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

For sake of discussion we will just work in the non-periodic domain , , although the arguments here can be adapted without much difficulty to the periodic setting. We will only work with solutions in which the pressure is normalised in the usual fashion:

Formally, the Euler equations (with normalised pressure) arise as the vanishing viscosity limit of the Navier-Stokes equations

that was studied in previous notes. However, because most of the bounds established in previous notes, either on the lifespan of the solution or on the size of the solution itself, depended on , 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 lying in for , but the bound on the norm is and so one could lose this regularity in the limit , at which point it is not clear how to ensure that the nonlinear term 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 . 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

becomes infinite at the final time , where 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.

** — 1. A priori bounds — **

We now develop some *a priori* bounds for very smooth solutions to Navier-Stokes that are uniform in the viscosity . Define an function to be a function that lies in every space; similarly define an function to be a function that lies in for every . Given divergence-free initial data , an mild solution to the Navier-Stokes initial value problem (3) is a solution that is an mild solution for all . From the (non-periodic version of) Corollary 40 of Notes 1, we know that for any divergence-free initial data , there is unique maximal Cauchy development , with infinite if is finite.

Here are our first bounds:

Theorem 1 (A priori bound)Let be an maximal Cauchy development to (3) with initial data .

- (i) For any integer , we have
Furthermore, if for a sufficiently small constant depending only on , then

- (ii) For any and integer , one has

The hypothesis that is integer can be dropped by more heavily exploiting the theory of paraproducts, but we shall restrict attention to integer for simplicity.

We now prove this theorem using the energy method. Using the Navier-Stokes equations, we see that and all lie in for any ; an easy iteration argument then shows that the same is true for all higher derivatives of also. This will make it easy to justify the differentiation under the integral sign that we shall shortly perform.

Let be an integer. For each time , we introduce the energy-type quantity

Here we think of as taking values in the Euclidean space . This quantity is of course comparable to , up to constants depending on . It is easy to verify that is continuously differentiable in time, with derivative

where we suppress explicit dependence on in the integrand for brevity. We now try to bound this quantity in terms of . We expand the right-hand side in coordinates using (3) to obtain

where

For , we can integrate by parts to move the operator onto and use the divergence-free nature of to conclude that . Similarly, we may integrate by parts for to move one copy of over to the other factor in the integrand to conclude

so in particular (note that as we are seeking bounds that are uniform in , we can’t get much further use out of beyond this bound). Thus we have

Now we expand out using the Leibniz rule. There is one dangerous term, in which all the derivatives in fall on the factor, giving rise to the expression

But we can locate a total derivative to write this as

and then an integration by parts using as before shows that this term vanishes. Estimating the remaining contributions to using the triangle inequality, we arrive at the bound

At this point we now need a variant of Proposition 35 from Notes 1:

Exercise 2Let be integers. For any , show that(

Hint:for or , use Hölder’s inequality. Otherwise, use a suitable Littlewood-Paley decomposition.)

Using this exercise and Hölder’s inequality, we see that

By Gronwall’s inequality we conclude that

for any and , which gives part (ii).

Now assume . Then we have the Sobolev embedding

which when inserted into (4) yields the differential inequality

or equivalently

for some constant (strictly speaking one should work with for some small which one sends to zero later, if one wants to avoid the possibility that vanishes, but we will ignore this small technicality for sake of exposition.) Since , we conclude that stays bounded for a time interval of the form ; this, together with the blowup criterion that must go to infinity as , gives part (i).

As a consequence, we can now obtain local existence for the Euler equations from smooth data:

Corollary 3 (Local existence for smooth solutions)Let be divergence-free. Let be an integer, and setThen there is a smooth solution , to (1) with all derivatives of in for appropriate . Furthermore, for any integer , one has

*Proof:* We use the compactness method, which will be more powerful here than in the last section because we have much higher regularity uniform bounds (but they are only local in time rather than global). Let be a sequence of viscosities going to zero. By the local existence theory for Navier-Stokes (Corollary 40 of Notes 1), for each we have a maximal Cauchy development , to the Navier-Stokes initial value problem (3) with viscosity and initial data . From Theorem 1(i), we have for all (if is small enough), and

for all . By Sobolev embedding, this implies that

and then by Theorem 1(ii) one has

for every integer . Thus, for each , is bounded in , uniformly in . By repeatedly using (3) and product estimates for Sobolev spaces, we see the same is true for , and for all higher derivatives of . In particular, all derivatives of are equicontinuous.

Using weak compactness (Proposition 2 of Notes 2), one can pass to a subsequence such that converge weakly to some limits , such that and all their derivatives lie in on ; in particular, are smooth. From the Arzelá-Ascoli theorem (and Proposition 3 of Notes 2), and converge locally uniformly to , and similarly for all derivatives of . One can then take limits in (3) and conclude that solve (1). The bound (5) follows from taking limits in (6).

Remark 4We are able to easily pass to the zero viscosity limit here because our domain has no boundary. In the presence of a boundary, we cannot freely differentiate in space as casually as we have been doing above, and one no longer has bounds on higher derivatives on and near the boundary that are uniform in the viscosity. Instead, it is possible for the fluid to form a thin boundary layer that has a non-trivial effect on the limiting dynamics. We hope to return to this topic in a future set of notes.

We have constructed a local smooth solution to the Euler equations from smooth data, but have not yet established uniqueness or continuous dependence on the data; related to the latter point, we have not extended the construction to larger classes of initial data than the smooth class . To accomplish these tasks we need a further *a priori* estimate, now involving *differences* of two solutions, rather than just bounding a single solution:

Theorem 5 (A priori bound for differences)Let , let be an integer, and let be divergence-free with norm at most . Letwhere is sufficiently small depending on . Let and be an solution to (1) with initial data (this exists thanks to Corollary 3), and let and be an solution to (1) with initial data . Then one has

Note the asymmetry between and in (8): this estimate requires control on the initial data in the high regularity space in order to be usable, but has no such requirement on the initial data . This asymmetry will be important in some later applications.

*Proof:* From Corollary 3 we have

Now we need bounds on the difference . Initially we have , where . To evolve later in time, we will need to use the energy method. Subtracting (1) for and , we have

By hypothesis, all derivatives of and lie in on , which will allow us to justify the manipulations below without difficulty. We introduce the low regularity energy for the difference:

Arguing as in the proof of Proposition 1, we see that

where

As before, the divergence-free nature of ensures that vanishes. For , we use the Leibniz rule and again extract out the dangerous term

which again vanishes by integration by parts. We then use the triangle inequality to bound

Using Exercise 2 and Hölder, we may bound this by

which by Sobolev embedding gives

Applying (9) and Gronwall’s inequality, we conclude that

for , and (7) follows.

Now we work with the high regularity energ

Arguing as before we have

Using Exercise 2 and Hölder, we may bound this by

Using Sobolev embedding we thus have

By the chain rule, we obtain

(one can work with in place of and then send later if one wishes to avoid a lack of differentiability at ). By Gronwall’s inequality, we conclude that

for all , and (8) follows.

By specialising (7) (or (8)) to the case where , we see the solution constructed in Corollary 3 is unique. Now we can extend to wider classes of initial data than initial data. The following result is essentially due to Kato and to Swann (with a similar result obtained by different methods by Ebin-Marsden):

Proposition 6Let be an integer, and let be divergence-free. Setwhere is sufficiently small depending on . Let be a sequence of divergence-free vector fields converging to in norm (for instance, one could apply Littlewood-Paley projections to ). Let , be the associated solutions to (1) provided by Corollary 3 (these are well-defined for large enough). Then and converge in norm on to limits , respectively, which solve (1) in a distributional sense.

