A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation
aa r X i v : . [ m a t h . A P ] M a y A QUANTITATIVE FORMULATION OF THE GLOBALREGULARITY PROBLEM FOR THE PERIODICNAVIER-STOKES EQUATION
TERENCE TAO
Abstract.
The global regularity problem for the periodic Navier-Stokes system ∂ t u + ( u · ∇ ) u = ∆ u − ∇ p ∇ · u = 0 u (0 , x ) = u ( x )for u : R + × ( R / Z ) → R and p : R + × ( R / Z ) → R asks whetherto every smooth divergence-free initial datum u : ( R / Z ) → R there exists a global smooth solution. In this note we observe (usinga simple compactness argument) that this qualitative question isequivalent to the more quantitative assertion that there exists anon-decreasing function F : R + → R + for which one has a local-in-time a priori bound k u ( T ) k H x (( R / Z ) ) ≤ F ( k u k H x (( R / Z ) ) )for all 0 < T ≤ u : [0 , T ] × ( R / Z ) → R to the Navier-Stokes system. We also show that this local-in-timebound is equivalent to the corresponding global-in-time bound. Introduction
Throughout this paper, Ω := ( R / Z ) will denote the standard three-dimensional torus. This note is concerned with solutions to the periodicNavier-Stokes system ∂ t u + ( u · ∇ ) u = ∆ u − ∇ p ∇ · u = 0 u (0 , x ) = u ( x ) (1)where u : R + × Ω → R , p : R + × Ω → R is smooth, and u : Ω → R is smooth and divergence-free. As is well known, the pressure p can be Mathematics Subject Classification. One can place a viscosity factor ν > u term, but this can beeasily normalised away by the change of variables ˜ u ( t, x ) := νu ( νt, x ) and ˜ p ( t, x ) := ν p ( νt, x ). eliminated from this system via Leray projections, and so we view thisequation as an evolution equation for u alone.We have the following well-known unsolved conjecture (see e.g. [6]): Conjecture 1.1 (Global regularity for periodic Navier-Stokes) . Let u : Ω → R be smooth and divergence free. Then there exists a globalsmooth solution u : R + × Ω → R , p : R + × Ω → R to (1) . There is of course an enormous literature on this and related problemswhich we will not attempt to survey here; see for instance [11] forfurther discussion.It is well known (e.g. by standard energy methods, see Section 2 below)that the Navier-Stokes system (1) is locally well-posed in the Sobolevspace H x (Ω), with a time of existence depending on the norm k u k H x (Ω) of the initial data, and with the solution smooth on this time intervalif the initial data was smooth. Because of this, a positive answer toConjecture 1.1 would follow immediately (from standard continuityarguments) from the following long-term a priori bound: Conjecture 1.2 (Long-term a priori bound for periodic Navier-S-tokes) . There exists a non-decreasing function F : R + → R + for whichone has an a priori bound k u ( T ) k H x (Ω) ≤ F ( k u k H x (Ω) ) (2) for all < T < ∞ and all smooth solutions u : [0 , T ] × Ω → R and p : [0 , T ] × Ω → R to (1) . Alternatively, one could be more modest and only ask for the a prioribound up to time 1:
Conjecture 1.3 (Short-term a priori bound for periodic Navier-S-tokes) . There exists a non-decreasing function G : R + → R + for whichone has an a priori bound k u ( T ) k H x (Ω) ≤ G ( k u k H x (Ω) ) (3) for all < T ≤ and all smooth solutions u : [0 , T ] × Ω → R and p : [0 , T ] × Ω → R to (1) . The main result of this paper is
Theorem 1.4 (Equivalence of qualitative and quantitative regularityconjectures) . Conjecture 1.1, Conjecture 1.2, and Conjecture 1.3 areall equivalent.
UANTITATIVE FORMULATION OF PERIODIC NAVIER-STOKES 3
This observation is not very deep, being based on well-known compact-ness properties of the Navier-Stokes flow, and would be unsurprisingto the experts , but the author was not able to find it in the previousliterature . The H x (Ω) norm could be replaced by a variety of othersubcritical norms (and probably some critical norms also) but we willnot pursue such generalisations here. We stress that Theorem 1.4 does not make any serious progress on Conjecture 1.1 itself, but it does sug-gest that this conjecture cannot be solved by purely “soft” methods;some quantitative estimates must be involved. For instance, it empha-sises the (already well understood) point that if one seeks to establishConjecture 1.1 by a regularisation method (approximating u by regu-larised solutions), it is essential to be able to control those solutionsin a subcritical norm such as H x (Ω) with a bound which is uniform inthe choice of regularisation parameter. Theorem 1.4 also shows thatin order to disprove Conjecture 1.1, it suffices to demonstrate “normexplosion” of H x , in the sense of [3]: a sequence of solutions which arebounded in H x at time zero, but are unbounded in H x at later times.1.5. Notation.
