p}$ to the Euler equations on a spacetime slab ${[0,T) \times {\bf R}^d_E}$ in Eulerian coordinates; assume furthermore that the velocity field ${u}$ is uniformly bounded. We introduce another copy ${{\bf R}^d_L}$ of ${{\bf R}^d}$, which we call Lagrangian space or labels space; we use symbols such as ${a}$ to refer to elements of this space, ${da}$ to denote Lebesgue measure on ${{\bf R}^d_L}$, and ${a^1,\dots,a^d}$ to refer to the ${d}$ coordinates of ${a}$. We use indices such as ${\alpha,\beta,\gamma}$ to index these coordinates, thus for instance ${\partial_\alpha}$ denotes partial differentiation along the ${a^\alpha}$ coordinate. We will use summation conventions for both the Eulerian coordinates ${i,j,k}$ and the Lagrangian coordinates ${\alpha,\beta,\gamma}$, with an index being summed if it appears as both a subscript and a superscript in the same term. While ${{\bf R}^d_L}$ and ${{\bf R}^d_E}$ are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time ${t=0}$ in order to fix the initialisation of Lagrangian coordinates.

Given a smooth and bounded velocity field ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$, define a trajectory map for this velocity to be any smooth map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ that obeys the ODE

$\displaystyle \partial_t X(t,a) = u( t, X(t,a) ); \ \ \ \ \ (3)$

in view of (2), this describes the trajectory (in ${{\bf R}^d_E}$) of a particle labeled by an element ${a}$ of ${{\bf R}^d_L}$. From the Picard existence theorem and the hypothesis that ${u}$ is smooth and bounded, such a map exists and is unique as long as one specifies the initial location ${X(0,a)}$ assigned to each label ${a}$. Traditionally, one chooses the initial condition

$\displaystyle X(0,a) = a \ \ \ \ \ (4)$

for ${a \in {\bf R}^d_L}$, so that we label each particle by its initial location at time ${t=0}$; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels ${a \in {\bf R}^d_L}$ by an arbitrary diffeomorphism: if ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a trajectory map, and ${\pi: {\bf R}^d_L \rightarrow{\bf R}^d_L}$ is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map ${X \circ \pi: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is also a trajectory map, albeit one with different initial conditions ${X(0,a)}$.

Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space ${{\bf R}^d_E}$ from the Lagrangian space ${{\bf R}^d_L}$, as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric ${d\eta^2 = dx^1 dx^1 + \dots + dx^d dx^d}$ is an important feature of Eulerian space ${{\bf R}^d_E}$, it is not a geometrically natural structure to use in Lagrangian space ${{\bf R}^d_L}$. We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded.

• If ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a smooth map, show that there exists a unique smooth trajectory map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ with initial condition ${X(0,a) = X_0(a)}$ for all ${a \in {\bf R}^d_L}$.
• Show that if ${X_0}$ is a diffeomorphism and ${t \in [0,T)}$, then the map ${X(t): a \mapsto X(t,a)}$ is also a diffeomorphism.

Remark 2 The first of the Euler equations (1) can now be written in the form

$\displaystyle \frac{d^2}{dt^2} X(t,a) = - (\nabla p)( t, X(t,a) ) \ \ \ \ \ (5)$

which can be viewed as a continuous limit of Newton’s first law ${m^{(a)} \frac{d^2}{dt^2} x^{(a)}(t) = F^{(a)}(t)}$.

