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Frequency localized regularity criteria for the 3D Navier-Stokes equations 6 1 0 Z. Bradshaw and Z. Grujic´ 2 v o November 22, 2016 N 1 2 Abstract ] Two regularity criteria are established to highlight which Littlewood-Paley fre- P A quenciesplayanessentialroleinpossiblesingularityformationinaLeray-Hopfweak . solution to the Navier-Stokes equations in three spatial dimensions. One of these is h a frequency localized refinement of known Ladyzhenskaya-Prodi-Serrin-type regu- t a larity criteria restricted to a finite window of frequencies the lower bound of which m divergesto+∞astapproachesaninitialsingulartime. [ 2 v 3 1 Introduction 4 0 1 0 TheNavier-Stokes equationsgoverning the evolution of aviscous, incompressible flow’s 1. velocity field u inR3 ×(0,T)read 0 5 ∂ u+u·∇u = −∇p+ν∆u+f in R3 ×(0,T) 1 t (3DNSE) : ∇·u = 0 in R3 ×(0,T), v i X whereν istheviscositycoefficient, pisthepressure,andf istheforcing. Forconvenience r a we take f to be zero and set ν = 1. The flow evolves from an initial vector field u0 taken in an appropriate function space. Theregularity ofLeray-Hopfweaksolutions (i.e. distributional solutions foru ∈ L2 that 0 satisfytheglobalenergyinequalityandbelongtoL∞(0,T;L2)∩L2(0,T;H1)foranyT > 0) remainsanopenproblem. Thebestresultsavailablerelyoncriticalquantitiesbeingfinite, thatisquantitieswhichareinvariantgiventhenaturalscalingassociatedwiththeNavier- Stokes equations. In this note we provide several regularity criteria which highlight the essential role ofhigh frequenciesin a possibly singular Leray-Hopf weaksolution. Frequencies are interpreted in the Littlewood-Paley sense. Let λ = 2j for j ∈ Z be j measured in inverse length scales and let B denote the ball of radius r centered at the r origin. Fix a non-negative, radial cut-off function χ ∈ C∞(B ) so that χ(ξ) = 1 for all 0 1 ξ ∈ B . Letφ(ξ) = χ(λ−1ξ)−χ(ξ)andφ (ξ) = φ(λ−1)(ξ). Suppose thatuisavector field 1/2 1 j j 1 oftempereddistributions andlet∆ u = F−1φ ∗ufor j ∈ N and∆ = F−1χ∗u. Then,u j j −1 can be written as u = ∆ u. j jX≥−1 If F−1φ ∗ u → 0 as j → −∞ in the space of tempered distributions, then for j ∈ Z we j define∆˙ u = F−1φ ∗u andhave j j u = ∆˙ u. j Xj∈Z For s ∈ R, 1 ≤ p,q ≤ ∞ the homogeneous Besov spaces include tempered distributions modulo polynomials forwhich the norm 1/q kuk := (cid:18) j∈Z λsjk∆˙ jukLp(Rn) q(cid:19) ifq < ∞ , B˙ps,q suPpj∈Zλ(cid:0)sjk∆˙ jukLp(Rn) (cid:1) ifq = ∞   isfinite. See [2]for more details. GivenaLeray-Hopfweaksolution uthatbelongstoC(0,T;B˙−ǫ )forsomeǫin(0,1),we ∞,∞ definethe following upperandlower endpointfrequencies: for tin (0,T)let 1/(1−ǫ) J (t) = log c ku(t)k , (1) high 2(cid:20) 1 B˙∞−ǫ,∞ (cid:21) and J (t) = log c ku(t)kB˙∞−ǫ,∞ 2/(3−2ǫ) , (2) low 