For any s ∈ R , we use H s (Ω) to denote the Ba-nach space of distributions f of mean zero on Ω whose Fourier trans-form ˆ f ( k ) := R Ω e − ik · x f ( x ) dx is such that the norm k f k H s (Ω) :=( P k ∈ Z \{ } | k | s | ˆ f ( k ) | ) / is finite. In practice, these functions willbe vector-valued, living in R (or in some cases, R ⊗ R ≡ R ).If I is a compact time interval, s ∈ R , and 1 ≤ p < ∞ , we use L pt H s ( I × Ω) to denote the closure of the smooth mean zero functionson I × Ω under the norm k u k L pt H s ( I × Ω) := ( Z I k u ( t ) k pH s (Ω) dt ) /p . For other examples of equivalences between qualitative global existence and apriori bounds, see e.g. [2], [12]. One could make the case, however, that such compactness results are alreadyimplicitly present in the literature on compact attractors for nonlinear parabolicequations. For instance, in view of such results as [5], [7], it would be natural to considerthe critical norm L x . In particular, it does not address at all the fundamental issue of turbulenceand supercriticality, since the estimate (3) is subcritical and thus out of reach ofall known methods. We remark though that for spherically symmetric classicalsolutions to the logarithmically supercritical equation (cid:3) u = u log(2 + u ) in R ,one has a similarly subcritical a priori estimate k u ( T ) k H x ( R ) ≤ F ( k u (0) k H x ( R ) + k u t (0) k H x ( R ) ); see [13]. TERENCE TAO
Similarly, we let C t H s ( I × Ω) ⊂ L ∞ t H s ( I × Ω) be the closure of thesmooth mean zero functions on I × Ω under the norm k u k C t H s ( I × Ω) := k u k L ∞ t H s ( I × Ω) := sup t ∈ I k u ( t ) k H s (Ω) . We use X . Y to denote the estimate X ≤ CY for an absolute constant C . If we need C to depend on a parameter, we shall indicate this bysubscripts, thus for instance X . s Y denotes the estimate X ≤ C s Y for some C s depending on s .1.6. Acknowledgements.
The author is supported by NSF ResearchAward CCF-0649473 and a grant from the MacArthur Foundation.The author also thanks Isabelle Gallagher, Nets Katz and Igor Rodni-anski for helpful discussions, and Shuanglin Shao for corrections.2.
Review of H theory In solving (1) we first make a well-known reduction. For smooth solu-tions to (1), the mean u = u ( t ) := Z Ω u ( t, x ) dx is easily seen by integration by parts to be an invariant of the flow(this is just conservation of momentum), thus u = u . By making thechange of variables ˜ u ( t, x ) := u ( t, x − u t ) we thus easily reduce to themean zero case u = u = 0. In that case, as is well-known, we can useLeray projections to eliminate the pressure p from (1) and arrive at theequation ∂ t u = ∆ u + D ( u ⊗ u )where D is an explicit first order (tensor-valued) Fourier multiplier (orpseudo-differential operator) whose exact form does not need to bemade explicit for our arguments, although we will note that the imageof D consists entirely of divergence-free mean-zero vector fields.By Duhamel’s formula we have u ( t ) = e t ∆ u + Z t e ( t − t ′ )∆ ( D ( u ⊗ u )) dt ′ . (4)Conversely, if u and u is smooth and obeys (4), with u divergencefree, then it is easy to see that u solves (1) with an appropriate choiceof pressure p .Now suppose u is not smooth, but is merely in H (Ω). We definea strong H solution of (4) on a time interval [0 , T ] to be a function UANTITATIVE FORMULATION OF PERIODIC NAVIER-STOKES 5 u ∈ X T which obeys (4) for all 0 ≤ t ≤ T , where for any s ∈ R , X sT isthe Banach space C t H s ([0 , T ] × Ω) ∩ L t H s +10 ([0 , T ] × Ω) with norm k u k X T := k u k C t H s ([0 ,T ] × Ω) + k u k L t H s +10 ([0 ,T ] × Ω) . We observe that if u is divergence-free then a strong H solution u of(4) must be divergence-free also.We recall the following standard energy estimate: Lemma 2.1 (Energy estimate) . Let ≤ T < ∞ and s ∈ R . If F ∈ L t H s − ([0 , T ] × Ω) has spatial mean zero and u ∈ H s (Ω) , thenthe function u ( t ) := e t ∆ u + R t e ( t − t ′ )∆ F dt ′ lies in X sT with k u k X sT . s k u k H s (Ω) + k F k L t H s − ([0 ,T ] × Ω) . Proof.