Call a diffeomorphism ${Y: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ (oriented) volume preserving if one has the equation

$\displaystyle \mathrm{det}( \nabla Y )(a) = 1 \ \ \ \ \ (6)$

for all ${a \in {\bf R}^d_L}$, where the total differential ${\nabla Y}$ is the ${d \times d}$ matrix with entries ${\partial_\alpha Y^i}$ for ${\alpha = 1,\dots,d}$ and ${i=1,\dots,d}$, where ${Y^1,\dots,Y^d:{\bf R}^d_L \rightarrow {\bf R}}$ are the components of ${Y}$. (If one wishes, one can also view ${\nabla Y}$ as a linear transformation from the tangent space ${T_a {\bf R}^d_L}$ of Lagrangian space at ${a}$ to the tangent space ${T_{Y(a)} {\bf R}^d_E}$ of Eulerian space at ${Y(a)}$.) Equivalently, ${Y}$ is orientation preserving and one has a Jacobian-free change of variables formula

$\displaystyle \int_{{\bf R}^d_F} f( Y(a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${f \in C_c({\bf R}^d_E \rightarrow {\bf R})}$, which is in turn equivalent to ${Y(E) \subset {\bf R}^d_E}$ having the same Lebesgue measure as ${E}$ for any measurable set ${E \subset {\bf R}^d_L}$.

The divergence-free condition ${\nabla \cdot u = 0}$ then can be nicely expressed in terms of volume-preserving properties of the trajectory maps ${X}$, in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded, let ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be a volume-preserving diffeomorphism, and let ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be the trajectory map. Then the following are equivalent:

• ${\nabla \cdot u = 0}$ on ${[0,T) \times {\bf R}^d_E}$.
• ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is volume-preserving for all ${t \in [0,T)}$.

Proof: Since ${X_0}$ is orientation-preserving, we see from continuity that ${X(t)}$ is also orientation-preserving. Suppose that ${X(t)}$ is also volume-preserving, then for any ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ we have the conservation law

$\displaystyle \int_{{\bf R}^d_L} f( X(t,a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${t \in [0,T)}$. Differentiating in time using the chain rule and (3) we conclude that

$\displaystyle \int_{{\bf R}^d_L} (u(t) \cdot \nabla f)( X(t,a)) \ da = 0$

for all ${t \in [0,T)}$, and hence by change of variables

$\displaystyle \int_{{\bf R}^d_E} (u(t) \cdot \nabla f)(x) \ dx = 0$

which by integration by parts gives

$\displaystyle \int_{{\bf R}^d_E} (\nabla \cdot u(t,x)) f(x)\ dx = 0$

for all ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ and ${t \in [0,T)}$, so ${u}$ is divergence-free.

To prove the converse implication, it is convenient to introduce the labels map ${A:[0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_L}$, defined by setting ${A(t): {\bf R}^d_E \rightarrow {\bf R}^d_L}$ to be the inverse of the diffeomorphism ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$, thus

$\displaystyle A(t, X(t,a)) = a$

for all ${(t,a) \in [0,T) \times {\bf R}^d_L}$. By the implicit function theorem, ${A}$ is smooth, and by differentiating the above equation in time using (3) we see that

$\displaystyle D_t A(t,x) = 0$

where ${D_t}$ is the usual material derivative

$\displaystyle D_t := \partial_t + u \cdot \nabla \ \ \ \ \ (7)$

acting on functions on ${[0,T) \times {\bf R}^d_E}$. If ${u}$ is divergence-free, we have from integration by parts that

$\displaystyle \partial_t \int_{{\bf R}^d_E} \phi(t,x)\ dx = \int_{{\bf R}^d_E} D_t \phi(t,x)\ dx$

for any test function ${\phi: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$. In particular, for any ${g \in C^\infty_c({\bf R}^d_L \rightarrow {\bf R})}$, we can calculate

$\displaystyle \partial_t \int_{{\bf R}^d_E} g( A(t,x) )\ dx = \int_{{\bf R}^d_E} D_t (g(A(t,x)))\ dx$

$\displaystyle = \int_{{\bf R}^d_E} 0\ dx$

and hence

$\displaystyle \int_{{\bf R}^d_E} g(A(t,x))\ dx = \int_{{\bf R}^d_E} g(A(0,x))\ dx$

for any ${t \in [0,T)}$. Since ${X_0}$ is volume-preserving, so is ${A(0)}$, thus

