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FINITE TIME BLOWUP FOR AN AVERAGED

THREE-DIMENSIONAL NAVIER-STOKES EQUATION

TERENCE TAO

Abstract.The Navier-Stokes equation on the Euclidean spaceR3can be expressed in the formBtuuBpu;uq, whereBis a certain bilinear operator on divergence-free vector eldsuobeying the cancellation propertyxBpu;uq;uy 0 (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modicationBtuu~Bpu;uqof this equation, where~Bis an averaged version of the bilinear operatorB(where the average involves rotations, dilations and Fourier multi- pliers of order zero), and which also obeys the cancellation conditionx~Bpu;uq;uy 0 (so that it obeys the usual energy identity). By analysing a system of ODE related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such an averaged Navier-Stokes equa- tion which blows up in nite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use ner structure on the nonlinear portionBpu;uqof the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.

1.Introduction

1.1.Statement of main result.The purpose of this paper is to formalise the \super-

criticality" barrier for the (infamous) global regularity problem for the Navier-Stokes equation, using a blowup solution to a certain averaged version of Navier-Stokes equa- tion to demonstrate that any proposed positive solution to the regularity problem which does not use the ner structure of the nonlinearity cannot possibly be successful. This barrier also suggests a possible route to provide a negative answer to this problem, that is to say it suggests a program for constructing a blowup solution to the true

Navier-Stokes equations.

The barrier is not particularly sensitive to the precise formulation

1of the regularity

problem, but to state the results in the cleanest fashion we will take the homogeneous global regularity problem in the Euclidean setting in three spatial dimensions as our formulation: Conjecture 1.1(Navier-Stokes global regularity).[13, (A)]Let¡0, and letu0: R

3ÑR3be a divergence-free vector eld in the Schwartz class. Then there exist a2010Mathematics Subject Classication.35Q30.

1See [41] for an analysis of the relationship between dierent formulations of the Navier-Stokes reg-

ularity problem in three dimensions. It is likely that our main results also extend to higher dimensions

than three, although we will not pursue this matter here.

1arXiv:1402.0290v3 [math.AP] 1 Apr 2015

2 TERENCE TAO

smooth vector eldu:r0;8q R3ÑR3(the velocity eld) and smooth function p:R3ÑR(the pressure eld) obeying the equations B tu purquurp ru0 up0;q u0(1.1) as well as the nite energy conditionuPL8tL2xpr0;Ts R3qfor every0 T 8. By applying the rescaling ~upt;xq:upt;xq, ~ppt;xq:ppt;xqwe may normalise

1 (note that there is no smallness requirement on the initial datau0), and we shall

do so henceforth. To study this conjecture, we perform some standard computations to eliminate the role of the pressurep, and to pass from the category of smooth (classical) solutions to the closely related category ofmildsolutions in a high regularity class. It will not matter too much what regularity class we take here, as long as it is subcritical, but for sake of concreteness (and to avoid some very minor technicalities) we will take a quite high regularity space, namely the Sobolev spaceH10dfpR3qof (distributional) vector elds u:R3ÑR3withH10regularity (thus the weak derivativesrjuare square-integrable forj0;:::;10) and which are divergence free in the distributional sense:ru0.

By using theL2inner product2

xu;vy:» R

3uv dx

on vector eldsu;v:R3ÑR3, the dualH10dfpR3qmay be identied with the negative- order Sobolev spaceH10 dfpR3qof divergence-free distributionsu:R3ÑR3ofH10 regularity. We introduce theEuler bilinear operatorB:H10dfpR3qH10dfpR3q ÑH10dfpR3q via duality as xBpu;vq;wy: 12 R

3pppurqvq wq pppvrquq wqdx

foru;v;wPH10dfpR3q; it is easy to see from Sobolev embedding that this operator is well dened. More directly, we can write

Bpu;vq 12

Prpurqv pvrqus

wherePis theLeray projectiononto divergence-free vector elds, dened on square- integrableu:R3ÑR3by the formula Pu i:ui1BiBjuj with the usual summation conventions, where

1BiBjis dened as the Fourier mul-

tiplier with symbol ij||2. Note thatBpu;vqtakes values inL2pR3q(and not just in H

10dfpR3q) whenu;vPH10dfpR3q. We refer to the formpu;v;wq ÞÑ xBpu;vq;wyas the

Euler trilinear form. As is well known, we have the important cancellation law xBpu;uq;uy 0 (1.2)2 We will not use theH10dfinner product in this paper, thus all appearances of thex;ynotation should be interpreted in theL2sense.