The estimate commutes with fractional differentiation opera-tors, so we may take s = 1. By a limiting argument we may assumethat u and F (and hence u ) are smooth, thus u solves the heat equa-tion u t + ∆ u = F with initial data u (0) = u . A standard integrationby parts computation then reveals that ∂ t k u ( t ) k H (Ω) ≤ − c k u ( t ) k H (Ω) + C Z Ω |∇ u ( t ) || F ( t ) | dt for some absolute constants c, C > ≤ t ≤ T . By Cauchy-Schwarz we thus have ∂ t k u ( t ) k H (Ω) ≤ − c ′ k u ( t ) k H (Ω) + C ′ k F ( t ) k L (Ω) for some other absolute constants c ′ , C ′ >
0. Integrating this in t , weobtain the claim. (cid:3) We also recall the smoothing estimate k e t ∆ f k H s + δ (Ω) . s,δ t − δ/ k f k H s (Ω) (5)for all 0 < t < ∞ , s ∈ R , and δ >
0, which follows easily from Fourieranalysis.We now review some well known local well-posedness theory for theequation (4) in the space H (Ω). Proposition 2.2 (Subcritical parabolic theory) . Let
A > , and set T := cA − for some small absolute constant c > . Then the followingstatements hold. Much better local well-posedness results are known, of course; see for instance[9], [7], [1]. Also, the uniqueness aspect of this proposition can be significantlystrengthened using the “weak-strong uniqueness” theorems in the literature; see [8]for the most recent results in this direction.
TERENCE TAO (i) (Existence and uniqueness) If u ∈ H (Ω) with k u k H (Ω) ≤ A ,then there exists a unique strong solution u ∈ X T to (4) , withthe bound k u k X T . A. (6) Furthermore the solution map u u is Lipschitz continuousfrom the ball { u ∈ H (Ω) : k u k H (Ω) ≤ A } to X T . (ii) (Instantaneous regularity) The solution u in (i) is smooth forall positive times t > . (iii) (Compactness) Let u and u be as in (i). Let u ( n )0 ∈ H (Ω) be asequence with k u ( n )0 k H (Ω) ≤ A which converges weakly in H (Ω) to u , and let u ( n ) be the associated strong solutions. Then forany < ε < T , u ( n ) converges strongly in C t H ([ ε, T ] × Ω) to u .Proof. Fix A , and set T := cA − for some sufficiently small absoluteconstant c > u and let Φ : X T → X T denote the nonlinear mapΦ( u )( t ) := e t ∆ u + Z t e ( t − t ′ )∆ ( D ( u ⊗ u )) dt ′ , thus the task is to establish that Φ has a unique fixed point whichdepends in a Lipschitz manner on u .To see that Φ actually maps X T to X T , we use Lemma 2.1 (with s = 1)followed by the H¨older and Sobolev inequalities (and the fact that D is of order 1) to compute k Φ( u ) k X T . k u k H (Ω) + k D ( u ⊗ u ) k L t L x ([0 ,T ] × Ω) . A + k∇ ( u ⊗ u ) k L t L x ([0 ,T ] × Ω) . A + T / k| u ||∇ u |k L t L x ([0 ,T ] × Ω) . A + c / A − k u k L ∞ t L x k∇ u k / L ∞ t L x k∇ u k / L t L x . A + c / A − k u k X T . If c is small enough, we see that Φ thus maps the ball of radius CA in X T to itself for some absolute constant C , and a similar argumentthen shows that Φ is a contraction on that ball if c is small enough,thus yielding the desired unique fixed point, which also obeys (6) andobeys the required Lipschitz property. Strictly speaking, this argument only shows uniqueness of the fixed point insidea ball in X T , however one can use continuity arguments (using the fact that a strongsolution lies in C t H and thus varies continuously in time in the H topology) toextend the uniqueness to all of X T . Alternatively one can modify the argumentsused to prove property (iii) below. UANTITATIVE FORMULATION OF PERIODIC NAVIER-STOKES 7