$\displaystyle \int_{{\bf R}^d_E} g \circ A(t)\ dx = \int_{{\bf R}^d_L} g\ da.$

Thus ${A(t)}$ is volume-preserving, and hence ${X(t)}$ is also. $\Box$

Exercise 4 Let ${M: [0,T) \rightarrow \mathrm{GL}_d({\bf R})}$ be a continuously differentiable map from the time interval ${[0,T)}$ to the general linear group ${\mathrm{GL}_d({\bf R})}$ of invertible ${d \times d}$ matrices. Establish Jacobi’s formula

$\displaystyle \partial_t \det(M(t)) = \det(M(t)) \mathrm{tr}( M(t)^{-1} \partial_t M(t) )$

and use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5 One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function ${f: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime, one has

$\displaystyle \frac{d}{dt} f(t,X(t,a)) = (D_t f)(t,X(t,a))$

and hence any transport equation that in Eulerian coordinates takes the form

$\displaystyle D_t f = g$

for smooth functions ${f,g: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime is equivalent to the ODE

$\displaystyle \frac{d}{dt} F = G$

where ${F,G: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}}$ are the smooth functions of Lagrangian spacetime defined by

$\displaystyle F(t,a) := f(t,X(t,a)); \quad G(t,a) := g(t,X(t,a)).$

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as ${{\bf R}^d_E}$ and ${{\bf R}^d_L}$, and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6 One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian ${\Delta}$ 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.

Note: this post is not required reading for this course, or for the sequel course in the winter quarter.

In a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes equations. We did not quite follow Leray’s original proof, in that the notes relied more heavily on the machinery of Littlewood-Paley projections, which have become increasingly common tools in modern PDE. On the other hand, we did use the same “exploiting compactness to pass to weakly convergent subsequence” strategy that is the standard one in the PDE literature used to construct weak solutions.

As I discussed in a previous post, the manipulation of sequences and their limits is analogous to a “cheap” version of nonstandard analysis in which one uses the Fréchet filter rather than an ultrafilter to construct the nonstandard universe. (The manipulation of generalised functions of Columbeau-type can also be comfortably interpreted within this sort of cheap nonstandard analysis.) Augmenting the manipulation of sequences with the right to pass to subsequences whenever convenient is then analogous to a sort of “lazy” nonstandard analysis, in which the implied ultrafilter is never actually constructed as a “completed object“, but is instead lazily evaluated, in the sense that whenever membership of a given subsequence of the natural numbers in the ultrafilter needs to be determined, one either passes to that subsequence (thus placing it in the ultrafilter) or the complement of the sequence (placing it out of the ultrafilter). This process can be viewed as the initial portion of the transfinite induction that one usually uses to construct ultrafilters (as discussed using a voting metaphor in this post), except that there is generally no need in any given application to perform the induction for any uncountable ordinal (or indeed for most of the countable ordinals also).

On the other hand, it is also possible to work directly in the orthodox framework of nonstandard analysis when constructing weak solutions. This leads to an approach to the subject which is largely equivalent to the usual subsequence-based approach, though there are some minor technical differences (for instance, the subsequence approach occasionally requires one to work with separable function spaces, whereas in the ultrafilter approach the reliance on separability is largely eliminated, particularly if one imposes a strong notion of saturation on the nonstandard universe). The subject acquires a more “algebraic” flavour, as the quintessential analysis operation of taking a limit is replaced with the “standard part” operation, which is an algebra homomorphism. The notion of a sequence is replaced by the distinction between standard and nonstandard objects, and the need to pass to subsequences disappears entirely. Also, the distinction between “bounded sequences” and “convergent sequences” is largely eradicated, particularly when the space that the sequences ranged in enjoys some compactness properties on bounded sets. Also, in this framework, the notorious non-uniqueness features of weak solutions can be “blamed” on the non-uniqueness of the nonstandard extension of the standard universe (as well as on the multiple possible ways to construct nonstandard mollifications of the original standard PDE). However, many of these changes are largely cosmetic; switching from a subsequence-based theory to a nonstandard analysis-based theory does not seem to bring one significantly closer for instance to the global regularity problem for Navier-Stokes, but it could have been an alternate path for the historical development and presentation of the subject.

