We consider the incompressible Euler equations on the (Eulerian) torus , which we write in divergence form as

where is the (inverse) Euclidean metric. Here we use the summation conventions for indices such as (reserving the symbol for other purposes), and are retaining the convention from 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

is formally conserved by the flow, where is the Euclidean metric. Indeed, if one assumes that are continuously differentiable in both space and time on , then one can multiply the equation (1) by and contract against to obtain which rearranges using (2) and the product rule to and then if one integrates this identity on and uses Stokes’ theorem, one obtains the required energy conservation law 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 is not a test function and so one cannot immediately integrate (1) against . And indeed, as we shall soon see, it is now known that once the regularity of 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 for , defined as the space of continuous functions whose norm

is finite. The space lies between the space of continuous functions and the space of continuously differentiable functions, and informally describes a space of functions that is “ times differentiable” in some sense. The above derivation of the energy conservation law involved the integral 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 which suggests that the integral can be made sense of for once . More precisely, one can make

Conjecture 1 (Onsager’s conjecture)Let and , and let .

- (i) If , then any weak solution to the Euler equations (in the Leray form ) obeys the energy conservation law (3).
- (ii) If , then there exist weak solutions 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 (using -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 in three and higher dimensions ; the endpoint remains open. (In two dimensions it may be the case that the positive results extend to a larger range than Onsager’s conjecture predicts; see 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 for close to , which is slightly easier to establish, though still rather intricate. Finally, we discuss the modifications of that construction needed to establish (ii), though we shall stop short of a full proof of that part of the conjecture.

We thank Phil Isett for some comments and corrections.

** — 1. Energy conservation for sufficiently regular weak solutions — **

We now prove the positive part (i) of Onsager’s conjecture, which turns out to be a straightforward application of Littlewood-Paley theory. We need the following relation between Hölder spaces and Littlewood-Paley projections:

Exercise 2Let for some and . Establish the bounds

Let be a weak solution to the Euler equations for some , thus

To show (3), it will suffice from dominated convergence to show that as . Applying to (4), we have From Bernstein’s inequality we conclude that , and thus . Thus solves the PDE in the classical sense. We can then apply the fundamental theorem of calculus to write and so it will suffice to show that We can integrate by parts to place the Leray projection onto the divergence-free factor , at which point it may be removed. Moving the derivative over there as well, we now reduce to showing that On the other hand, the expression is a total derivative (as is divergence-free), and thus has vanishing integral. Thus it will remain to show that From Bernstein’s inequality, Exercise 2, and the triangle inequality one has for any time that as is finite, it thus suffices to establish the pointwise bound We split the left-hand side into the sum of and where we use the fact that .To treat (7), we use Exercise 2 to conclude that

and so the quantity (7) is , which is acceptable since . Now we turn to (5). This is a commutator of the form where . Observe that this commutator would vanish if were replaced by , thus we may write this commutator as where . If we write for a suitable Schwartz function of total mass one, we have writing , we thus have the bound But from Bernstein’s inequality and Exercise 2 we have and and so we see that (5) is also of size , which is acceptable since . A similar argument gives (6), and the claim follows.As shown by Constantin, E, and Titi, the Hölder regularity in the above result can be relaxed to Besov regularity, at least in non-endpoint cases:

Exercise 3Let . Define the Besov space to be the space of functions such that the Besov space norm Show that if is a weak solution to the Euler equations, the energy is conserved in time.

The endpoint case of the above exercise is still open; however energy conservation in the slightly smaller space is known thanks to the work of Cheskidov, Constantin, Friedlander, and Shvydkoy (see also this paper of Isett for further discussion, and this paper of Isett and Oh for an alternate argument that also works on Riemannian manifolds).

As observed by Isett (see also the recent paper of Colombo and de Rosa), the above arguments also give some partial information about the energy in the low regularity regime:

Exercise 4Let be a weak solution to the Euler equations for .

- (i) If , show that the energy is a function of time. (
Hint:express the energy as the uniform limit of .)- (ii) If , show that the energy is a function of time.

Exercise 5Let be a weak solution to the Navier-Stokes equations for some with initial data . Establish the energy identity

Remark 6An alternate heuristic derivation of the threshold for the Onsager conjecture is as follows. If , then from Exercise 2 we see that the portion of that fluctuates at frequency has amplitude at most ; in particular, the amount of energy at frequencies is at most . On the other hand, by the heuristics in Remark 11 of 254A Notes 3, the time needed for the portion of the solution at frequency to evolve to a higher frequency scale such as is of order . Thus the rate ofenergy fluxat frequency should be . For , the energy flux goes to zero as , and so energy cannot escape to frequency infinity in finite time.

Remark 7Yet another alternate heuristic derivation of the threshold arises by considering the dynamics of individual Fourier coefficients. Using a Fourier expansion the Euler equations may be written as In particular, the energy at a single Fourier mode evolves according to the equation If , then we have for any , hence by Plancherel’s theorem which suggests that (up to logarithmic factors) one would expect to be of magnitude about . Onsager posited that for typically “turbulent” or “chaotic” flows, the main contributions to (8) come when have magnitude roughly comparable to that of , and that the various summands should not be correlated strongly to each other. For , one might expect about significant terms in the sum, which according to standard “square root cancellation heuristics” (cf. the central limit theorem) suggests that the sum is about as large as times the main term. Thus the total flux of energy in or out of a single mode would be expected to be of size , and so the total flux in or out of the frequency range (which consists of modes ) should be about . As such, for the energy flux should decay in and so there is no escape of energy to frequency infinity, whereas for such an escape should be possible. Related heuristics can also support Kolmogorov’s 1941 model of the distribution of energy in the vanishing viscosity limit; see this blog post for more discussion. On the other hand, while Onsager did discuss the dynamics of individual Fourier coefficients in his paper, it appears that he arrived at the threshold by a more physical space based approach, a rigorous version of which was eventually established by Duchon and Robert; see this survey of Eyink and Sreenivasan for more discussion.