To show the regularity property (ii), we observe from the above com-putations that k D ( u ⊗ u ) k L t L x ([0 ,T ] × Ω) . A . Now from Lemma 2.1 we have k Z t e ( t − t ′ )∆ F dt ′ k C t H ([0 ,T ] × Ω) . k F k L t L x ([0 ,T ] × Ω) for any test function F , while from (5) and Minkowski’s inequality weeasily verify that k Z t e ( t − t ′ )∆ F dt ′ k C t H − σ ([0 ,T ] × Ω) . σ k F k L ∞ t L x ([0 ,T ] × Ω) for all σ >
0, so by complex interpolation we see that k Z t e ( t − t ′ )∆ F dt ′ k C t H − σ ([0 ,T ] × Ω) . σ k F k L t L x ([0 ,T ] × Ω) for all σ >
0. Applying this with F := D ( u ⊗ u ) and using (5) we seethat k u k C t H − σ ([ ε,T ] × Ω) . σ,ε,A < ε < T and σ >
0. Thus we have obtained a smoothing effectof almost half a derivative. One can continue iterating this argument(using the fractional Leibnitz rule instead of the ordinary one, or alter-natively working in H¨older spaces C k,α and using Schauder theory) toget an arbitrary amount of regularity; we omit the standard details.Now we show the compactness property (iii). Write v ( n ) := u ( n ) − u ,then we see that v ( n ) (0) converges weakly to H (Ω) and thus strongly in L x (Ω) (by the Rellich compactness theorem). Also, by (4), v ( n ) solvesthe difference equation v ( n ) ( t ) = e t ∆ v ( n ) (0) + Z t e ( t − t ′ )∆ ( D ( v ( n ) ⊗ u ) + D ( u ( n ) ⊗ v ( n ) )) dt ′ and so by Lemma 2.1 with s = 0, followed by H¨older, Sobolev, and (6)(for both u and u ( n ) ) we have k v ( n ) k X T . k v ( n ) (0) k L x (Ω) + k D ( v ( n ) ⊗ u ) + D ( u ( n ) ⊗ v ( n ) ) k L t H − ([0 ,T ] × Ω) . k v ( n ) (0) k L x (Ω) + T / ( k v ( n ) ⊗ u k L t L x ([0 ,T ] × Ω) + k v ( n ) ⊗ u ( n ) k L t L x ([0 ,T ] × Ω) ) . k v ( n ) (0) k L x (Ω) + T / k v ( n ) k / L ∞ t L x ([0 ,T ] × Ω) k v ( n ) k / L t L x ([0 ,T ] × Ω) × ( k u k L ∞ t L x ([0 ,T ] × Ω) + k u ( n ) k L ∞ t L x ([0 ,T ] × Ω) ) . k v ( n ) (0) k L x (Ω) + c / k v ( n ) k X T . Since v ( n ) has a finite X T norm, we conclude (if c is small enough) that k v ( n ) k X T . k v ( n ) (0) k L x (Ω) . TERENCE TAO
Now, the above argument in fact shows that k D ( v ( n ) ⊗ u ) + D ( u ( n ) ⊗ v ( n ) ) k L t H − ([0 ,T ] × Ω) . A,T k v ( n ) (0) k L x (Ω) and so by using the interpolation argument as before (but at one lowerderivative of regularity) one concludes that k v ( n ) k C t H − σ ([ ε,T ] × Ω) . σ,ε,A,T k v ( n ) (0) k L x (Ω) for all σ > < ε < T . Iterating this argument a finite numberof times (using the fractional Leibnitz rule) we eventually obtain k v ( n ) k C t H ([ ε,T ] × Ω) . ε,A,T k v ( n ) (0) k L x (Ω) for any 0 < ε < T . Since the right-hand side goes to zero as n goes toinfinity, the claim (iii) follows. (cid:3) Proof of Theorem 1.4
By the discussion of the preceding section we may assume throughoutthat we are in the mean zero case.3.1.
Derivation of Conjecture 1.1 from Conjecture 1.2.
Thisfollows immediately from iterating Proposition 2.2.3.2.
Derivation of Conjecture 1.2 from Conjecture 1.3.