In any case, I would like to present below the fold this nonstandard analysis perspective, quickly translating the relevant components of real analysis, functional analysis, and distributional theory that we need to this perspective, and then use it to re-prove Leray’s theorem on existence of global weak solutions to Navier-Stokes.

I’ve just uploaded to the arXiv my paper “Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion“, submitted to Revista Matematica Iberoamericana. This paper concerns the extent to which one can accurately embed the metric structure of the Heisenberg group

$\displaystyle H := \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix}$

into Euclidean space, which we can write as ${\{ [x,y,z]: x,y,z \in {\bf R} \}}$ with the notation

$\displaystyle [x,y,z] := \begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}.$

Here we give ${H}$ the right-invariant Carnot-Carathéodory metric ${d}$ coming from the right-invariant vector fields

$\displaystyle X := \frac{\partial}{\partial x} + y \frac{\partial}{\partial z}; \quad Y := \frac{\partial}{\partial y}$

but not from the commutator vector field

$\displaystyle Z := [Y,X] = \frac{\partial}{\partial z}.$

This gives ${H}$ the geometry of a Carnot group. As observed by Semmes, it follows from the Carnot group differentiation theory of Pansu that there is no bilipschitz map from ${(H,d)}$ to any Euclidean space ${{\bf R}^D}$ or even to ${\ell^2}$, since such a map must be differentiable almost everywhere in the sense of Carnot groups, which in particular shows that the derivative map annihilate ${Z}$ almost everywhere, which is incompatible with being bilipschitz.

On the other hand, if one snowflakes the Heisenberg group by replacing the metric ${d}$ with ${d^{1-\varepsilon}}$ for some ${0 < \varepsilon < 1}$, then it follows from the general theory of Assouad on embedding snowflaked metrics of doubling spaces that ${(H,d^{1-\varepsilon})}$ may be embedded in a bilipschitz fashion into ${\ell^2}$, or even to ${{\bf R}^{D_\varepsilon}}$ for some ${D_\varepsilon}$ depending on ${\varepsilon}$.

Of course, the distortion of this bilipschitz embedding must degenerate in the limit ${\varepsilon \rightarrow 0}$. From the work of Austin-Naor-Tessera and Naor-Neiman it follows that ${(H,d^{1-\varepsilon})}$ may be embedded into ${\ell^2}$ with a distortion of ${O( \varepsilon^{-1/2} )}$, but no better. The Naor-Neiman paper also embeds ${(H,d^{1-\varepsilon})}$ into a finite-dimensional space ${{\bf R}^D}$ with ${D}$ independent of ${\varepsilon}$, but at the cost of worsening the distortion to ${O(\varepsilon^{-1})}$. They then posed the question of whether this worsening of the distortion is necessary.

The main result of this paper answers this question in the negative:

Theorem 1 There exists an absolute constant ${D}$ such that ${(H,d^{1-\varepsilon})}$ may be embedded into ${{\bf R}^D}$ in a bilipschitz fashion with distortion ${O(\varepsilon^{-1/2})}$ for any ${0 < \varepsilon \leq 1/2}$.

To motivate the proof of this theorem, let us first present a bilipschitz map ${\Phi: {\bf R} \rightarrow \ell^2}$ from the snowflaked line ${({\bf R},d_{\bf R}^{1-\varepsilon})}$ (with ${d_{\bf R}}$ being the usual metric on ${{\bf R}}$) into complex Hilbert space ${\ell^2({\bf C})}$. The map is given explicitly as a Weierstrass type function