** — 2. The (local) isometric embedding problem — **

Before we develop the convex integration method for fluid equations, we first discuss the simpler (and historically earlier) instance of this technique for the isometric embedding problem for Riemannian manifolds. To avoid some topological technicalities that are not the focus of the present set of notes, we only consider the *local problem* of embedding a small neighbourhood of the origin in into Euclidean space .

Let be an open neighbourhood of in . A (smooth) Riemannian metric on , when expressed in coordinates, is a family of smooth maps for , such that for each point , the matrix is symmetric and positive definite. Any such metric gives the structure of an (incomplete) Riemannian manifold . An *isometric embedding* of this manifold into a Euclidean space is a map which is continuously differentiable, injective, and obeys the equation

The *isometric embedding problem* asks, given a Riemannian manifold such as , whether there is an isometric embedding from this manifold to a Euclidean space ; for simplicity we only discuss the simpler *local isometric embedding problem* of constructing an isometric immersion of into for some sufficiently small neighbourhood of the origin. In particular for the local problem we do not need to worry about injectivity since (9) ensures that the derivative map is injective at the origin, and hence is injective near the origin by the inverse function theorem (indeed it is an immersion near the origin).

It is a celebrated theorem of Nash (discussed in this previous blog post) that the isometric embedding problem is possible in the smooth category if the dimension is large enough. For sake of discussion we just present the local version:

Theorem 8 (Nash embedding theorem)Suppose that is sufficiently large depending on . Then for any smooth metric on a neighbourhood of the origin, there is a smooth local isometric embedding on some smaller neighbourhood of the origin.

The optimal value of depending on is not completely known, but it grows roughly quadratically in . Indeed, in this paper of Günther it is shown that one can take

In the other direction, one cannot take below :

Proposition 9Suppose that . Then there exists a smooth Riemannian metric on an open neighbourhood of the origin in such that there is no smooth embedding from any smaller neighbourhood of the origin to .

*Proof:* Informally, the reason for this is that the given field has degrees of freedom (which is the number of independent fields after accounting for the symmetry ), but there are only degrees of freedom for the unknown . To make this rigorous, we perform a Taylor expansion of both and around the origin up to some large order , valid for a sufficiently small neighbourhood :

Remark 10If one replaces “smooth” with “analytic”, one can reverse the arguments here using the Cauchy-Kowaleski theorem and show that any analytic metric on can be locally analytically embedded into ; this is a classical result of Cartan and Janet.

Apart from the slight gap in dimensions, this would seem to settle the question of when a -dimensional metric may be locally isometrically embedded in . However, all of the above arguments required the immersion map to be smooth (i.e., ), whereas the definition of an isometric embedding only required the regularity of .

It is a remarkable (and somewhat counter-intuitive) result of Nash and Kuiper that if one only requires the embedding to be in , then one can embed into a much lower dimensional space:

Theorem 11 (Nash-Kuiper embedding theorem)Let . Then for any smooth metric on a neighbourhood of the origin, there is a local isometric embedding on some smaller neighbourhood of the origin.

Nash originally proved this theorem with the slightly weaker condition ; Kuiper then obtained the optimal condition . The case fails due to curvature obstructions; for instance, if the Riemannian metric has positive scalar curvature, then small Riemannian balls of radius will have (Riemannian) volume slightly less than their Euclidean counterparts, whereas any embedding into will preserve both Riemannian length and volume, preventing such an isometric embedding from existing.

Remark 12One striking illustration of the distinction between the and smooth categories comes when considering isometric embeddings of the round sphere (with the usual metric) into Euclidean space . It is a classical result (see e.g., Spivak’s book) that the only isometric embeddings of in are the obvious ones coming from composing the inclusion map with an isometry of the Euclidean space; however, the Nash-Kuiper construction allows one to create an embedding of into an arbitrarily small ball! Thus the embedding problem lacks the “rigidity” of the embedding problem. This is an instance of a more general principle that nonlinear differential equations such as (10) can become much less rigid when one weakens the regularity hypotheses demanded on the solution.

To prove this theorem we work with a relaxation of the isometric embedding problem. We say that is a *short isometric embedding* on if , solve the equation

Proposition 13Let , and let be a smooth Riemannian metric on a neighbourhood of the origin in . There is at least one short isometric embedding .

*Proof:* Set and for a sufficiently small , where is the standard embedding, and the Euclidean metric on ; this will be a short isometric embedding on some neighbourhood of the origin.

To create a true isometric embedding , we will first construct a sequence of short embeddings with converging to zero in a suitable sense, and then pass to a limit. The key observation is then that by using the fact that the positive matrices lie in the convex hull of the rank one matrices, one can add a high frequency perturbation to the first component of a short embedding to largely erase the error term , replacing it instead with a much higher frequency error.

We now prove Theorem 11. The key iterative step is

Theorem 14 (Iterative step)Let , let be a closed ball in , let be a smooth Riemannian metric on , and let be a short isometric embedding on . Then for any , one can find a short isometric embedding to (11) on with

Suppose for the moment that we had Theorem 14. Starting with the short isometric embedding on a ball provided by Proposition 13, we can iteratively apply the above theorem to obtain a sequence of short isometric embeddings on with

for . From this we see that converges uniformly to zero, while converges in norm to a limit , which then solves (10) on , giving Theorem 11. (Indeed, this shows that the space of isometric embeddings is dense in the space of short maps in the topology.)We prove Theorem 14 through a sequence of reductions. Firstly, we can rearrange it slightly:

Theorem 15 (Iterative step, again)Let , let be a closed ball in , let be a smooth immersion, and let be a smooth Riemannian metric on . Then there exists a sequence of smooth immersions for obeying the bounds uniformly on for , where denotes a quantity that goes to zero as (for fixed choices of ).

Let us see how Theorem 15 implies Theorem 14. Let the notation and hypotheses be as in Theorem 14. We may assume to be small. Applying Theorem 15 with replaced by (which will be positive definite for small enough), we obtain a sequence of smooth immersions obeying the estimates

If we set then is smooth, symmetric, and (from (13)) will be positive definite for large enough. By construction, we thus have solving (11), and Theorem 14 follows.To prove Theorem 15, it is convenient to break up the metric into more “primitive” pieces that are rank one matrices:

Lemma 16 (Rank one decomposition)Let be a closed ball in , and let be a smooth Riemannian metric on . Then there exists a finite collection of unit vectors in , and smooth functions , such that for all . Furthermore, for each , at most of the are non-zero.