Thisshall exploit an observation of Leray that any finite energy solution toNavier-Stokes becomes well-behaved after a bounded amount of time,thanks to energy dissipation.Let u : [0 , T ] × Ω → R and p : [0 , T ] × Ω → R be a smooth solutionto (1) for some time 0 < T < ∞ . Write E := k u k H x (Ω) . From thewell-known energy identity ∂ t Z Ω | u ( t, x ) | dx = − Z Ω |∇ u ( t, x ) | dx we see that k u k L ∞ t L x ([0 ,T ] × Ω) + k∇ u k L t L x ([0 ,T ] × Ω) . E . Let ε > E to be chosen later.Suppose for now that T ≥ /ε . Then by the pigeonhole principle, wecan thus find a time 0 ≤ T ′ ≤ /ε such that k∇ u ( T ′ ) k L x (Ω) . E ε. Since u has mean zero, this implies (by Poincar´e’s inequality) that k u ( T ′ ) k H x (Ω) . E ε. (7) UANTITATIVE FORMULATION OF PERIODIC NAVIER-STOKES 9
Now let T ′′ be any time between T ′ and min( T ′ + 1 , T ). From Propo-sition 2.2, we have (for ε small enough) that k u k C t H x ([ T ′ ,T ′′ ] × Ω) + k u k L t H x ([ T ′ ,T ′′ ] × Ω) . E k u ( T ′ ) k H x (Ω) . (8)Repeating the computations in that proposition, we conclude in par-ticular that k D ( u ⊗ u ) k L t L x ([ T ′ ,T ′′ ] × Ω) . E k u ( T ′ ) k H x (Ω) . Using (4) and (part of) Lemma 2.1, we conclude that k u ( T ′′ ) − e ( T ′′ − T ′ )∆ u ( T ′ ) k H x (Ω) . E k u ( T ′ ) k H x (Ω) . The least non-trivial eigenvalue of − ∆ on the torus Ω is 1; since u ( T ′ )has mean zero, we thus have k e ( T ′′ − T ′ )∆ u ( T ′ ) k L x (Ω) ≤ e − ( T ′′ − T ′ ) k u ( T ′ ) k L x (Ω) for some λ >
0. In particular, if T ′′ = T ′ + 1 then from the triangleinequality we have k u ( T ′ + 1) k H x (Ω) ≤ e − k u ( T ′ ) k H x (Ω) + C E k u ( T ′ ) k H x (Ω) for some constant C E depending only on E . From (7) we conclude (if ε is small enough depending on E ) that k u ( T ′ + 1) k H x (Ω) ≤ k u ( T ′ ) k H x (Ω) . E ε. Iterating this inequality as far as we can, and then using (8), we con-clude that k u k C t H x ([ T ′ ,T ] × Ω) . E ε. (9)On the other hand, since T ′ ≤ /ε , we can iterate Conjecture 1.3 andobtain k u k C t H x ([0 ,T ′ ] × Ω) . E,ε . (10)Combining (9) and (10) gives Conjecture 1.2 in the case T ≥ /ε . Inthe case T < /ε , we omit all the steps leading up to (9) and simplyiterate Conjecture 1.3 as in (10). The claim follows. Remark . In the mean zero case, the above argument in fact allowsus to strengthen (2) to k u ( T ) − u k H x (Ω) ≤ e − T ˜ F ( k u − u k H x (Ω) ) (11)for some function ˜ F : R + → R + and all solutions u : [0 , T ] × Ω → R , p : [0 , T ] × Ω → R to (1). We leave the details to the reader. Derivation of Conjecture 1.3 from Conjecture 1.1.