$\displaystyle \Phi(x) := \sum_{k \in {\bf Z}} 2^{-\varepsilon k} (\phi_k(x) - \phi_k(0))$

where for each ${k}$, ${\phi_k: {\bf R} \rightarrow \ell^2}$ is the function

$\displaystyle \phi_k(x) := 2^k e^{2\pi i x / 2^k} e_k.$

and ${(e_k)_{k \in {\bf Z}}}$ are an orthonormal basis for ${\ell^2({\bf C})}$. The subtracting of the constant ${\phi_k(0)}$ is purely in order to make the sum convergent as ${k \rightarrow \infty}$. If ${x,y \in {\bf R}}$ are such that ${2^{k_0-2} \leq d_{\bf R}(x,y) \leq 2^{k_0-1}}$ for some integer ${k_0}$, one can easily check the bounds

$\displaystyle |\phi_k(x) - \phi_k(y)| \lesssim d_{\bf R}(x,y)^{(1-\varepsilon)} \min( 2^{-(1-\varepsilon) (k_0-k)}, 2^{-\varepsilon (k-k_0)} )$

with the lower bound

$\displaystyle |\phi_{k_0}(x) - \phi_{k_0}(y)| \gtrsim d_{\bf R}(x,y)^{(1-\varepsilon)}$

at which point one finds that

$\displaystyle d_{\bf R}(x,y)^{1-\varepsilon} \lesssim |\Phi(x) - \Phi(y)| \lesssim \varepsilon^{-1/2} d_{\bf R}(x,y)^{1-\varepsilon}$

as desired.

The key here was that each function ${\phi_k}$ oscillated at a different spatial scale ${2^k}$, and the functions were all orthogonal to each other (so that the upper bound involved a factor of ${\varepsilon^{-1/2}}$ rather than ${\varepsilon^{-1}}$). One can replicate this example for the Heisenberg group without much difficulty. Indeed, if we let ${\Gamma := \{ [a,b,c]: a,b,c \in {\bf Z} \}}$ be the discrete Heisenberg group, then the nilmanifold ${H/\Gamma}$ is a three-dimensional smooth compact manifold; thus, by the Whitney embedding theorem, it smoothly embeds into ${{\bf R}^6}$. This gives a smooth immersion ${\phi: H \rightarrow {\bf R}^6}$ which is ${\Gamma}$-automorphic in the sense that ${\phi(p\gamma) = \phi(p)}$ for all ${p \in H}$ and ${\gamma \in \Gamma}$. If one then defines ${\phi_k: H \rightarrow \ell^2 \otimes {\bf R}^6}$ to be the function

$\displaystyle \phi_k(p) := 2^k \phi( \delta_{2^{-k}}(p) ) \otimes e_k$

where ${\delta_\lambda: H \rightarrow H}$ is the scaling map

$\displaystyle \delta_\lambda([x,y,z]) := [\lambda x, \lambda y, \lambda^2 z],$

then one can repeat the previous arguments to obtain the required bilipschitz bounds

$\displaystyle d(p,q)^{1-\varepsilon} \lesssim |\Phi(p) - \Phi(q) \lesssim \varepsilon^{-1/2} d(p,q)^{1-\varepsilon}$

for the function

$\displaystyle \Phi(p) :=\sum_{k \in {\bf Z}} 2^{-\varepsilon k} (\phi_k(p) - \phi_k(0)).$

To adapt this construction to bounded dimension, the main obstruction was the requirement that the ${\phi_k}$ took values in orthogonal subspaces. But if one works things out carefully, it is enough to require the weaker orthogonality requirement

$\displaystyle B( \phi_{k_0}, \sum_{k>k_0} 2^{-\varepsilon(k-k_0)} \phi_k ) = 0$

for all ${k_0 \in {\bf Z}}$, where ${B(\phi, \psi): H \rightarrow {\bf R}^2}$ is the bilinear form

