ON THE REGULARITY SET AND ANGULAR INTEGRABILITY FOR THE NAVIER–STOKES EQUATION PIEROD’ANCONAANDRENATOLUCA` Abstract. We investigate the size of the regular set for suitable weak solu- 5 tionsoftheNavier–Stokesequation,inthesenseofCaffarelli–Kohn–Nirenberg 1 [2]. We consider initial data in weighted Lebesgue spaces with mixed radial- 0 angular integrability, and we prove that the regular set increases if the data 2 havehigherangular integrability,invadingthewholehalfspace{t>0}inan n appropriate limit. In particular, we obtain that if the L2 norm with weight −1 a |x| 2 of the data tends to 0, the regular set invades {t > 0}; this result J improvesTheoremDof[2]. 0 3 ] P 1. Introduction and main results A We consider the Cauchy problem for the Navier–Stokes equation on R+×R3 . h ∂ u+(u·∇)u−∆u = −∇P t at ∇·u = 0 (1.1)  m u(x,0) = u (x).  0 [ describing a viscous incompressible fluid in the absence of external forces, where  as usual u is the velocity field of the fluid and P the pressure, and the initial data 1 v satisfy the compatibility condition ∇·u0 = 0. We use the same notation for the 0 norm of scalar, vector or tensor quantities: 8 7 kPkL2 :=( P2dx)12, kuk2L2 := jkujk2L2, k∇uk2L2 := j,kk∂kujk2L2 7 0 and we writeRsimply L2(R3) instead oPf [L2(R3)]3, or S′(R3) instePad of [S′(R3)]3 . and so on. Regularity of the global weak solutions constructed in [17, 21] is a 1 notorious open problem, although many partial results exist. 0 5 The case of small data is well understood. In the proofs of well posedness for 1 smalldata,the equationis regardedasa linearheatequationperturbed by asmall : nonlinearterm(u·∇)u,andthenaturalapproachisafixedpointargumentaround v i the heat propagator. More precisely, one rewrites the problem in integral form X u=et∆u − te(t−s)∆P∇·(u⊗u)(s) ds in R+×R3 (1.2) r 0 0 a where P is the Leray projRection Pf :=f −∇∆−1(∇·f), and then the Picard iteration scheme is defined by u :=et∆u , u :=et∆u − te(t−s)∆P∇·(u ⊗u )(s) ds. (1.3) 1 0 n 0 0 n−1 n−1 Once the velocity is known the presRsure can be recovered at each time by P = −∆−1∇⊗∇(u⊗u). Small data results fit in the following abstract framework: Date:February2,2015. 2000 Mathematics Subject Classification. 35Q30,35K55,42B20. Theauthors arepartiallysupported by the ItalianProject FIRB 2012 “Dispersivedynamics: Fourier Analysis and Variational Methods”. The second author is supported by the ERC grant 277778andMINECOgrantSEV-2011-0087(Spain). 1 ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 2 Proposition 1.1 ([20]). Let X ⊂ L2L2 ((0,s)×R3)1 be a Banach space s<∞ t uloc,x such that the bilinear form T B(u,v):= te(t−s)∆P∇·(u⊗v)(s) ds (1.4) 0 is bounded from X ×X to X: R kB(u,v)k ≤C kuk kvk . X X X X Moreover, let X ⊂S′(R3) be a normed space such that et∆ :X →X is bounded: 0 0 ket∆fk ≤A kfk . X X0,X X0 Then foreverydatau suchthatku k <1/4C A thesequenceu isCauchy 0 0 X0 X X0,X n in X and converges to a solution u of the integral equation (1.2). The solution satisfies kuk ≤2A ku k . X X0,X 0 X0 The space X is usually called an admissible (path) space, while X is called an 0 adapted space. ManyadaptedspacesX havebeenstudied: L3 [18], Morreyspaces 0 [16, 33], Besov spaces [4, 14, 24] and several others. The largest space in which Picard iteration has been proved to converge is BMO−1 [19]. Acrucialingredientinthetheoryissymmetryinvariance. Thenaturalsymmetry of the Navier–Stokes equation is the translation-scaling u (x)7→λu (λ(x−x )), λ∈R+, x ∈R3, 0 0 0 0 andindeedallthespacesX mentionedaboveareinvariantforthistransformation. 