*Proof:* We use a variant of Kato’s argument (see also the paper of Bona and Smith for a related technique). It will suffice to show that the form a Cauchy sequence in , since the algebra properties of then give the same for , and one can then easily take limits (in this relatively high regularity setting) to obtain the limiting solution that solves (1) in a distributional sense.

Let be a large dyadic integer. By Corollary 3, we may find an solution be the solution to the Euler equations (1) with initial data (which lies in ). From Theorem 5, one has

Applying the triangle inequality and then taking limit superior, we conclude that

But by Plancherel’s theorem and dominated convergence we see that

as , and hence

giving the claim.

Remark 7Since the sequence can converge to at most one limit , we see that the solution to (1) is unique in the class of distributional solutions that are limits of smooth solutions (with initial data of those solutions converging to in ). However, this leaves open the possibility that there are other distributional solutions that do not arise as the limits of smooth solutions (or as limits of smooth solutions whose initial data only converge to in a weaker sense). It is possible to recover some uniqueness results for fairly weak solutions to the Euler equations if one also assumes some additional regularity on the fields (or on related fields such as the vorticity ). In two dimensions, for instance, there is a celebrated theorem of Yudovich that weak solutions to 2D Euler are unique if one has an bound on the vorticity. In higher dimensions one can also obtain uniqueness results if one assumes that the solution is in a high-regularity space such as , . See for instance this paper of Chae for an example of such a result.

Exercise 8 (Continuous dependence on initial data)Let be an integer, let , and set , where is sufficiently small depending on . Let be the closed ball of radius around the origin of divergence-free vector fields in . The above proposition provides a solution to the associated initial value problem. Show that the map from to is a continuous map from to .

Remark 9The continuity result provided by the above exercise is not as strong as in Navier-Stokes, where the solution map is in fact Lipschitz continuous (see e.g., Exercise 43 of Notes 1). In fact for the Euler equations, which is classified as a “quasilinear” equation rather than a “semilinear” one due to the lack of the dissipative term in the equation, the solution map is not expected to be uniformly continuous on this ball, let alone Lipschitz continuous. See this previous blog post for some more discussion.

Exercise 10 (Maximal Cauchy development)Let be an integer, and let be divergence free. Show that there exists a unique and unique , with the following properties:

- (i) If and is divergence-free and converges to in norm, then for large enough, there is an solution to (1) with initial data on , and furthermore and converge in norm on to .
- (ii) If , then as .
- (iii) If , then we have the
weak Beale-Kato-Majda criterionFurthermore, show that do not depend on the particular choice of , in the sense that if belongs to both and for two integers then the time and the fields produced by the above claims are the same for both and .

We will refine part (iii) of the above exercise in the next section. It is a major open problem as to whether the case (i.e., finite time blowup) can actually occur. (It is important here that we have some spatial decay at infinity, as represented here by the presence of the norm; when the solution is allowed to diverge at spatial infinity, it is not difficult to construct smooth solutions to the Euler equations that blow up in finite time; see e.g., this article of Stuart for an example.)

Remark 11The condition that recurs in the above results can be explained using the heuristics from Section 5 of Notes 1. Assume that a given time , the velocity field fluctuates at a spatial frequency , with the fluctuations being of amplitude . (We however permit the velocity field to contain a “bulk” low frequency component which can have much higher amplitude than ; for instance, the first component of might take the form where is a quantity much larger than .) Suppose one considers the trajectories of two particles whose separation at time zero is comparable to the wavelength of the frequency oscillation. Then the relative velocities of will differ by about , so one would expect the particles to stay roughly the same distance from each other up to time , and then exhibit more complicated and unpredictable behaviour after that point. Thus the natural time scale here is , so one only expects to have a reasonable local well-posedness theory in the regime

On the other hand, if lies in , and the frequency fluctuations are spread out over a set of volume , the heuristics from the previous notes predict that

The uncertainty principle predicts , and so

Thus we force the regime (11) to occur if , and barely have a chance of doing so in the endpoint case , but would not expect to have a local theory (at least using the sort of techniques deployed in this section) for .