$\displaystyle B(\phi,\psi) := (X \phi \cdot X \psi, Y \phi \cdot Y \psi ).$

One can then try to construct the ${\phi_k: H \rightarrow {\bf R}^D}$ for bounded dimension ${D}$ by an iterative argument. After some standard reductions, the problem becomes this (roughly speaking): given a smooth, slowly varying function ${\psi: H \rightarrow {\bf R}^{D}}$ whose derivatives obey certain quantitative upper and lower bounds, construct a smooth oscillating function ${\phi: H \rightarrow {\bf R}^{D}}$, whose derivatives also obey certain quantitative upper and lower bounds, which obey the equation

$\displaystyle B(\phi,\psi) = 0. \ \ \ \ \ (1)$

We view this as an underdetermined system of differential equations for ${\phi}$ (two equations in ${D}$ unknowns; after some reductions, our ${D}$ can be taken to be the explicit value ${36}$). The trivial solution ${\phi=0}$ to this equation will be inadmissible for our purposes due to the lower bounds we will require on ${\phi}$ (in order to obtain the quantitative immersion property mentioned previously, as well as for a stronger “freeness” property that is needed to close the iteration). Because this construction will need to be iterated, it will be essential that the regularity control on ${\phi}$ is the same as that on ${\psi}$; one cannot afford to “lose derivatives” when passing from ${\psi}$ to ${\phi}$.

This problem has some formal similarities with the isometric embedding problem (discussed for instance in this previous post), which can be viewed as the problem of solving an equation of the form ${Q(\phi,\phi) = g}$, where ${(M,g)}$ is a Riemannian manifold and ${Q}$ is the bilinear form

$\displaystyle Q(\phi,\psi)_{ij} = \partial_i \phi \cdot \partial_j \psi.$

The isometric embedding problem also has the key obstacle that naive attempts to solve the equation ${Q(\phi,\phi)=g}$ iteratively can lead to an undesirable “loss of derivatives” that prevents one from iterating indefinitely. This obstacle was famously resolved by the Nash-Moser iteration scheme in which one alternates between perturbatively adjusting an approximate solution to improve the residual error term, and mollifying the resulting perturbation to counteract the loss of derivatives. The current equation (1) differs in some key respects from the isometric embedding equation ${Q(\phi,\phi)=g}$, in particular being linear in the unknown field ${\phi}$ rather than quadratic; nevertheless the key obstacle is the same, namely that naive attempts to solve either equation lose derivatives. Our approach to solving (1) was inspired by the Nash-Moser scheme; in retrospect, I also found similarities with Uchiyama’s constructive proof of the Fefferman-Stein decomposition theorem, discussed in this previous post (and in this recent one).

To motivate this iteration, we first express ${B(\phi,\psi)}$ using the product rule in a form that does not place derivatives directly on the unknown ${\phi}$:

$\displaystyle B(\phi,\psi) = \left( W(\phi \cdot W \psi) - \phi \cdot WW \psi\right)_{W = X,Y} \ \ \ \ \ (2)$

This reveals that one can construct solutions ${\phi}$ to (1) by solving the system of equations

$\displaystyle \phi \cdot W \psi = \phi \cdot WW \psi = 0 \ \ \ \ \ (3)$

for ${W \in \{X, Y \}}$. Because this system is zeroth order in ${\phi}$, this can easily be done by linear algebra (even in the presence of a forcing term ${B(\phi,\psi)=F}$) if one imposes a “freeness” condition (analogous to the notion of a free embedding in the isometric embedding problem) that ${X \psi(p), Y \psi(p), XX \psi(p), YY \psi(p)}$ are linearly independent at each point ${p}$, which (together with some other technical conditions of a similar nature) one then adds to the list of upper and lower bounds required on ${\psi}$ (with a related bound then imposed on ${\phi}$, in order to close the iteration). However, as mentioned previously, there is a “loss of derivatives” problem with this construction: due to the presence of the differential operators ${W}$ in (3), a solution ${\phi}$ constructed by this method can only be expected to have two degrees less regularity than ${\psi}$ at best, which makes this construction unsuitable for iteration.