Remark 17Strictly speaking, the unit vectors should belong to the dual space of rather than itself, in order to have the index appear as subscripts instead of superscripts. A similar consideration occurs for the frequency vectors from Remark 7. However, we will not bother to distinguish between and here (since they are identified using the Euclidean metric).

*Proof:* Fix a point in . Then the matrix is symmetric and positive definite; one can thus write , where is an orthonormal basis of (column) eigenvectors of and are the eigenvalues (we suppress for now the dependence of these objects on ). Using the parallelogram identity

Remark 18Informally, Lemma 16 lets one synthesize a metric as a “convex integral” of rank one pieces, so that if the problem at hand has the freedom to “move” in the direction of each of these rank one pieces, then it also has the freedom to move in the direction , at least if one is working in low enough regularities that one can afford to rapidly change direction from one rank one perturbation to another. This convex integration technique was formalised by Gromov in his interpretation of the Nash-Kuiper method as part of his “-principle“, which we will not discuss further here.

One can now deduce Theorem 15 from

Theorem 19 (Iterative step, rank one version)Let , let be a closed ball in , let be a smooth immersion, let be a unit vector, and let be smooth. Then there exists a sequence of smooth immersions for obeying the bounds uniformly on for . Furthermore, the support of is contained in the support of .

Indeed, suppose that Theorem 19 holds, and we are in the situation of Theorem 15. We apply Lemma 16 to obtain the decomposition

with the stated properties. On taking traces we see that for all . Applying Theorem 19 times (and diagonalising the sequences as necessary), we obtain sequences of smooth immersions for such that and one has an such that the support of is contained in that of . The claim then follows from the triangle inequality, noting that the implied constant in (12) will not depend on because of the bounded overlap in the supports of the .It remains to prove Theorem 19. We note that the requirement that be an immersion will be automatic from (15) for large enough since was already an immersion, making the matrix positive definite uniformly for , and this being unaffected by the addition of the perturbation and the positive semi-definite rank one matrix .

Writing , it will suffice to find a sequence of smooth maps supported in the support of and obeying the approximate difference equation

and the bounds uniformly on .
To locate these functions , we use the *method of slow and fast variables*. First we observe by applying a rotation that we may assume without loss of generality that is the unit vector , thus . We then use the ansatz

Remark 20Informally, one should think of functions that are independent of the fast variable as being of “low frequency”, and conversely functions that have mean zero in the fast variable (thus for all ) as being of “high frequency”. Thus for instance any smooth function on can be uniquely decomposed into a “low frequency” component and a “high frequency” component, with the two components orthogonal to each other. In later sections we will start inverting “fast derivatives” on “high frequency” functions, which will effectively gain important factors of in the analysis. See also the table below for the dictionary between ordinary physical coordinates and fast-slow coordinates.

Position | Slow variable |

Fast variable | Fast variable |

Function | Function |

Low-frequency function | Function independent of |

High-frequency function | Function mean zero in |

N/A | Slow derivative |

N/A | Fast derivative |

If we expand out using the chain rule, using and to denote partial differentiation in the coordinates of the slow and fast variables respectively, and noting that all terms with at least one power of can be absorbed into the error, we see that we will be done as long as we can construct to obey the bounds

and solve the exact equation where are viewed as functions of the slow variable only. The original approach of Nash to solve this equation was to use a function that was orthogonal to the entire gradient of , thus for . Taking derivatives in one would conclude that and similarly and one now just had to solve the equation For this, Nash used a “spiral” construction where were unit vectors varying smoothly with respect to the slow variable; this obeys (22) and (19), and would also obey (21) if the vectors and were both always orthogonal to the entire gradient of . This is not possible in (as cannot then support linearly independent vectors), but there is no obstruction for :

Lemma 21 (Constructing an orthogonal frame)Let be an immersion. If , then there exist smooth vector fields such that at every point , are unit vectors orthogonal to each other and to for .

*Proof:* Applying the Gram-Schmidt process to the linearly independent vectors for , we can find an orthonormal system of vectors , depending smoothly on , whose span is the same as the span of the . Our task is now to find smooth functions solving the system of equations

For this is possible at the origin from the Gram-Schmidt process. Now we extend in the direction to the line segment . To do this we evolve the fields by the parallel transport ODE

on this line segment. From the Picard existence and uniqueness theorem we can uniquely extend smoothly to this segment with the specified initial data at , and a simple calculation using Gronwall’s inequality shows that the system of equations (23), (24), (25) is preserved by this evolution. Then, one can extend to the disk by using the previous extension to the segment as initial data and solving the parallel transport ODE Iterating this procedure we obtain the claim.This concludes Nash’s proof of Theorem 11 when . Now suppose that . In this case we cannot locate two unit vector fields orthogonal to each other and to the entire gradient of ; however, we may still obtain one such vector field by repeating the above arguments. By Gram-Schmidt, we can then locate a smooth unit vector field which is orthogonal to and to for , but for which the quantity is positive. If we use the “Kuiper corrugation” ansatz

for some smooth functions , one is reduced to locating such functions that obey the bounds and the ODEThis can be done by an explicit construction:

Exercise 22 (One-dimensional corrugation)For any positive and any , show that there exist smooth functions solving the ODE and which vary smoothly with (even at the endpoint ), and obey the bounds (Hint:one can renormalise . The problem is basically to locate a periodic function mapping to the circle of mean zero and Lipschitz norm that varies smoothly with . Choose for some smooth and small that is even and compactly supported in with mean zero on each interval, and then choose to be odd.)

This exercise supplies the required functions , completing Kuiper’s proof of Theorem 11 when .