BLOWUP FOR AVERAGED NAVIER-STOKES 3

for alluPH10dfpR3q, as can be seen by a routine integration by parts exploiting the divergence-free nature ofu, with all manipulations being easily justied due to the high regularity ofu. It will also be convenient to express the Euler trilinear form in terms of the Fourier transform ^upq:³ R

3upxqe2ixdxas

xBpu;vq;wy i» 1230

1;2;3p^up1q;^vp2q;^wp3qq(1.3)

for allu;v;wPH10dfpR3q, where we adopt the shorthand»

1230Fp1;2;3q:»

R 3» R

3Fp1;2;12qd1d2

and

1;2;3:K1K2K3ÑRis the trilinear form

1;2;3pX1;X2;X3q: pX12qpX2X3q pX21qpX1X3q;(1.4)

dened for vectorsXiin the orthogonal complementKi: tXiPR3:Xii0uof ifori1;2;3; note the divergence-free condition ensures that ^up1q PK1for (almost) all1PR3, and similarly forvandw. This also provides an alternate way to establish (1.2). Given a Schwartz divergence-free vector eldu0:R3ÑR3and a time intervalI€ r0;8qcontaining 0, we dene amildH10solution to the Navier-Stokes equations(or mild solutionfor short) with initial datau0to be a continuous mapu:IÑH10dfpR3q obeying the integral equation uptq etu0» t 0 eptt1qBpupt1q;upt1qqdt1(1.5) for alltPI, whereetare the usual heat propagators (dened onL2pR3q, for instance); formally,(1.5) implies the projected Navier-Stokes equation B tuuBpu;uq up0;q u0(1.6) in a distributional sense at least (actually, at theH10dflevel of regularity it is not dicult to justify (1.6) in the classical sense for mild solutions). The distinction between smooth nite energy solutions andH10dfmild solutions is es- sentially non-existent (at least

3for Schwartz initial data), and the reader may wish

to con ate the two notions on a rst reading. More rigorously, we can reformulate Conjecture 1.1 as the following logically equivalent conjecture: Conjecture 1.2(Navier-Stokes global regularity, again).Letu0:R3ÑR3be a divergence-free vector eld in the Schwartz class. Then there exists a mild solution u:r0;8q ÑH10dfpR3qto the Navier-Stokes equations with initial datau0. Lemma 1.3.Conjecture 1.1 and Conjecture 1.2 are equivalent.3 For data which is only inH10df, there is a technical distinction between the two solution concepts, due to a lack of unlimited time regularity at the initial timet0 that is ultimately caused by the non-local eects of the divergence-free conditionru0, requiring one to replace the notion of a

smooth solution with that of analmost smooth solution; see [41] for details. However, in this paper we

will only concern ourselves with Schwartz initial data, so that this issue does not arise.

4 TERENCE TAO

Proof.We use the results from [41], although this equivalence is essentially classical and was previously well known to experts. Let us rst show that Conjecture 1.1 implies Conjecture 1.2. Letu0:R3ÑR3be a Schwartz divergence-free vector eld, then by Conjecture 1.1 we may nd a smooth vector eldu:r0;8q R3ÑR3and smooth functionp:R3ÑRobeying the equations (1.1) and the nite energy condition. By [41, Corollary 11.1],uis anH1 solution, that is to sayuPL8tH1xpr0;TsR3qfor all niteT. By [41, Corollary 4.3], we then have the integral equation (1.5), and by [41, Theorem 5.4(ii)],uPL8tHkxpr0;Ts R

3qfor everyk, which easily implies (from (1.5)) thatuis a continuous map from

r0;8qtoH10dfpR3q. This gives Conjecture 1.2. Conversely, if Conjecture 1.2 holds, andu0:R3ÑR3is a Schwartz class solution, we may nd a mild solutionu:r0;8q ÑH10dfpR3qwith this initial data. By [41, Theorem

5.4(ii)],uPL8tHkxpr0;Ts R3qfor everyk. If we dene the normalised pressure

p: 1BiBjpuiujq then by [41, Theorem 5.4(iv)],uandpare smooth onr0;8qR3, and for eachj;k¥0, the functionsBj tu;Bj tplie inL8tHkxpr0;Ts R3qfor all niteT. By dierentiating (1.5), we have B tuuBpu;uq u purqurp; and Conjecture 1.1 follows. If we take the inner product of (1.6) withuand integrate in time using (1.2), we arrive at