Supposefor contradiction that Conjecture 1.3 failed. Carefully unwrapping allthe quantifiers and using the axiom of choice , we conclude that thereexists A > u ( n ) : [0 , T ( n ) ] × Ω → R , p ( n ) : [0 , T ( n ) ] × Ω → R to (1) (and hence (4)) with smooth meanzero divergence-free initial data u ( n ) (0) = u ( n )0 , where 0 < T ( n ) < k u ( n )0 k H (Ω) ≤ A for all n , and such thatlim n →∞ k u ( n ) ( T ( n ) ) k H (Ω) = ∞ . (12)By passing to a subsequence if necessary we may assume that lim n →∞ T ( n ) = T for some 0 ≤ T ≤
1. The assertion T = 0 would contradict (6), sowe may assume 0 < T ≤ u ( n )0 are weakly convergent in H (Ω) to a limit u ∈ H (Ω), which is alsodivergence-free. Applying Proposition 2.2, we can obtain a strong so-lution u to the Navier-Stokes equation (4) with initial datum u forsome short time [0 , ε ] with ε >
0, and that u is smooth, divergence-freeand mean zero for all times 0 < t ≤ ε . If we then apply Conjecture1.1 to the initial datum u ( ε ), we thus see u can be continued globallyas a smooth mean zero solution for all 0 < t < ∞ . In particular, u ∈ C t H ([0 , T ] × Ω), and so there must exist
A < A ′ < ∞ such thatsup ≤ t ≤ T k u ( t ) k H (Ω) ≤ A ′ . (13)Now recall that u ( n )0 converges weakly in H (Ω) to u and have thebound k u ( n )0 k H (Ω) ≤ A ≤ A ′ . Then by repeated application of Propo-sition 2.2 (splitting [0 , T ] up into intervals of length T (2 A ′ ) >
0, say,and only considering sufficiently large n ) we see that u ( n ) convergesstrongly in C t H ( I × Ω) to u for all compact intervals I ⊂ (0 , T ).But this is inconsistent with (12) and (13). Theorem 1.4 follows.4. Remarks
It will be clear to the experts that in our main theorem, the torus Ωcould easily be replaced by any other compact surface, and that reason-able boundary conditions can also be added. It is tempting to extendthis result also to the non-periodic case, but the translation invari-ance causes some failure of compactness which needs to be addressed.In principle, this can be solved either by concentration compactness, It is not difficult to eliminate the use of this axiom if desired, for instance byusing an explicit (and explicitly well-ordered) countable dense subset of the smoothfunctions; we omit the details.
UANTITATIVE FORMULATION OF PERIODIC NAVIER-STOKES 11 or by requiring that the data be compactly supported or rapidly de-creasing; the latter is in fact used in the standard formulations of thenon-periodic regularity problem (see e.g. [6]) but if one were to ap-ply the above arguments to that setting, one encounters the annoyingfact that the rapid decrease of the initial data is not preserved by theNavier-Stokes flow (due to the non-local nature of the Leray projectionused to eliminate the pressure term). We will not pursue these matters.Define the monotone non-decreasing function F : R + → [0 , + ∞ ] by F ( A ) := sup {k u k C t H x ([0 ,T ] × Ω) : k u k H x (Ω) ≤ A ; 0 ≤ T < ∞} where u ranges over all local smooth solutions u : [0 , T ] × Ω → R to (1) with an initial datum u of norm at most A . By Theorem 1.4,the periodic Navier-Stokes global regularity conjecture is equivalent tothe assertion that F ( A ) is finite for all A . One can show (by modi-fying Proposition 2.2) that F is finite for small A , and is also right-continuous; thus if the global regularity conjecture fails, there exists acritical value 0 < A c < ∞ such that F ( A c ) = + ∞ and F ( A ) < ∞ forall A < A c . It is tempting to then use induction-on-energy arguments(as in e.g. [4]) to try to analyse solutions at this critical value of the H norm, but unfortunately the lack of any conservation law at the H level seems to make this idea fruitless. (In any event, there is nothingspecial about the H norm in this argument; a large number of othersubcritical norms would also work here.)Our arguments yield no information as to the rate of growth of F . Thearguments in [10] (see also [3]) should at least yield that F ( A ) /A goes toinfinity as A → ∞ , but it seems difficult to even obtain F ( A ) & A ε for some ε >
0. It seems of interest to understand this growth ratebetter, even at a heuristic level.It is amusing to note that for any fixed 0 < A, M < ∞ , the statement F ( A ) < M , if true, can be verified in finite time, by constructingsufficiently accurate and sufficiently numerous numerical solutions, andusing some quantitative version of the compactness phenomenon inProposition 2.2 to then rigorously establish F ( A ) < M ; we omit thedetails. A similar verifiability result holds for the statement F ( A ) > M .Unfortunately, if F ( A ) is infinite, there does not appear to be anyobvious way to verify this in finite time.As a final observation, we note from our results that if the Navier-Stokes regularity conjecture is true, then the solution map u u isdefined as a map from H (Ω) to C t H ([0 , ∞ ) × Ω); this essentiallyfollows from (2) and Proposition 2.2. Furthermore, by working more We are indebted to Jean Bourgain for this observation. carefully with the proof of Proposition 2.2 and using the exponentialdecay (11) one can show that this map is Lipschitz continuous onany bounded subset of H (Ω); we omit the details. Thus one canview the Navier-Stokes regularity conjecture as an assertion that theNavier-Stokes flow enjoys global non-perturbative stability in the H topology. References [1] P. Auscher, S. Dubois, P. Tchamitchian,
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E-mail address : [email protected] It is important here that the mean u0