Remark 23For sake of discussion let us restrict attention to the surface case . For the local isometric embedding problem, we have seen that we have rigidity at regularities at or above , but lack of regularity at . The precise threshold at which rigidity occurs is not completely known at present: a result of Borisov (also reproven here) gives rigidity at the level for , while a result of de Lellis, Inauen, and Szekelyhidi (building upon a series of previous results) establishes non-rigidity when . For recent results in higher dimensions, see this paper of Cao and Szekelyhidi.

** — 3. Low regularity weak solutions to Navier-Stokes in high dimensions — **

We now turn to constructing solutions (or near-solutions) to the Euler and Navier-Stokes equations. For minor technical reasons it is convenient to work with solutions that are periodic in both space and time, and normalised to have zero mean at every time (although the latter restriction is not essential for our arguments, since one can always reduce to this case after a Galilean transformation as in 254A Notes 1). Accordingly, let denote the periodic spacetime

and let denote the space of smooth periodic functions that have mean zero and are divergence-free at every time , thus and We use as an abbreviation for for various vector spaces (the choice of which will be clear from context).Let (for now, our discussion will apply both to the Navier-Stokes equations and the Euler equations ). Smooth solutions to Navier-Stokes equations then take the form

for some and smooth . Here of course denotes the spatial Laplacian.Much as we replaced the equation (10) in the previous section with (11), we will consider the relaxed version

of the Navier-Stokes equations, where we have now introduced an additional field , known as the*Reynolds stress*(cf. the Cauchy stress tensor from 254A Notes 0). If , , are smooth solutions to (26), (27), (28), with having mean zero at every time, then we call a

*Navier-Stokes-Reynolds flow*(or

*Euler-Reynolds flow*, if ). Note that if then we recover a solution to the true Navier-Stokes equations. Thus, heuristically, the smaller is, the closer and should become to a solution to the true Navier-Stokes equations. (The Reynolds stress tensor here is a rank tensor, as opposed to the rank tensor used in the previous section to measure the failure of isometric embedding, but this will not be a particularly significant distinction.)

Note that if is a Navier-Stokes-Reynolds flow, and , , are smooth functions, then will also be a Navier-Stokes-Reynolds flow if and only if has mean zero at every time, and obeys the difference equation

When this occurs, we say that is a*difference Navier-Stokes-Reynolds flow*at .

It will be thus of interest to find, for a given , difference Navier-Stokes-Reynolds flows at with small, as one could hopefully iterate this procedure and take a limit to construct weak solutions to the true Euler equations. The main strategy here will be to choose a highly oscillating (and divergence-free) correction velocity field such that approximates up to an error which is also highly oscillating (and somewhat divergence-free). The effect of this error can then eventually be absorbed efficiently into the new Reynolds stress tensor . Of course, one also has to manage the other terms , , , appearing in (29). In high dimensions it turns out that these terms can be made very small in norm, and can thus be easily disposed of. In three dimensions the situation is considerably more delicate, particularly with regards to the and terms; in particular, the transport term term is best handled by using a local version of Lagrangian coordinates. We will discuss these subtleties in later sections.

To execute above strategy, it will be convenient to have an even more flexible notion of solution, in which is no longer required to be perfectly divergence-free and mean zero, and is also allowed to be slightly inaccurate in solving (29). We say that is an *approximate difference Navier-Stokes-Reynolds flow* at if , , are smooth functions obeying the system

If the error terms , as well as the mean of , are all small, one can correct an approximate difference Navier-Stokes-Reynolds flow to a true difference Navier-Stokes-Reynolds flow with only small adjustments:

Exercise 24 (Removing the error terms)Let be a Navier-Stokes-Reynolds flow, and let be an approximate difference Navier-Stokes-Reynolds flow at . Show that is an approximate difference Navier-Stokes-Reynolds flow at , where and is the mean of , thus (Hint:one will need at some point to show that has mean zero in space at every time; this can be achieved by integrating (32) in space.)

Because of this exercise we will be able to tolerate the error terms if they (and the mean ) are sufficiently small.

As a simple corollary of Exercise 24, we have the following analogue of Proposition 13:

Proposition 25Let . Then there exist smooth fields , such that is a Navier-Stokes-Reynolds flow. Furthermore, if is supported in for some compact time interval , then can be chosen to also be supported in this region.

*Proof:* Clearly is an approximate difference Navier-Stokes-Reynolds flow at , where

Now, we show that, in sufficiently high dimension, a Navier-Stokes-Reynolds flow can be approximated (in an sense) as the limit of Navier-Stokes-Reynolds flows , with the Reynolds stress going to zero.

Proposition 26 (Weak improvement of Navier-Stokes-Reynolds flows)Let , and let be sufficiently large depending on . Let be a Navier-Stokes-Reynolds flow. Then for sufficiently large , there exists a Navier-Stokes-Reynolds flow obeying the estimates for all , and such that Furthermore, if is supported in for some interval , then one can arrange for to be supported on for any interval containing in its interior (at the cost of allowing the implied constants in the above to depend also on ).

This proposition can be viewed as an analogue of Theorem 14. For an application at the end of this section it is important that the implied constant in (36) is uniform in the choice of initial flow . The estimate (35) can be viewed as asserting that the new velocity field is oscillating at frequencies , at least in an sense. In the next section, we obtain a stronger version of this proposition with more quantitative estimates that can be iterated to obtain higher regularity weak solutions.

To simplify the notation we adopt the following conventions. Given an -dimensional vector of differential operators, we use to denote the -tuple of differential operators with . We use to denote the -tuple formed by concatenating for . Thus for instance the estimate (35) can be abbreviated as

for all . Informally, one should read the above estimate as asserting that is bounded in with norm , and oscillates with frequency in time and in space (or equivalently, with a temporal wavelength of and a spatial wavelength of ).
*Proof:* We can assume that is non-zero, since if we can just take . We may assume that is supported in for some interval (which may be all of ), and let be an interval containing in its interior. To abbreviate notation, we allow all implied constants to depend on .