4the fundamentalenergy identity

12 R

3|upT;xq|2dx»

T 0» R

3|rupt;xq|2dxdt12

R

3|u0pxq|2dx(1.7)

for any mild solution to the Navier-Stokes equation. If one was unaware of the supercritical nature of the Navier-Stokes equation, one might attempt to obtain a positive solution to Conjecture 1.1 or Conjecture 1.2 by combining (1.7) (or equivalently, (1.2)) with various harmonic analysis estimates for the inhomo- geneous heat equation B tuuF up0;q u0 (or, in integral form,uptq etu0³t

0eptt1qFpt1qdt1), together with harmonic analysis

estimates for the Euler bilinear operatorB, a simple example of which is the estimate One has to justify the integration by parts of course, but this is routine under the hypothesis of a mild solution; we omit the (standard) details.

BLOWUP FOR AVERAGED NAVIER-STOKES 5

for some absolute constantC. Such an approach succeeds for instance if the initial data u

0is suciently small5in a suitable critical norm (see [29] for an essentially optimal re-

sult in this direction), or if the dissipative operator is replaced by a hyperdissipative operatorpqfor some¥5{4 (see [25]) or with very slightly less hyperdissipative operators (see [39]). Unfortunately, standard scaling heuristics (see e.g. [40,x2.4]) have long indicated to the experts that the energy estimate (1.7) (or (1.2)), together with the harmonic analysis estimates available for the heat equation and for the Euler bilin- ear operatorB, are not sucient by themselves to armatively answer Conjecture 1.1. However, these scaling heuristics are not formalised as a rigorous barrier to solvability, and the above mentioned strategy to solve the Navier-Stokes global regularity problem continues to be attempted on occasion. The most conclusive way to rule out such a strategy would of course be to demonstrate 6 a mild solution to the Navier-Stokes equation that develops a singularity in nite time, in the sense that theH10dfnorm ofuptqgoes to innity astapproaches a nite time T . Needless to say, we are unable to produce such a solution. However, we will in this paper obtain a nite time blowup (mild) solution to anaveragedequation B tuu~Bpu;uq up0;q u0;(1.9) where ~B:H10dfpR3qH10dfpR3q ÑH10dfpR3qwill be a (carefully selected) averaged version

ofBthat has equal or lesser \strength" from a harmonic analysis point of view (indeed,~Bobeys slightlymoreestimates thanBdoes), and which still obeys the fundamen-

tal cancellation property (1.2). Thus, any successful method to armatively answer Conjecture 1.1 (or Conjecture 1.2) must either use ner structure of the Navier-Stokes equation beyond the general form (1.6), or else must rely crucially on some estimate or other property of the Euler bilinear operatorBthat is not shared by the averaged operator~B. We pause to mention some previous blowup results in this direction. If one drops the cancellation requirement (1.2), so that one no longer has the energy identity (1.7), then blowup solutions for various Navier-Stokes type equations have been constructed in the literature. For instance, in [33] nite time blowup for a \cheap Navier-Stokes equation"Btuu?pu2q(withunow a scalar eld) was constructed in the one- dimensional setting, with the results extended to higher dimensions in [17]. As remarked in that latter paper, it is essential to the methods of proof that no energy identity is available. In a slightly dierent direction, nite time blowup was established in [7] for a complexied version of the Navier-Stokes equations, in which the energy identity was again unavailable (or more precisely, it is available but non-coercive). These models are not exactly of the type (1.9) considered in this paper, but are certainly very similar in spirit.5 One can of course also consider other perturbative regimes, in which the solutionuis expected

to be close to some other special solution than the zero solution. There is a vast literature in these

directions, see e.g. [9] and the references therein.

6It is a classical fact that mild solutions to a given initial data are unique, see e.g. [41, Theorem

5.4(iii)].

6 TERENCE TAO

Further models of Navier-Stokes type, which obey an energy identity, were introduced by Plechac and Sverak [35], [36], by Katz and Pavlovic [26], and by Hou and Lei [21]; of these three, the model in [26] is the most relevant for our work and will be discussed in detail in Section 1.2 below. These models dier from each other in several respects, but interestingly, in all three cases there is substantial evidence of blowup in ve and higher dimensions, but not in three or four dimensions; indeed, for all three of the models mentioned above there are global regularity results in three dimensions, even in the presence of blowup results for the corresponding inviscid model. Numerical evidence for blowup for the Navier-Stokes equations is currently rather scant (except in the innite energy setting, see [20], [34]); the blowup evidence is much stronger in the case of the Euler equations (see [23] for a recent result in this direction, and [22] for a survey), but it is as yet unclear