To get around this obstacle (which also prominently appears when solving (linearisations of) the isometric embedding equation ${Q(\phi,\phi)=g}$), we instead first construct a smooth, low-frequency solution ${\phi_{\leq N_0} \colon H \rightarrow {\bf R}^{D}}$ to a low-frequency equation

$\displaystyle B( \phi_{\leq N_0}, P_{\leq N_0} \psi ) = 0 \ \ \ \ \ (4)$

where ${P_{\leq N_0} \psi}$ is a mollification of ${\psi}$ (of Littlewood-Paley type) applied at a small spatial scale ${1/N_0}$ for some ${N_0}$, and then gradually relax the frequency cutoff ${P_{\leq N_0}}$ to deform this low frequency solution ${\phi_{\leq N_0}}$ to a solution ${\phi}$ of the actual equation (1).

We will construct the low-frequency solution ${\phi_{\leq N_0}}$ rather explicitly, using the Whitney embedding theorem to construct an initial oscillating map ${f}$ into a very low dimensional space ${{\bf R}^6}$, composing it with a Veronese type embedding into a slightly larger dimensional space ${{\bf R}^{27}}$ to obtain a required “freeness” property, and then composing further with a slowly varying isometry ${U(p) \colon {\bf R}^{27} \rightarrow {\bf R}^{36}}$ depending on ${P_{\leq N_0}}$ and constructed by a quantitative topological lemma (relying ultimately on the vanishing of the first few homotopy groups of high-dimensional spheres), in order to obtain the required orthogonality (4). (This sort of “quantitative null-homotopy” was first proposed by Gromov, with some recent progress on optimal bounds by Chambers-Manin-Weinberger and by Chambers-Dotterer-Manin-Weinberger, but we will not need these more advanced results here, as one can rely on the classical qualitative vanishing ${\pi^k(S^d)=0}$ for ${k < d}$ together with a compactness argument to obtain (ineffective) quantitative bounds, which suffice for this application).

To perform the deformation of ${\phi_{\leq N_0}}$ into ${\phi}$, we must solve what is essentially the linearised equation

$\displaystyle B( \dot \phi, \psi ) + B( \phi, \dot \psi ) = 0 \ \ \ \ \ (5)$

of (1) when ${\phi}$, ${\psi}$ (viewed as low frequency functions) are both being deformed at some rates ${\dot \phi, \dot \psi}$ (which should be viewed as high frequency functions). To avoid losing derivatives, the magnitude of the deformation ${\dot \phi}$ in ${\phi}$ should not be significantly greater than the magnitude of the deformation ${\dot \psi}$ in ${\psi}$, when measured in the same function space norms.

As before, if one directly solves the difference equation (5) using a naive application of (2) with ${B(\phi,\dot \psi)}$ treated as a forcing term, one will lose at least one derivative of regularity when passing from ${\dot \psi}$ to ${\dot \phi}$. However, observe that (2) (and the symmetry ${B(\phi, \dot \psi) = B(\dot \psi,\phi)}$) can be used to obtain the identity

$\displaystyle B( \dot \phi, \psi ) + B( \phi, \dot \psi ) = \left( W(\dot \phi \cdot W \psi + \dot \psi \cdot W \phi) - (\dot \phi \cdot WW \psi + \dot \psi \cdot WW \phi)\right)_{W = X,Y} \ \ \ \ \ (6)$

and then one can solve (5) by solving the system of equations

$\displaystyle \dot \phi \cdot W \psi = - \dot \psi \cdot W \phi$

for ${W \in \{X,XX,Y,YY\}}$. The key point here is that this system is zeroth order in both ${\dot \phi}$ and ${\dot \psi}$, so one can solve this system without losing any derivatives when passing from ${\dot \psi}$ to ${\dot \phi}$; compare this situation with that of the superficially similar system