Assume is sufficiently large. Using the ansatz

and the triangle inequality, it suffices to construct a difference Navier-Stokes-Reynolds flow at supported on and obeying the bounds for all .
It will in fact suffice to construct an *approximate* difference Navier-Stokes-Reynolds flow at supported on and obeying the bounds

To construct this approximate solution, we again use the method of fast and slow variables. Set , and introduce the fast-slow spacetime , which we coordinatise as ; we use to denote partial differentiation in the coordinates of the slow variable , and to denote partial differentiation in the coordinates of the fast variable . We also use as shorthand for . Define an *approximate fast-slow solution* to the difference Navier-Stokes-Reynolds equation at (at the frequency scale ) to be a tuple of smooth functions , , that obey the system of equations

Let denote the tuple . Suppose that for any sufficiently large , we can construct an approximate fast-slow solution to the difference equations at supported on supported on that obeys the bounds

for all . (Informally, the presence of the derivatives means that the fields involved are allowed to oscillate in time at wavelength , in the slow variable at wavelength , and in the fast variable at wavelength .) From (46) and the choice of we then have for all , and similarly for (48), (49). (Note here that there was considerable room in the estimates with regards to regularity in the variable; this room will be exploited more in the next section.) For any shift , we see from the chain rule that is an approximate difference Navier-Stokes-Reynolds flow at supported on , where Also from (46) and Fubini’s theorem we have and similarly for all . By Markov’s inequality and (52), we see that for each , we have for all outside of an exceptional set of measure (say) . Similarly for the other equations above. Applying the union bound, we can then find a such that obeys all the required bounds (37)–(41) simultaneously for all . (This is an example of the probabilistic method, originally developed in combinatorics; one can think of probabilistically as a shift drawn uniformly at random from the torus , in order to relate the fast-slow Lebesgue norms to the original Lebesgue norms .)
It remains to construct an approximate fast-slow solution supported on with the required bounds (46)–(51). Actually, in this high-dimensional setting we can afford to simplify the situation here by removing some of the terms (and in particular eliminating the role of the reference velocity field ). Define a *simplified fast-slow solution* at to be a tuple of smooth functions on obeying the simplified equations

Now we need to construct a simplified fast-slow solution supported on obeying the bounds (56)–(60). We do this in stages, first finding a solution that cancels off the highest order terms and , and also such that has mean zero in the fast variable (so that it is “high frequency” in the sense of Remark 20). This still leads to fairly large values of and , but we will then apply a “divergence corrector” to almost completely eliminate , followed by a “stress corrector” that almost completely eliminates , at which point we will be done.

We turn to the details. Our preliminary construction of the velocity field will be a “Mikado flow”, consisting of flows along narrow tubes. (Earlier literature used other flows, such as Beltrami flows; however, Mikado flows have the advantage of being localisable to small subsets of spacetime, which is particularly useful in high dimensions.) We need the following modification of Lemma 16:

Exercise 27Let be a compact subset of the space of positive definite matrices . Show that there exist non-zero lattice vectors and smooth functions for some such that for all . (This decomposition is essentially due to de Lellis and Szekelyhidi. The subscripting and superscripting here is reversed from that in Lemma 16; this is because we are now trying to decompose a rank tensor rather than a rank tensor.)

We would like to apply this exercise to the matrix with entries . We thus need to select the pressure so that this matrix is positive definite. There are many choices available for this pressure; we will take

where is the Frobenius norm of . Then is smooth and is positive definite on all of the compact spacetime (recall that we can assume to not be identically zero), and in particular ranges in a compact subset of positive definite matrices. Applying the previous exercise and composing with the function , we conclude that there exist non-zero lattice vectors and smooth functions for some such that for all . As depend only on , and is a component of , all norms of these quantities are bounded by ; they are independent of . Furthermore, on taking traces and integrating on , we obtain the estimate (note here that the implied constant is uniform in , ). By applying a smooth cutoff in time that equals on and vanishes outside of to (and applying to ), we may assume that the are supported in ; this redefines slightly to but this does not significantly affect the estimate (64).Now for each , the closed subgroup is a one-dimensional subset of , so the -neighbourhood of this subgroup has measure ; crucially, this will be a large negative power of when is very large. let be a translate of this -neighbourhood such that all the are disjoint; this is easily accomplished by the probabilistic method for large enough, translating each of the by an independent random shift and noting that the probability of a collision goes to zero as (here we need the fact that we are in at least three dimensions).

Let be a large integer (depending on ) to be chosen later. For any , let be a scalar test function supported on that is constant in the direction, thus

and is not identically zero, which implies that the iterated Laplacian of is also not identically zero (thanks to the unique continuation property of harmonic functions). We can normalise so that and we can also arrange to have the bounds for all (basically by first constructing a version of on a standard cylinder and the applying an affine transformation to map onto ).Let denote the function

intuitively this represents the velocity field of a fluid traveling along the tube , with the presence of the Laplacian ensuring that this function is extremely well balanced (for instance, it will have mean zero, and thus “high frequency” in the sense of Remark 20). Clearly is divergence free, and one also has the steady-state Euler equation and the normalisation and for all and . If we then set then one easily checks that is a simplified fast-slow solution supported in . Direct calculation using the Leibniz rule then gives the bounds for all , while from (64) one has (note here that the implied constant is uniform in ).This looks worse than (56)–(60). However, observe that is supported on the set , which has measure , which for large enough can be taken to be (say) . Thus by Cauchy-Schwarz one has

for all . Also, from construction (particularly (66)) we see that is of mean zero in the variable (thus it is “high frequency” in the sense of Remark 20).We are now a bit closer to (56)–(60), but our bounds on are not yet strong enough. We now apply a “divergence corrector” to make much smaller. Observe from construction that where

and is supported on and obeys the estimates for all . Observe that We abbreviate the differential operator as . Iterating the above identity times, we obtain where and In particular, is supported in . Observe that is a simplified fast-slow solution supported in , where From (72) we have so in particular for large enough for any . Meanwhile, another appeal to (72) yields for any , and hence by (67) and the triangle inequality Similarly one has Since continues to be supported on the thin set , we can apply Hölder as before to conclude that for any . Also, from (73) and Hölder we haveWe have now achieved the bound (59); the remaining estimate that needs to be corrected for is (60). This we can do by a modification of the previous argument, where we now work to reduce the size of rather than . Observe that as is “high frequency” (mean zero in the variable), one can write