2(cid:20)(cid:18) 2kukL∞(0,T;L2)(cid:19) (cid:21) where c and c are universal constants (their values will become clear in Section 2). Our 1 2 first regularity criterion shows J and J determine the Littlewood-Paley frequencies low high which, if well behaved at a finite number of times prior to a possible blow-up time, pre- ventsingularity formation. Theorem1. Fixǫ ∈ (0,1)andT > 0,and assumethatu ∈ C(0,T;B˙−ǫ )isaLeray-Hopfweak ∞,∞ solutionto 3DNSE on[0,T]. If there existst ∈ (0,T)suchthat 0 sup λ−j ǫk∆˙ ju(ti)kL∞ ≤ ku(t0)kB˙∞−ǫ,∞, (3) Jlow(t0)≤j≤Jhigh(t0) where{t }k ⊂ (t ,T)is afinitecollectionof k timessatisfying i i=1 0 2/(1−ǫ) c 3 t −t > (i = 0,...,k −1), i+1 i (cid:18)ku(t0)kB˙∞−ǫ,∞(cid:19) and 2/(1−ǫ) 2c 3 T −t < k (cid:18)ku(t0)kB˙∞−ǫ,∞(cid:19) for auniversalconstant c , thenu canbesmoothly extendedbeyond timeT. 3 2 The novelty here is that the solution remains finite provided only a finite range of frequencies remain subdued at a finite number of uniformly spaced times. If u is not in the energy class then a partial result can be formulated since J does not depend on high kukL∞(0,T;L2). Inparticular, we just needto replace(3)with sup λ−j ǫk∆˙ ju(ti)kL∞ ≤ ku(t)kB˙∞−ǫ,∞, j≤Jhigh(t) andassumeuisthe mildsolution foru ∈ B˙−ǫ whichisastrong solution on [0,T)(note 0 ∞,∞ that alocal-in-time existence theory for mildsolution isavailablein B˙−ǫ ). ∞,∞ Oursecondresultisarefinementofawellknownclassofregularitycriteria(see,e.g.,[7]): ifuis aLeray-Hopf weaksolution to 3DNSE on R3 ×[0,T]satisfying T kukq dt < ∞, Z Lp 0 for pairs(p,q)where 3 ≤ p ≤ ∞,2 ≤ q ≤ ∞, and 2 3 + = 1, q p then u is smooth. This is the Ladyzhenskaya-Prodi-Serrin class for non-endpoint values of (p,q). The case p = ∞ is the Beale-Kato-Majda regularity criteria. The case p = 3 was only (relatively) recently proven in [5]. Similar criteria can be formulated for a variety of spaces larger than Lp when p > 3. For example, Cheskidov and Shvydkoy give the following Ladyzhenskaya-Prodi-Serrin-type regularity criteria in Besov spaces (see [3]): if u is a Leray-Hopf solution and u ∈ L2/(1−ǫ)(0,T;B˙−ǫ ), then u is regular on (0,T]. A ∞,∞ regularity criterion for weakly time integrable Besov norms in critical classes appears in [1]. In the endpoint case when ǫ = −1, smallness is needed either over all frequencies (see [3]) or over high frequencies provided a Beale-Kato-Majda-type bound holds for the projection onto low frequencies (see [4]). Our result is essentially a refinement of the non-endpoint regularity criteria given in[3]. Theorem2. Fixǫ ∈ (0,1)andT > 0,and assumethatu ∈ C(0,T;B˙−ǫ )isaLeray-Hopfweak ∞,∞ solutionto 3DNSE on[0,T]. If T 2/(1−ǫ) sup λ−ǫk∆˙ u(t)k dt < ∞, Z (cid:18) j j ∞(cid:19) 0 Jlow(t)≤j≤Jhigh(t) thenuis regularon (0,T]. ClearlyJ blowsupmorerapidlythanJ ast → T− andthereforeanincreasingnum- high low ber of frequencies are relevant as we approach the possible blow-up time. It is unlikely thatthiscanbeimprovedforweaksolutionsinsupercriticalclasseslikeLeray-Hopfsolu- tions. On one hand, the upper cutoff is available because of local well-posedness for the subcritical quantity ku(t)kB˙∞−ǫ,∞ which suppresses high frequencies at times close to and 3 after t. On the other hand, the supercritical quantity kukL∞(0,T;L2) plays a crucial role in suppressing low frequencies. Any supercritical quantity is sufficient; for example, if we replace L∞L2 with L∞Lp for some 2 < p < 3, then the lower cutoff function is J (t) = log ku(t)kB˙∞−ǫ,∞ p/(3−pǫ) . low 2(cid:20)(cid:18)ckukL∞(0,T;Lp)(cid:19) (cid:21) Note that p/(3 − pǫ) = 1/(1 − ǫ) only when p = 3, i.e. the exponents in the cutoffs will match only when wereach acritical class L∞(0,T;L3). 2 Technical lemmas LocalexistenceofstrongsolutionsfordatainthesubcriticalspaceB˙−ǫ isknown,see[7]. ∞,∞ Resultsinspacesclose toB˙−ǫ are givenin[6,9]. Indeed,theproof of[6,Theorem 1]can ∞,∞ be modified to show that if a ∈ B˙−ǫ , then the Navier-Stokes equations have a unique ∞,∞ strong solution u which persists atleastuntil time 2/(1−ǫ) c 0 T = , (4) ∗ (cid:18)kakB˙∞−ǫ,∞(cid:19) for a universal constant c . Moreover we have 0 ku(t)kB˙∞−ǫ,∞ ≤ c0kakB˙∞−ǫ,∞, (5) and t1/2k∇u(t)kB˙∞−ǫ,∞ ≤ c0kakB˙∞−ǫ,∞, (6) for any t ∈ (0,T ) (the value of c changes from line to line but always represents a uni- ∗ 0 versal constant). Since the proof of this is nearly identical to the proof of [6, Theorem 1] it isomitted. Note that by [7, Proposition 3.2], the left hand side of (6)can be replaced by t1/2kukB˙∞1−,∞ǫ . Givenasolutionuandatimetsothatu(t) ∈ B˙−ǫ ,lett′ = t+T /2andt′′ = t+T whereT ∞,∞ ∗ ∗ ∗ isasin (4)with a = u(t). Wenow state and prove several (short) technical lemmas. Lemma 3. Fix ǫ ∈ [0,3/2) and T > 0. If u is a Leray-Hopf weak solution to 3D NSE on [0,T] and u(t) ∈ B˙−ǫ forsomet ∈ [0,T], thenforany M > 0we have ∞,∞ λ−ǫk∆˙ u(t)k ≤ M, j j ∞ provided 2/(3−2ǫ) M j ≤ log c 2(cid:20)(cid:18) kukL∞(0,T;L2)(cid:19) (cid:21) for asuitableuniversal constant c. 4 Proof. AssumeuisaLeray-Hopfweaksolutionon[0,T]andt ∈ [0,T]suchthatku(t)kB˙∞−ǫ,∞ < ∞. By Bernstein’sinequalitieswe have k∆˙ u(t)k ≤ λ3/2k∆˙ u(t)k . j ∞ j j 2 Since u ∈ L∞(0,T;L2) = L∞(0,T;B˙0 ),for anyj ∈ Z, 2,2 λ−j ǫk∆˙ juk∞ ≤ cλ3j/2−ǫkukL∞(0,T;L2). Let 2/(3−2ǫ) M J(t) = log ; 2(cid:20)(cid:18)ckukL∞(0,T;L2)(cid:19) (cid:21) then supλ−ǫk∆˙ uk ≤ M. j j ∞ j≤J Lemma4. Fix ǫ ∈ (0,1)and T > 0, and assume u is a Leray-Hopf weak solution to 3D NSE on [0,T]belonging toC(0,T;B˙−ǫ ). Then, forany t ∈ (0,T)and all t ∈ [t′,t′′]we have ∞,∞ 1 1 1 1 {j∈Z:j≤Jlsowuporj≥Jhigh}k∆˙ ju(t)kL∞ ≤ 2ku(t1)kB˙∞−ǫ,∞, whereJ and J