7whether these blowup results have direct implications for Navier-

Stokes in the three-dimensional setting, due to the relatively signicant strength of the dissipation. Finally, we mention work [18], [3], [12] establishing nite time blowup for supercrit- ical fractal Burgers equations; such equations are not exactly of Navier-Stokes type, being scalar one-dimensional equations rather than incompressible vector-valued three- dimensional ones, but from a scaling perspective the results are of the same type, namely a demonstration of blowup whenever the norms controlled by the conservation and monotonicity laws are all supercritical. We now describe more precisely the type of averaged operator ~B:H10dfpR3qH10dfpR3q Ñ H

10dfpR3qwe will consider. We consider three types of symmetries onH10dfpR3qthat we

will average over. Firstly, we have rotation symmetry: ifRPSOp3qis a rotation matrix onR3anduPH10dfpR3q, then the rotated vector eld Rot

Rpuqpxq:RupR1xq

is also inH10dfpR3q; note that the Fourier transform also rotates by the same law,

RotRpuqpq R^upR1q:

Clearly, these rotation operators are uniformly bounded onH10dfpR3q, and also on every

Sobolev spaceWs;ppR3qwithsPRand 1 p 8.

Next, dene a (complex)Fourier multiplier of order0 to be an operatormpDqdened on (the complexicationH10dfpR3q bCof)H10dfpR3qby the formula mpDqupq:mpq^upq wherem:R3ÑCis a function that is smooth away from the origin, with the seminorms }m}k:sup

0||k|rkmpq|(1.10)

being nite for every natural numberk. We say thatmpDqisrealif the symbolm obeys the symmetrympq mpqfor allPR3zt0u, thenmpDqmapsH10dfpR3qto itself. From the Hormander-Mikhlin multiplier theorem (see e.g. [38]), complex Fourier multipliers of order 0 are also bounded on (the complexications of) every Sobolev7 However, in [24], nite time blowup for a three-dimensional \partially viscous" Navier-Stokes type model, in which some but not all of the elds are subject to a viscosity term, was established.

BLOWUP FOR AVERAGED NAVIER-STOKES 7

spaceWs;ppR3qfor allsPRand 1 p 8, with an operator norm that depends linearly on nitely many of the}m}k. We letM0denote the space of all real Fourier multipliers of order 0, so thatM0bCis the space of complex Fourier multipliers (note that every complex Fourier multipliermpDqof order 0 can be uniquely decomposed asmpDq m1pDq im2pDqwithm1pDq;m2pDqreal Fourier multipliers of order 0). Fourier multipliers of order 0 do not necessarily commute with the rotation operators Rot R, but the group of rotation operators normalises the algebraM0, and hence also the complexicationM0bC.

Finally, we will average

8over the dilation operators

Dil puqpxq:3{2upxq(1.11) for¡0. These operators do not quite preserve theH10dfpR3qnorm, but ifis re- stricted to a compact subset ofp0;8qthen these operators (and their inverses) will be uniformly bounded onH10dfpR3q. We now dene anaveraged Euler bilinear operatorto be an operator~B:H10dfpR3q H

10dfpR3q ÑH10dfpR3q, dened via duality by the formula

x (1.12) for allu;v;wPH10dfpR3q, wherem1pDq;m2pDq;m3pDqare random real Fourier multi- pliers of order 0,R1;R2;R3are random rotations, and1;2;3are random dilations, obeying the moment bounds

E}m1}k1}m2}k2}m3}k3 8

and C almost surely for any natural numbersk1;k2;k3and some niteC. To phrase this denition without probabilistic notation, we have x ~Bpu;vq;wy » @Bm

1;!pDqRotR1;!Dil1u;m2;!pDqRotR2;!Dil2v;m

3;!pDqRotR3;!Dil3wDdp!q

(1.13) for some probability spacep ;qand some measurable mapsRi;:

ÑSOp3q,i;:

Ñ p0;8qandmi;pDq:

ÑM0, whereM0is given the Borel-algebra coming

from the seminorms}}k, and one has» }m1;!}k1}m2;!}k2}m3;!}k3dp!q 8 and C for all natural numbersk1;k2;k3. One can also express~Bpu;vqwithout duality by the formula

Bpu;vq »

Dil1

3;!RotR1

3;!m

3;!pDqBm

In an earlier version of this manuscript, no averaging over dilations was assumed, but it was pointed

out to us by the referee that the non-degeneracy condition (3.24) failed if one did not introduce dilation

averaging.