0 Ontheotherhand,inresultsoflocalregularityarolemaybeplayedbysomespaces which are scaling but not translation invariant, like the weighted Lp spaces with norm k|x|1−p3u(x)kLp(R3). When p = 2 this is the weighted L2 space with norm k|x|−21u(x)kL2, used in the classical regularity results of [2]. We recall a key definition from that paper: Definition 1.2. Apoint(t ,x )∈R+×R3 is regular for asolutionu(t,x)of(1.1) 0 0 if u is essentially bounded on a neighbourhood of (t ,x ). It follows that u(t,x) is 0 0 smoothnear (t ,x ) (see for instance [28]). A subset of R+×R3 is regular if all its 0 0 points are regular. The following result (Theorem D in [2]) applies to the special class of suitable weak solutions, which are, roughly speaking, solutions with bounded energy; see the beginning of Section 2 for the precise definition. We use the notation |x|2 Π := (t,x)∈R+×R3 : t> α α (cid:26) (cid:27) to denote the paraboloidof aperture α in the upper half space R+×R3; note that Π is increasing in α. α Theorem 1.3 (Caffarelli–Kohn–Nirenberg). There exists a constant ε > 0 such 0 that the following holds. Let u be a suitable weak solution of Problem (1.1) with divergence free initial data u ∈L2(R3). If 0 k|x|−1/2u k2 =ε<ε 0 L2(R3) 0 then the paraboloid |x|2 Π ≡ (t,x) : t> ε0−ε ε −ε (cid:26) 0 (cid:27) 1The space L2 consists of the functions that are uniformly locally square-integrable (see uloc [20] Definition 11.3). The operator (1.4) is well-defined on Ts<∞L2tL2uloc,x((0,s) × R3) × Ts<∞L2tL2uloc,x((0,s)×R3). Wereferto[20],Chapter 11,formoredetails. ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 3 is a regular set. ThetheoremstatesthatiftheweightedL2 normofthedataissufficientlysmall, thenthesolutionissmoothonacertainparaboloidwithvertexattheorigin. Ifthe size of the data tends to 0, the regular set increases and invades a limit paraboloid Π , which is strictly contained in the half space t>0. ε0 It is reasonable to expect that the regular set actually invades the whole upper half space t>0 when the size of the data tends to 0. This is indeed a special case of our main result, see Theorem 1.5 below and in particular Corollary 1.6. However our main goal is a more general investigation of the influence on the regular set of additional angular integrability of the data. We measure angular regularity using the following mixed norms: 1 kfkLp Lpe := 0+∞kf(ρ · )kpLpe(S2)ρ2dρ p , (1.5) |x| θ kfkL∞Lpe := s(cid:16)uRpρ>0kf(ρ · )kLpe(S2). (cid:17) |x| θ The idea of separating radial and angular regularity is not new; it proved useful especially in the context of Strichartz estimates and dispersive equations (see [5], [8], [13], [23], [26] [34]). The Lp Lpescale includes the usual Lp norms when p=p: |x| θ kukLp Lp =kukLp(R3). |x| θ e Note also that for radial functions the value of p is irrelevant, in the sense that u radial =⇒ kukLp Lpe ≃kukLp(R3) ∀p,p∈[1,∞] |x| θ e while for generic functions we have only2 e kukLp Lpe .kukLp Lpe1 if p≤p1. |x| θ |x| θ With respectto