where is the linear operator on smooth vector-valued functions on of mean zero defined by the formula Note that also has mean zero. We can thus iterate and obtain where and is a smooth function whose exact form is explicit but irrelevant for our argument. We then see that is a simplified fast-slow solution supported in . Since is bounded in , we see from (69) that and if is large enough. Thus obeys the required bounds (56)–(60), concluding the proof.
As an application of this proposition we construct a low-regularity weak solution to high-dimensional Navier-Stokes that does not obey energy conservation. More precisely, for any , let be the Banach space of periodic functions which are divergence free, and of mean zero at every time. For , define a *time-periodic weak solution* of the Navier-Stokes (or Euler, if ) equations to be a function that solves the equation

Corollary 28 (Low regularity non-trivial weak solutions)Assume that the dimension is sufficiently large. Then for any , there exists a periodic weak solution to Navier-Stokes which equals zero at time , but is not identically zero. In particular, periodic weak solutions are not uniquely determined by their initial data, and do not necessarily obey the energy inequality

*Proof:* Let be an element of that is supported on and is not identically zero (it is easy to show that such an element exists). By Proposition 25, we may then find a Navier-Stokes-Reynolds flow also supported on . Let be sufficiently large. By applying Proposition 26 repeatedly (with say ) and with a sufficiently rapidly increasing sequence , we can find a sequence of Navier-Stokes-Reynolds flows supported on (say) obeying the bounds

** — 4. High regularity weak solutions to Navier-Stokes in high dimensions — **

Now we refine the above arguments to give a higher regularity version of Corollary 28, in which we can give the weak solutions almost half a derivative of regularity in the Sobolev scale:

Theorem 29 (Non-trivial weak solutions)Let , and assume that the dimension is sufficiently large depending on . Then for any , there exists a periodic weak solution which equals zero at time , but is not identically zero. In particular, periodic weak solutions are not uniquely determined by their initial data, and do not necessarily obey the energy inequality (74).

This result is inspired by a three-dimensional result of Buckmaster and Vicol (with a small value of ) and a higher dimensional result of Luo (taking , and restricting attention to time-independent solutions). In high dimensions one can create fairly regular functions which are large in type norms but tiny in type norms; when using the Sobolev scale to control the solution (and type norms to measure an associated stress tensor), this has the effect of allowing one to treat as negligible the linear terms in (variants of) the Navier-Stokes equation, as well as interaction terms between low and high frequencies. As such, the analysis here is simpler than that required to establish the Onsager conjecture. The construction used to prove this theorem shows in fact that periodic weak solutions are in some sense “dense” in , but we will not attempt to quantify this fact here.

In the proof of Corollary 28, we took the frequency scales to be extremely rapidly growing in . This will no longer be good enough for proving Theorem 29, and in fact we need to take a fairly dense set of frequency scales in which for a small . In order to do so, we have to replace Proposition 26 with a more quantitative version in which the dependence of bounds on the size of the original Navier-Stokes-Reynolds flow is made much more explicit.

We turn to the details. We select the following parameters:

- A regularity ;
- A quantity , assumed to be sufficiently small depending on ;
- An integer , assumed to be sufficiently large depending on ; and
- A dimension , assumed to be sufficiently large depending on .

As before, we abbreviate as . We write for the spatial gradient to distinguish it from the time derivative .

The main iterative statement (analogous to Theorem 14) starts with a Navier-Stokes-Reynolds flow oscillating at spatial scales up to some small wavelength , and modifies it to a Navier-Stokes-Reynolds flow oscillating at a slightly smaller wavelength , with a smaller Reynolds stress. It can be viewed as a more quantitative version of Proposition 26.

Theorem 30 (Iterative step)Let be sufficiently large depending on the parameters . Set . Suppose that one has a Navier-Stokes-Reynolds flow obeying the estimates for some . Set . Then there exists a Navier-Stokes-Reynolds flow obeying the estimates Furthermore, if is supported on for some interval , then one can ensure that is supported in , where is the -neighbourhood of .

Let us assume Theorem 30 for the moment and establish Theorem 29. Let be chosen to be supported on (say) and not be identically zero. By Proposition 25, we can then find a Navier-Stokes-Reynolds flow supported on . Let be a sufficiently large parameter, and set , then the hypotheses (75), (76) will be obeyed for large enough. Set for all . By iteratively applying Theorem 30, we may find a sequence of Navier-Stokes-Reynolds flows, all supported on (say) , obeying the bounds

for . In particular, the converge weakly to zero on , and we have the bound from Plancherel’s theorem, and hence by Sobolev embedding in time Thus converges strongly in (and in particular also in for some ) to some limit ; as the are all divergence-free, is also. From applying Leray projections to (26) one has Taking weak limits we conclude that is a weak solution to Navier-Stokes. Also, from construction one has (say), and so for large enough is not identically zero. This proves Theorem 29.It remains to establish Theorem 30. It will be convenient to introduce the intermediate frequency scales

where is slightly larger than , and is slightly smaller than (and constrained to be integer).Before we begin the rigorous argument, we first give a heuristic explanation of the numerology. The initial solution has about degrees of regularity controlled at . For technical reasons we will upgrade this to an infinite amount of regularity, at the cost of worsening the frequency bound slightly from to . Next, to cancel the Reynolds stress up to a smaller error , we will perturb by some high frequency correction , basically oscillating at spatial frequency (and temporal frequency ), so that is approximately equal to (minus a pressure term) after averaging at spatial scales . Given the size bound (76), one expects to achieve this with of norm about . By exploiting the small gap between and , we can make concentrate on a fairly small measure set (of density something like ), which in high dimension allows us to make linear terms such as and (as well as the correlation terms and ) negligible in size (as measured using type norms) when compared against quadratic terms such as (cf. the proof of Proposition 26). The defect will then oscillate at frequency , but can be selected to be of size about in norm, because can choose to cancel off all the high-frequency (by which we mean or greater) contributions to this term, leaving only low frequency contributions (at frequencies or below). Using the ellipticity of the Laplacian, we can then express this defect as where the norm of is of order

When , this is slightly less than , allowing one to close the argument.
We now turn to the rigorous details. In a later part of the argument we will encounter a *loss of derivatives*, in that the new Navier-Stokes-Reynolds flow has lower amounts of controlled regularity (in both space and time) than the Navier-Stokes-Reynolds flow used to construct it. To counteract this loss of derivatives we need to perform an initial *mollification step*, which improves the amount of regularity from derivatives in space and time to an unlimited number of derivatives in space and time, at the cost of worsening the estimates on slightly (basically by replacing with ).