are definedby (1)and (2). high low Proof. Using subcritical local well-posedness in B˙−ǫ at t we have that there exists a ∞,∞ 1 mild/strong solution v defined on [t ,t′′]. By(6)we have 1 1 (t−t1)1/2kv(t)kB˙∞1−,∞ǫ ≤ c0kv(t1)kB˙∞−ǫ,∞ for all t ∈ (t ,t′′). Since v(t ) = u(t ) ∈ L2 and since the strong solution v is smooth, 1 1 1 1 integration by parts verifies that v is also a Leray-Hopf weak solution to 3D NSE. The weak-strong uniqueness result of [8] then guarantees that u = v on [t ,t′′]. Thus, for any 1 1 t ∈ [t′,t′′], 1 1 λ−ǫk∆˙ u(t)k ≤ cλ−1ku(t )k1/(1−ǫ)+1 j j ∞ j 1 B˙∞−ǫ,∞ for all j ∈ Z. By (1)weconclude that 1 sup λ−j ǫk∆˙ ju(t)k∞ ≤ 2ku(t1)kB˙∞−ǫ,∞. j≥Jhigh The lowmodesare eliminated usingLemma3 with M = ku(t1)kB˙∞−ǫ,∞/2. Definition 5. WesaythattisanescapetimeifthereexistssomeM > 0suchthatt = sup{s ∈ (0,T) : ku(s)kB˙∞−ǫ,∞ < M}. 5 Lemma 6. Fix ǫ ∈ (0,1) and T > 1, and assume u is a Leray-Hopf weak solution to 3D NSE on [0,T]belonging to C(0,T;B˙−ǫ ). Let E denote the collectionof escape times in (0,T) and let ∞,∞ I = ∪ (t′,t′′). Then t∈E T 2/(1−ǫ) ku(t)k dt = ∞, (7) Z B˙∞−ǫ,∞ 0 ifand only if 2/(1−ǫ) ku(t)k dt = ∞. (8) Z B˙∞−ǫ,∞ I Proof. Itisobvious that (8)implies(7). Assume (7). Let {tk}k∈N ⊂ (0,T) be an increasing sequence of escape times which con- verge to T at k → ∞. Clearly ku(tk)kB˙∞−ǫ,∞ blows up as k → ∞. Since u ∈ C(0,T;B˙∞−ǫ,∞), ku(tk1)kB˙∞−ǫ,∞ < ku(tk2)kB˙∞−ǫ,∞ for all k1 < k2. Wehave two casesdependingon the condition ∃t ∈ {t } such that ∀k ≥ k we havet′ ≤ t′′. (9) k0 k 0 k+1 k Case1: If(9)istrue, then [t′,T) = ∪ [t′,t′′).Inthis case letI = [t′,T). Clearly 0 k≥k0 k k 0 2/(1−ǫ) ku(t)k dt = ∞. Z B˙∞−ǫ,∞ I Case 2: If (9) is false then there exists an infinite sub-sequence of {t }, which we label k {sk}, such that s′k′ < s′k+1 for allk ∈ N. Inthis case letI = ∪k∈N[s′k,s′k′). Then, T∗(s ) c2/(1−ǫ) ku(t)k2/(1−ǫ)dt ≥ k ku(s )k2/(1−ǫ) = 0 = ∞. Z B˙∞−ǫ,∞ 2 k B˙∞−ǫ,∞ 2 I Xk∈N Xk∈N Ineither case,we have shown that (7)implies(8). 3 Proofs of Theorem 1 and Theorem 2 Proof of Theorem1. Fixǫ ∈ (0,1)andT > 0,andassumeu ∈ C(0,T;B˙−ǫ )isaLeray-Hopf ∞,∞ weaksolution to3DNSEon[0,T]. Assumet ,...,t areasinthestatementofthelemma. 0 k Itsuffices to show ku(tk)kB˙∞−ǫ,∞ ≤ ku(t0)kB˙∞−ǫ,∞, since then we re-solve at t and, by local-in-time well-posedness and the weak-strong 0 uniquenessof[8], see thatu isregular attime T. If k = 0, then we are done. Otherwise note that t ∈ (t′,t′′). Apply Lemma 4 at t to 1 0 0 0 conclude that ku(t1)kB˙∞−ǫ,∞ ≤ ku(t0)kB˙∞−ǫ,∞. 