Exercise 12Use similar heuristics to explain the relevance of quantities of the form that occurs in various places in this section.

Because the solutions constructed in Exercise 10 are limits (in rather strong topologies) of smooth solutions, it is fairly easy to extend estimates and conservation laws that are known for smooth solutions to these slightly less regular solutions. For instance:

Exercise 13Let be as in Exercise 10.

- (i) (Energy conservation) Show that for all .
- (ii) Show that
for all .

Exercise 14 (Vanishing viscosity limit)Let the notation and hypotheses be as in Corollary 3. For each , let , be the solution to (3) with this choice of viscosity and with initial data . Show that as , and converge locally uniformly to , and similarly for all derivatives of and . (In other words, there is actually no need to pass to a subsequence as is done in the proof of Corollary 3.)Hint:apply the energy method to control the difference .

Exercise 15 (Local existence for forced Euler)Let be divergence-free, and let , thus is smooth and for any and any integer and , . Show that there exists and a smooth solution to the forced Euler equation

Note:one will first need a local existence theory for the forced Navier-Stokes equation. It is also possible to develop forced analogues of most of the other results in this section, but we will not detail this here.

** — 2. The Beale-Kato-Majda blowup criterion — **

In Exercise 10 we saw that we could continue solutions, to the Euler equations indefinitely in time, unless the integral became infinite at some finite time . There is an important refinement of this blowup criterion, due to Beale, Kato, and Majda, in which the tensor is replaced by the vorticity two-form (or vorticity, for short)

that is to say is essentially the anti-symmetric component of . Whereas is the tensor field

is the anti-symmetric tensor field

Remark 16In two dimensions, is essentially a scalar, since and . As such, it is common in fluid mechanics to refer to the scalar field as the vorticity, rather than the two form . In three dimensions, there are three independent components of the vorticity, and it is common to view as a vector field rather than a two-form in this case (actually, to be precise would be a pseudovector field rather than a vector field, because it behaves slightly differently to vectors with respect to changes of coordinate). With this interpretation, the vorticity is now the curl of the velocity field . From a differential geometry viewpoint, one can view the two-form as an antisymmetric bilinear map from vector fields to scalar functions , and the relation between the vorticity two-form and the vorticity (pseudo-)vector field in is given by the relationfor arbitrary vector fields , where is the volume form on , which can be viewed in three dimensions as an antisymmetric trilinear form on vector fields. The fact that is a pseudovector rather than a vector then arises from the fact that the volume form changes sign upon applying a reflection.

The point is that vorticity behaves better under the Euler flow than the full derivative . Indeed, if one takes a smooth solution to the Euler equation in coordinates

and applies to both sides, one obtains

If one interchanges and then subtracts, the pressure terms disappear, and one is left with

which we can rearrange using the material derivative as

Writing and , this becomes the *vorticity equation*

The vorticity equation is particularly simple in two and three dimensions:

Exercise 17 (Transport of vorticity)Let be a smooth solution to Euler equation in , and let be the vorticity two-form.

- (i) If , show that
- (ii) If , show that
where is the vorticity pseudovector.

Remark 18One can interpret the vorticity equation in the language of differential geometry, which is a more covenient formalism when working on more general Riemann manifolds than . To be consistent with the conventions of differential geometry, we now write the components of the velocity field as rather than (and the coordinates of as rather than ). Define thecovelocity -formaswhere is the Euclidean metric tensor (in the standard coordinates, is the Kronecker delta, though can take other values than if one uses a different coordinate system). Thus in coordinates, ; the covelocity field is thus the musical isomorphism applied to the velocity field. The vorticity -form can then be interpreted as the exterior derivative of the covelocity, thus

or in coordinates

The Euler equations can be rearranged as

where is the Lie derivative along , which for -forms is given in coordinates as

and is the modified pressure

If one takes exterior derivatives of both sides of (14) using the basic differential geometry identities and , one obtains the vorticity equation

where the Lie derivative for -forms is given in coordinates as

and so we recover (13) after some relabeling.