$\displaystyle \dot \phi \cdot W \psi = - \phi \cdot W \dot \psi$

that one would obtain from naively linearising (3) without exploiting the symmetry of ${B}$. There is still however one residual “loss of derivatives” problem arising from the presence of a differential operator ${W}$ on the ${\phi}$ term, which prevents one from directly evolving this iteration scheme in time without losing regularity in ${\phi}$. It is here that we borrow the final key idea of the Nash-Moser scheme, which is to replace ${\phi}$ by a mollified version ${P_{\leq N} \phi}$ of itself (where the projection ${P_{\leq N}}$ depends on the time parameter). This creates an error term in (5), but it turns out that this error term is quite small and smooth (being a “high-high paraproduct” of ${\nabla \phi}$ and ${\nabla\psi}$, it ends up being far more regular than either ${\phi}$ or ${\psi}$, even with the presence of the derivatives) and can be iterated away provided that the initial frequency cutoff ${N_0}$ is large and the function ${\psi}$ has a fairly high (but finite) amount of regularity (we will eventually use the Hölder space ${C^{20,\alpha}}$ on the Heisenberg group to measure this).

We now turn to the local existence theory for the initial value problem for the incompressible Euler equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

$\displaystyle u(0,x) = u_0(x).$

For sake of discussion we will just work in the non-periodic domain ${{\bf R}^d}$, ${d \geq 2}$, although the arguments here can be adapted without much difficulty to the periodic setting. We will only work with solutions in which the pressure ${p}$ is normalised in the usual fashion:

$\displaystyle p = - \Delta^{-1} \nabla \cdot \nabla \cdot (u \otimes u). \ \ \ \ \ (2)$

Formally, the Euler equations (with normalised pressure) arise as the vanishing viscosity limit ${\nu \rightarrow 0}$ of the Navier-Stokes equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p + \nu \Delta u \ \ \ \ \ (3)$

$\displaystyle \nabla \cdot u = 0$

$\displaystyle p = - \Delta^{-1} \nabla \cdot \nabla \cdot (u \otimes u)$

$\displaystyle u(0,x) = u_0(x)$

that was studied in previous notes. However, because most of the bounds established in previous notes, either on the lifespan ${T_*}$ of the solution or on the size of the solution itself, depended on ${\nu}$, it is not immediate how to justify passing to the limit and obtain either a strong well-posedness theory or a weak solution theory for the limiting equation (1). (For instance, weak solutions to the Navier-Stokes equations (or the approximate solutions used to create such weak solutions) have ${\nabla u}$ lying in ${L^2_{t,loc} L^2_x}$ for ${\nu>0}$, but the bound on the norm is ${O(\nu^{-1/2})}$ and so one could lose this regularity in the limit ${\nu \rightarrow 0}$, at which point it is not clear how to ensure that the nonlinear term ${u_j u}$ still converges in the sense of distributions to what one expects.)

Nevertheless, by carefully using the energy method (which we will do loosely following an approach of Bertozzi and Majda), it is still possible to obtain local-in-time estimates on (high-regularity) solutions to (3) that are uniform in the limit ${\nu \rightarrow 0}$. Such a priori estimates can then be combined with a number of variants of these estimates obtain a satisfactory local well-posedness theory for the Euler equations. Among other things, we will be able to establish the Beale-Kato-Majda criterion – smooth solutions to the Euler (or Navier-Stokes) equations can be continued indefinitely unless the integral

$\displaystyle \int_0^{T_*} \| \omega(t) \|_{L^\infty_x( {\bf R}^d \rightarrow \wedge^2 {\bf R}^d )}\ dt$

becomes infinite at the final time ${T_*}$, where ${\omega := \nabla \wedge u}$ is the vorticity field. The vorticity has the important property that it is transported by the Euler flow, and in two spatial dimensions it can be used to establish global regularity for both the Euler and Navier-Stokes equations in these settings. (Unfortunately, in three and higher dimensions the phenomenon of vortex stretching has frustrated all attempts to date to use the vorticity transport property to establish global regularity of either equation in this setting.)

There is a rather different approach to establishing local well-posedness for the Euler equations, which relies on the vorticity-stream formulation of these equations. This will be discused in a later set of notes.