Proposition 31 (Mollification)Let the notation and hypotheses be as above. Then we can find a Navier-Stokes-Reynolds flow obeying the estimates and for all , and such that and Furthermore, if is supported on for some interval , then one can ensure that is supported in , where is the -neighbourhood of .

We remark that this sort of mollification step is now a standard technique in any iteration scheme that involves loss of derivatives, including the Nash-Moser iteration scheme that was first used to prove Theorem 8.

*Proof:* Let be a bump function (depending only on ) supported on the region of total mass , and define the averaging operator on smooth functions by the formula

Observe that are convolution operators and thus commute with each other and with the partial derivatives . If we apply the operator to (26), (27), (28), we see that is Navier-Stokes-Reynolds flow, where

Since , is a linear combination of the operators . In particular, we see that is supported on .We abbreviate . For any , we have

and therefore deduce the bounds for any , thanks to Young’s inequality. A similar application of Young’s inequality gives for all and .From (81) and decomposing as linear combinations of , we have

for any , and hence (77) follows from (75). In a similar spirit, from (82), (75) one has if is large enough; similar arguments give the same bound if is deleted. This gives (79), (80).Finally we prove (78). By the triangle inequality it suffices to show that

and for any . The claim (83) follows from (81), (76), after writing as a linear combination of and noting that . For (84), if we again write as a linear combination of and uses (81) and the Leibniz rule, one can bound the left-hand side of (84) by and hence by (75) (bounding by ) this is bounded by This gives (84) when . For , we rewrite the expression as The contribution of the first term to (84) can be bounded using (82), (81) (splitting ) by which by the Leibniz rule, bounding by , and (75) is bounded by which is again an acceptable contribution to (84) since is large. The other terms are treated similarly.We return to the proof of Theorem 30. We abbreviate . Let be the Navier-Stokes-Reynolds flow constructed by Proposition 31. By using the ansatz

and the triangle inequality, it will suffice to locate a difference Navier-Stokes-Reynolds flow at supported on , obeying the estimates From (77) we have so by the triangle inequality we can replace (85) by and then (85), (87) may then be replaced by the single estimate (say). By using Exercise 24 as in the previous section, it then suffices to construct an*approximate*difference Navier-Stokes-Reynolds flow to the difference equation at supported on supported in obeying the bounds (86), (88), (89) and

Now, we pass to fast and slow variables. Let denote the tuple

informally, the use of is consistent with oscillations in time of wavelength , in the slow variable of wavelength , and in the fast variable of wavelength .

Exercise 32By using the method of fast and slow variables as in the previous section, show that to construct the approximate Navier-Stokes-Reynolds flow at obeying the bounds (86), (88), (89), (90), (91), it suffices to locate an approximate fast-slow solution to the difference Navier-Stokes-Reynolds equation at (at frequency scale rather than ) and supported in that obey the bounds

As in the previous section, we can then pass to simplified fast-slow soutions:

Exercise 33Show that to construct the approximate fast-slow solution to the difference equation at obeying the estimates of the previous exercise, it will in fact suffice to locate a simplified fast-slow solution at (again at frequency scale ) supported on , obeying the bounds (92), (93), (95), (96) and (Hint:one will need the estimate from Proposition 31.)

Now we need to construct a simplified fast-slow solution at supported on obeying the bounds (92), (93), (95), (96), (97). As in the previous section, we do this in stages, first finding a solution that cancels off the top order terms and , and also such that is “ high frequency” (mean zero in ). Then we apply a divergence corrector to completely eliminate , followed by a stress corrector that almost completely eliminates .

As before, we need to select so that is positive definite. In the previous section we essentially took to be a large multiple of , but now we will need good control on the derivatives of , which requires a little more care. Namely, we will need the following technical lemma:

Lemma 34 (Smooth polar-type decomposition)There exists a factorisation , where , are smooth, supported on , and obey the estimates

*Proof:* We may assume that is not identically zero, since otherwise the claim is trivial. For brevity we write and . From (78) we have

Strictly speaking we are not quite done because is not supported in , but if one applies a smooth cutoff function in time that equals on (where is supported in time) and vanishes outside of , we obtain the required support property without significantly affecting the estimates.

Let be the factorisation given by the above lemma. If we set for a sufficiently large constant depending only on , then

For large enough, we see from (99) that the matrix with entries takes values in a compact subset of positive definite matrices) that depends only on . Applying Exercise 27, we conclude that there exist non-zero lattice vectors and smooth functions for some (depending only on ) such that for all , and furthermore (from (99) and the chain rule) we have the derivative estimates for . Setting , we thus have and from the Leibniz rule and (98) we have for .Let be the disjoint tubes in from the previous section, with width rather than . Construct the functions as in the previous section, and again set

Then as before, each is divergence free, and obeys the identities (65), (66) and the b ounds and the normalisation and for all and . As in the preceding section, we then set and one easily checks that is a simplified fast-slow solution supported in . Direct calculation using the Leibniz rule and (105), (107) then gives the bounds As before, is “high frequency” (mean zero in the variable). Also, is supported on the set , and for large enough the latter set has measure (say) . Thus by Cauchy-Schwarz (in just the variable) one hasThe divergence corrector can be applied without difficulty:

Exercise 35Show that there is a simplified fast-slow solution supported in obeying the estimates

The crucial thing here is the tiny gain in the third estimate, with the first factor coming from a “slow” derivative and the second factor coming from essentially inverting a “fast” derivative .