We now present the Beale-Kato-Majda condition.

Theorem 19 (Beale-Kato-Majda)Let be an integer, and let be divergence free. Let , be the maximal Cauchy development from Exercise 10, and let be the vorticity.

The double exponential in (i) is not a typo! It is an open question though as to whether this double exponential bound can be at all improved, even in the simplest case of two spatial dimensions.

We turn to the proof of this theorem. Part (ii) will be implied by part (i), since if is finite then part (i) gives a uniform bound on as , preventing finite time blowup. So it suffices to prove part (i). To do this, it suffices to do so for solutions, since one can then pass to a limit (using the strong continuity in ) to establish the general case. In particular, we can now assume that are smooth.

We would like to convert control on back to control of the full derivative . If one takes divergences of the vorticity using (12) and the divergence-free nature of , we see that

Thus, we can recover the derivative from the vorticity by the formula

where one can define via the Fourier transform as a multiplier bounded on every space.

If the operators were bounded in , then we would have

and the claimed bound (15) would follow from Theorem 1(ii) (with one exponential to spare). Unfortunately, is not quite bounded on . Indeed, from Exercise 18 of Notes 1 we have the formula

for any test function and , where is the singular kernel

If one sets to be a (smooth approximation) to the signum restricted to an annulus , we conclude that the operator norm of is at least as large as

But one can calculate using polar coordinaates that this expression diverges like in the limit , , giving unboundedness.

As it turns out, though, the Gronwall argument used to establish Theorem 1(ii) can just barely tolerate an additional “logarithmic loss” of the above form, albeit at the cost of worsening the exponential term to a double exponential one. The key lemma is the following result that quantifies the logarithmic divergence indicated by the previous calculaation, and is similar in spirit to a well known inequality of Brezis and Wainger.

Lemma 20 (Near-boundedness of )For any and , one has

The lower order terms will be easily dealt with in practice; the main point is that one can almost bound the norm of by that of , up to a logarithmic factor.

*Proof:* By a limiting argument we may assume that is a test function. We apply Littlewood-Paley decomposition to write

and hence by the triangle inequality we may bound the left-hand side of (17) by

where we omit the domain and range from the function space norms for brevity.

By Bernstein’s inequality we have

Also, from Bernstein and Plancherel we have

and hence by geometric series we have

for any . This gives an acceptable contribution if we select . This leaves remaining values of to control, so if one can bound

Observe from applying the scaling (that is, replacing with that to prove (18) for all it suffices to do so for . By Fourier analysis, the function is the convolution of with the inverse Fourier transform of the function

This function is a test function, so is a Schwartz function, and the claim now follows from Young’s inequality.

We return now to the proof of (15). We adapt the proof of Proposition 1(i). As in that proposition, we introduce the higher energy

We no longer have the viscosity term as , but that term was discarded anyway in the analysis. From (4) we have

Applying (16), (20) one thus has

From Exercise 13 one has

By the chain rule, one then has

and hence by Gronwall’s inequality one has

The claim (15) follows.

Remark 21The Beale-Kato-Majda criterion can be sharpened a little bit, by replacing the sup norm with slightly smaller norms, such as the bounded mean oscillation (BMO) norm of , basically by improving the right-hand side of Lemma 20 slightly. See for instance this paper of Planchon and the references therein.

Remark 22An inspection of the proof of Theorem 19 reveals that the same result holds if the Euler equations are replaced by the Navier-Stokes equations; the energy estimates acquire an additional “” term by doing so (as in the proof of Proposition 1), but the sign of that term is favorable.

We now apply the Beale-Kato-Majda criterion to obtain global well-posedness for the Euler equations in two dimensions:

Theorem 23 (Global well-posedness)Let be as in Exercise 10. If , then .

This theorem will be immediate from Theorem 19 and the following conservation law:

Proposition 24 (Conservation of vorticity distribution)Let be as in Exercise 10 with . Then one hasfor all and .