8 TERENCE TAO

where the integral is interpreted in the weak sense (i.e. the Gelfand-Pettis integral). However, we will not use this formulation of~Bhere. Remark 1.4.By the rotation symmetryxBpRotRu;RotRvq;RotRwy xBpu;vq;wy, we may eliminate one of the three rotation operators Rot

Ri;!in (1.13) if desired, and simi-

larly for the dilation operator. By some Fourier analysis (related to the fractional Leib- niz rule) it should also be possible to eliminate one of the Fourier multipliersmi;!pDq.

However, we will not attempt to do so here.

From duality, the triangle inequality (or more precisely, Minkowski's inequality for in- tegrals), and the Hormander-Mikhlin multiplier theorem, we see that every estimate on the Euler bilinear operatorBin Sobolev spacesWs;ppR3qwith 1 p 8implies a corresponding estimate for averaged Euler bilinear operators~B(but possibly with a larger constant). For instance, from (1.8) we have foru;vPH10dfpR3q, where the constantC~Bdepends9only on~B. A similar argument shows that the expectation in (1.12) (or the integral in (1.13)) is absolutely convergent for anyu;v;wPH10dfpR3q. Similar considerations hold for most other basic bilinear estimates

10onBin popular

function spaces such as Holder spaces, Besov spaces, or Morrey spaces. Because of this, the local theory (and related theory, such as the concentration-compactness theory) for (1.9) is essentially identical to that of (1.6) (up to changes in the explicit constants), although we will not attempt to formalise this assertion here. In particular, we may introduce the notion of amild solutionto the averaged Navier-Stokes equation (1.6) with initial datau0PH10dfpR3qon a time intervalI€ r0;8qcontaining 0, dened to be a continuous mapu:IÑH10dfpR3qobeying the integral equation uptq etu0» t 0 eptt1q~Bpupt1q;upt1qqdt1(1.15) uniqueness theory (see e.g. [41,x5]) for mild solutions of the Navier-Stokes equations, to mild solutions of the averaged Navier-Stokes equations, basically because of the previous observation that all the estimates onBused in that local theory continue to hold for~B.9 Note that by applying the transformationpu;~Bq Ñ pu;1~Bqto (1.9), we have the freedom to multiply ~Bby an arbitrary constant, and so the constantsC~Bappearing in any given estimate such as (1.14) can be normalised to any absolute constant (e.g. 1) if desired.

10There is a possible exception to this principle if the estimate involves endpoint spaces such asL1

andL8for which the Hormander-Mikhlin multiplier theorem is not available, or non-convex spaces such asL1;8for which the triangle inequality is not available. However, as the Leray projectionPis also badly behaved on these spaces, such endpoint spaces rarely appear in these sorts of analyses of the Navier-Stokes equation.

BLOWUP FOR AVERAGED NAVIER-STOKES 9

Because we have not imposed any symmetry or anti-symmetry hypotheses on the aver- aging measure, rotationsRj, and Fourier multipliersmjpDq, the analogue x ~Bpu;uq;uy 0 (1.16) of the cancellation condition (1.2) is not automatically satised. If however we have (1.16) for alluPH10dfpR3q, then mild solutions to (1.9) enjoy the same energy identity (1.7) as mild solutions to the true Navier-Stokes equation. We are now ready to state the main result of the paper. Theorem 1.5(Finite time blowup for an averaged Navier-Stokes equation).There exists a symmetric averaged Euler bilinear operator~B:H10dfpR3qH10dfpR3q ÑH10dfpR3q obeying the cancellation property(1.16)for alluPH10dfpR3q, and a Schwartz divergence- free vector eldu0, such that there is no global-in-time mild solutionu:r0;8q Ñ H

10dfpR3qto the averaged Navier-Stokes equation(1.9)with initial datau0.

In fact, the arguments used to prove the above theorem can be pushed a little further to construct a smooth mild solutionu:r0;Tq ÑH10dfpR3qfor some 0 T 8that blows up (at the spatial origin) astapproachesT(and with subcritical norms such as }uptq}H10dfpR3qdiverging to innity astÑT). Remark 1.6.One can also rewrite the averaged Navier-Stokes equation (1.9) in a form more closely resembling (1.1), namely B tuTpu;uq urp ru0 up0;q u0 whereTis an averaged version of the convection operatorpurqu, dened byT 12 pT12T21qwherequotesdbs_dbs5.pdfusesText_9

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