6 Ifk = 1, then weare done. Otherwise, werepeat the argument andeventuallyobtain ku(tk)kB˙∞−ǫ,∞ ≤ ku(t0)kB˙∞−ǫ,∞, which completes the proof. Proof of Theorem2. Assume u is a Leray-Hopf weak solution on [0,T] which belongs to C(0,T;B˙−ǫ ). ∞,∞ ByLemma3 with M = ku(t)kB˙∞−ǫ,∞/2it follows that 1 sup λ−j ǫk∆˙ ju(t)k∞ < 2ku(t)kB˙∞−ǫ,∞. (10) j≤Jlow(t) Ifuloses regularity attime T,local well-posedness inB˙−ǫ impliesthat ∞,∞ (1−ǫ)/2 c ∗ ku(t)kB˙∞−ǫ,∞ ≥ (cid:18)T −t(cid:19) , for a smalluniversal constant c . Therefore, ∗ T 2/(1−ǫ) ku(t)k dt = ∞. Z B˙∞−ǫ,∞ 0 LetE denote the collection ofescape timesin (0,T)and letI = ∪ (t′,t′′). By Lemma6 t∈E 2/(1−ǫ) ku(t)k dt = ∞. Z B˙∞−ǫ,∞ I For each t ∈ I there exists an escapetime t (t) sothat t ∈ (t′,t′′). Thus, 0 0 0 2/(1−ǫ) 2/(1−ǫ) 1 c c 0 0 ≤ t−t ≤ . 0 2(cid:18)ku(t0)kB˙∞−ǫ,∞(cid:19) (cid:18)ku(t0)kB˙∞−ǫ,∞(cid:19) Byre-solvingatt usingsubcriticalwell-posedness,inequality(6),andweak-strongunique- 0 ness(see [8]), we have (t−t0)1/2ku(t)kB˙∞1−,∞ǫ ≤ c0ku(t0)kB˙∞−ǫ,∞. Consequently, λ−ǫk∆˙ u(t)k ≤ 2c λ−1ku(t )k1+1/(1−ǫ) ≤ 2c λ−1ku(t)k1+1/(1−ǫ), j j ∞ 0 j 0 B˙∞−ǫ,∞ 0 j B˙∞−ǫ,∞ where we haveused the fact that t isan escapetime. Using(1) weobtain 0 sup λ−ǫk∆˙ u(t)k < ku(t)kB˙∞−ǫ,∞. (11) j j ∞ 2 j≥Jhigh(t) 7 Combining(10)and (11)yields 2/(1−ǫ) Z (cid:18) sup λ−j ǫk∆˙ ju(t)kL∞(cid:19) dt = ∞, I Jlow(t)≤j≤Jhigh(t) which proves Theorem 2. Remark7. Ifwe only wanted to eliminate low frequenciesin Theorem 2 then an alterna- tiveproofisavailablewhichwepresentlysketch. Decompose[0,T]intoadjacent,disjoint intervals [t ,t ) with t − t ∼ 2−kT. Then, a solution which is singular at T must k k+1 k+1 k satisfy 2k . ku(t ∼ t )k2/(1−ǫ). k B˙∞−ǫ,∞ Usingthe Bernstein inequalitieswe have tk+1 2/(1−ǫ) tk+1 2/(1−ǫ) sup λ−ǫk∆˙ u(t)k dt ≤ sup λ3/2−ǫk∆˙ u(t)k dt Z (cid:18) j j ∞(cid:19) Z (cid:18) j j 2(cid:19) tk j≤J0(t) tk j≤J0(t) . kuk2/(1−ǫ)λ(3−2ǫ)/(1−ǫ)(t −t ) L∞L2 J0(t) k+1 k . kuk2/(1−ǫ)2J0(t)(3−2ǫ)/(1−ǫ)2−k. L∞L2 Define J so that J (t)(3 − 2ǫ)/(1 −ǫ) = k/2 for t ∈ [t ,t + 1). Then, terms on the right 0 0 k k hand sideare summable and weobtain T 2/(1−ǫ) supλ−ǫk∆˙ u(t)k dt < ∞. Z (cid:18) j j ∞(cid:19) 0 j≤J0 Since the integral over allmodesmust be infinite ata first singular time, we conclude T 2/(1−ǫ) sup λ−ǫk∆˙ u(t)k dt = ∞. Z (cid:18) j j ∞(cid:19) 0 j≥J0(t) FurtheranalyzingthedefinitionofJ andthelower-boundfortheB˙−ǫ normweseethat 0 ∞,∞ 2/(3−2ǫ) J (t) ∼ log ku(t)k , 0 2(cid:18) B˙∞−ǫ,∞ (cid:19) which matches the rate found using the other approach. Acknowledgements. The authors are grateful to V. Šverák for his insightful comments which simplified the proofs. Z.G. acknowledges support of the Research Council of Norway via the grant 213474/F20 and the National ScienceFoundation via the grant DMS1212023. 8 References [1] H. Bae, A. Biswas, and E. Tadmor. Analyticity and decay estimates of the Navier- Stokes equations in critical Besov spaces. Arch. Ration. Mech. Anal., 205(3):963–991, 2012. [2] H. Bahouri, J.-Y. Chemin, and R. Danchin. 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