Finally, we apply a stress corrector:

Exercise 36Show that there is a simplified fast-slow solution supported in obeying the estimates

Again, we have a crucial gain of coming from applying a slow derivative and inverting a fast one.

Since

(with implied constant in the exponent uniform in ) and , we see (for small enough) that and the desired estimates (92), (93), (95), (96), (97) now follow.

** — 5. Constructing low regularity weak solutions to Euler — **

Throughout this section, we specialise to the Euler equations in the three-dimensional case (although all of the arguments here also apply without much modification to as well). In this section we establish an analogue of Corollary 28:

Proposition 37 (Low regularity non-trivial weak solutions)There exists a periodic weak solution to Euler which equals zero at time , but is not identically zero.

This result was first established by de Lellis and Szekelyhidi. Our approach will deviate from the one in that paper in a number of technical respects (for instance, we use Mikado flows in place of Beltrami flows, and we place more emphasis on the method of fast and slow variables). A key new feature, which was not present in the high-dimensional Sobolev-scale setting, is that the material derivative term in the difference Euler-Reynolds equations is no longer negligible, and needs to be treated by working with an ansatz in Lagrangian coordinates (or equivalently, an ansatz transported by the flow). (This use of Lagrangian coordinates is implicit in the thesis of Isett, this paper of de Lellis and Szekelyhidi, and in the later work of Isett.)

Just as Corollary 28 was derived from Proposition 26, the above proposition may be derived from

Proposition 38 (Weak improvement of Euler-Reynolds flows)Let be an Euler-Reynolds flow supported on a strict compact subinterval . Let be another interval in containing in its interior. Then for sufficiently large , there exists a Euler-Reynolds flow supported in obeying the estimates for all , and such that also, we have a decomposition where are smooth functions obeying the bounds

The point of the decomposition (115) is that it (together with the smallness bounds (116)) asserts that the velocity correction is mostly “high frequency” in nature, in that its low frequency components are small. Together with (112), the bounds roughly speaking assert that it is only the frequency components of that can be large in norm. Unlike the previous estimates, it will be important for our arguments that is supported in a strict subinterval of , because we will not be able to extend Lagrangian coordinates periodically around the circle. Actually the long-time instability of Lagrangian coordinates causes significant technical difficulties to overcome when one wants to construct solutions in higher regularity Hölder spaces , and in particular for close to ; we discuss this in the next section.

Exercise 39Deduce Proposition 37 from Proposition 38. (The decomposition (116) is needed to keep close to in a very weak topology – basically the topology – but one which is still sufficent to ensure that the limiting solution constructed is not identically zero.)

We now begin the proof of Proposition 38, repeating many of the steps used to prove Proposition 26. As before we may assume that is non-zero, and that is supported in . We can assume that is also a strict subinterval of .

Assume is sufficiently large; by rounding we may assume that is a natural number. Using the ansatz

and the triangle inequality, it suffices to construct a difference Euler-Reynolds flow at supported on and obeying the bounds for all , and for which we have a decomposition obeying (116).As before, we permit ourselves some error:

Exercise 40Show that it suffices to construct anapproximatedifference Euler-Reynolds flow at supported on and obeying the bounds for , and for which we have a decomposition obeying (116).

It still remains to construct the approximate difference Euler-Reynolds flow obeying the claimed estimates. By definition, has to obey the system of equations

with a decomposition As is divergence-free, the first equation (122) may be rewritten as where is the material Lie derivative of , thus The lower order terms in (126) will turn out to be rather harmless; the main new difficulty is dealing with the material Lie derivative term . We will therefore invoke Lagrangian coordinates in order to convert the material Lie derivative into the more tractable time derivative (at the cost of mildly complicating all the other terms in the system).We introduce a “Lagrangian torus” that is an isomorphic copy of the Eulerian torus ; as in the previous section, we parameterise this torus by , and adopt the usual summation conventions for the indices . Let be a trajectory map for , that is to say a smooth map such that for every time , the map is a diffeomorphism and one obeys the ODE

for all . The existence of such a trajectory map is guaranteed by the Picard existence theorem (it is important here that is not all of the torus ); see also Exercise 1 from Notes 1. From (the periodic version of) Lemma 3 of Notes 1, we can ensure that the map is volume-preserving, thusRecall from Notes 1 that

- (i) Any Eulerian scalar field on can be pulled back to a Lagrangian scalar field on by the formula
- (ii) Any Eulerian vector field on can be pulled back to a Lagrangian vector field on by the formula where is the inverse of the matrix , defined by and
- (iii) Any Eulerian rank tensor on , can be pulled back to a Lagrangian rank tensor on by the formula

*pushforward*by ). One can compute how these pullbacks interact with divergences:

Exercise 41 (Pullback and divergence)

- (i) If is a smooth Eulerian vector field, show that the pullback of the divergence of equals the divergence of the pullback of : (
Hint:you will need to use the fact that is volume-preserving. Similarly to Lemma 3 and Exercise 4 of Notes 1, one can establish this either using the laws of integration or the laws of differentiation.).- (ii) Show that there exist smooth functions for with the following property: for any smooth Eulerian rank tensor on , with divergence , one has (In fact, is the Christoffel symbol associated with the pullback of the Euclidean metric – but we will not need this precise formula. The right-hand side may also be written (in Penrose abstract index notation) as , where is the covariant derivative associated to .)
- (iii) Show that for any smooth Eulerian vector fields on with , one has (Hint: use (i), (ii), and the Leibniz rule.)

As remarked upon in the exercise, these calculations can be streamlined using the theory of the covariant derivative in Riemannian geometry; we will not develop this theory further here, but see for instance these two blog posts.

If one now applies the pullback operation to the system (126), (123), (124), (125) (and uses Exercise 16 from Notes 1 to convert the material Lie derivative into the ordinary time derivative) one obtain the equivalent system

where denotes the rank tensor