*Proof:* By a limiting argument it suffices to show the claim for , thus we need to show

By another limiting argument we can take to be an solution. By the monotone convergence theorem (and Sobolev embedding), it suffices to show that

whenever is a test function that vanishes in a neighbourhood of the origin . Note that as and all its derivatives are in on for every , it is Lipschitz in space and time, which among other things implies that the level sets are compact for every , and so is smooth and compactly supported in . We may therefore may differentiate under the integral sign to obtain

where we omit explicit dependence on for brevity. By Exercise 17(i), the right-hand side is

which one can write as a total derivative

which vanishes thanks to integration by parts and the divergence-free nature of . The claim follows.

The above proposition shows that in two dimensions, is constant, and so the integral cannot diverge for finite . Applying Theorem 19, we obtain Theorem 23. We remark that global regularity for two-dimensional Euler was established well before the Beale-Kato-Majda theorem, starting with the work of Wolibner.

One can adapt this argument to the Navier-Stokes equations:

Exercise 25Let be an integer, let , let be divergence-free, and let , be a maximal Cauchy development to the Navier-Stokes equations with initial data . Let be the vorticity.

- (i) Establish the vorticity equation .
- (ii) Show that for all and . (Note: to adapt the proof of Proposition 12, one should restrict attention to functions that are convex on the range of on, say, . The case of this inequality can also be established using the maximum principle for parabolic equations.)
- (iii) Show that .

Remark 26There are other ways to establish global regularity for two-dimensional Navier-Stokes (originally due to Ladyzhenskaya); for instance, the bound on the vorticity in Exercise 25(ii), combined with energy conservation, gives a uniform bound on the velocity field, which can then be inserted into (the non-periodic version of) Theorem 38 of Notes 1.

Remark 27If solve the Euler equations on some time interval with initial data , then the time-reversed fields solve the Euler equations on the reflected interval with initial data . Because of this time reversal symmetry, the local and global well-posedness theory for the Euler equations can also be extended backwards in time; for instance, in two dimensions any divergence free initial data leads to an solution to the Euler equations on the whole time interval . However, the Navier-Stokes equations are very muchnottime-reversible in this fashion.

## 6 comments

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9 October, 2018 at 9:57 pm

AnonymousThere is no expression in (2).

[Corrected, thanks – T.]10 October, 2018 at 11:19 am

AnonymousDear Terry,

Great post! I have a question on Remark 7. It seems to be an open problem if two dimensional Euler equation has unique weak solution assuming only L^p bound on the vorticity, for p<\infty.

[Oops, I meant here; now corrected. -T]12 October, 2018 at 12:49 am

yasin şaleDear Terry,

I would like to read your opinions about the proof of Riemann Hypothesis given by M. Atiyah.

Best regards.

12 October, 2018 at 9:46 pm

AnonymousWell, I can replace Pro Tao to response your question.According to our observation, Pro Tao seems to be difficult to answer your problem for public,if you meet him in private , he is willing to reply.Because there are many sensitive reasons,although he holds proofs of RH in your hands.On RH,he has 5 ways to approach to RH,every solution seems to constrast to Atiyah’s proof.The second reson , Pro.Tao is the member of Royal committee in England,while Atiyah is also a professor in England.The third reason, he is very old if compare to Tao,so he must respect to Atiyah.According to me,Clay millenium committee will invites Tao to remark Atiyah’sproof,because Pro Tao is the expert on RH,and only members in Clay institue know Atiyah’sproof right or wrong.

12 October, 2018 at 11:39 pm

AnonymousIn order to check Atiyah’s claimed proof of RH, in addition of being expert on RH, it is even more important to be an expert on Todd function and its related functional equations (e.g. Hirzebruch functional equation) on which Atiyah’s claimed proof is based.

13 October, 2018 at 2:13 am

AnonymousHello. You must see the comments on the entry “Polymath15, tenth thread: numerics update.”6 September 2018