scaling,the mixedradial-angularnoermLep|x|Lpθebehaveslike Lp and in particular we have for all p∈[1,∞] and all λ>0 3 k|x|αλu0(λx)kLp|x|Lpθe =ek|x|αu0(x)kLp|x|Lpθe, provided α=1− p. As a first application,we show thatfor initial data with small k|x|αu0kLp Lpe norm |x| θ and p ≥ 2p/(p−1), the problem has a global smooth solution. As we prove in −1+3/q Section 2, this norm controls the B norm (for q large enough), and this q,∞ spaceeis embedded in BMO−1, thus the existence part in Theorem 1.4 could be deduced from the more general results in [4, 19, 24]. However, the quantitative estimate (1.9) is new for suchinitial data,andit willbe a crucialtoolfor the proof of our main Theorem 1.5. Theorem 1.4. Let 1<p<5, p≥2p/(p−1), α=1−3/pandlet u ∈Lp Lpe 0 |x|αpd|x| θ be divergence free. Moreover, let 2p ≤eq <∞ if 1<p≤2 p−1 2p ≤q < 3p if 2≤p≤3 (1.6) p−1 p−2 p<q < 3p if 3≤p<5 p−2 and 2 3 + =1. (1.7) r q 2AsusualwewriteA.BifthereisaconstantC independentofA,BsuchthatA≤CBand A≃B ifA.B andB.A. ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 4 Then there exists an ε¯=ε¯(p,p,q)>0 such that, if k|x|αu0kLp Lpe <ε¯, (1.8) e |x| θ Problem (1.2) has a unique global smooth solution u satisfying3 kukLrtLqx ≤C¯k|x|αu0kLp|x|Lpθe. (1.9) for some constant C¯ =C¯(p,p,q) independent of u . 0 In the following we shall need only the special case corresponding to the choice e p=2, p=4, q =4. Thus, using the notations e ε :=ε¯(2,4,4), C :=C¯(2,4,4), (1.10) 1 1 we see in particular that for all divergence free initial data with k|x|−1/2u k <ε (1.11) 0 L2 L4 1 |x| θ there exists a unique global smooth solution u(t,x), which satisfies the estimate kuk ≤C k|x|−1/2u k . (1.12) L8L4 1 0 L2 L4 t x |x| θ To prepare for our last result, we introduce the notations θ (p):=(2pe−4)1−pe/4, θ (p):=(2pe−4)1−pe/2, p∈(2,4). 1 4−pe 2 4−pe It is easy to check that θ ,θ ∈[0,1] and actually 1 2 e e e lim θ =0, lim θ =1, (1.13) 1 1 pe→2+ pe→4− lim θ =1, lim θ =0. (1.14) 2 2 pe→2+ pe→4− Thus we may set θ (2)=0, θ (2)=1. We also define the norm 1 2 [u0]pe:=k|x|−p2eu0kLp2ep|−ex/|12Lpθek|x|−p1eu0k2L−pxep2e. (1.15) Note the following facts: (1) It is easy to construct initial data such that [u0]peis arbitrarily small while ku k isarbitrarilylarge. Indeed,fixatestfunctionφ∈C∞(R3)and 0 BMO−1 c denote with φ (x) := φ(x−Kξ) its translate in the direction ξ for some K |ξ|=1 and K >1; we have obviously k|x|−p1eφKkLpxe ≃K−p1e sincetheLpenormistranslationinvariant. Ontheotherhand,ifthesupport x of φ is contained in a sphere B(0,R), we have k|x|−p2eφKkpLe/pe2/2Lpe = 0+∞( S2|φ(θρ−Kξ)|pedSθ)12ρdρ. KK−+RRK−1ρdρ≃1 |x| θ R R R and we obtain [φK]pe.(1)p2e−1(K−p1e)2−p2e =K12−p2e. Thus, by the translation invariance of BMO−1, we conclude that if p ∈ [2,4) [φK]pe→0 while kφKkBMO−1 =const as K →∞. (1.e16) 3Here and in the following we use the notation kfkXYZ := kkkfkZkYkX for nested norms. WhenwewritekukLrtLqx wemeanthattheinegrationisextended toallthetimest>0. ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 5 (2) In the limit cases p=2 and p=4 we have simply [u ] =k|x|−1/2u k , [u ] =k|x|−1/2u k (1.17) 0 2 e 0 L2e 0 4 0 L2 L4 x |x| θ and actually the [ · ]pe norm arises as an interpolation norm between the two cases (see (4.2), (4.3) and (4.5) below). Wecannowstateourmainresult,whichinterpolatesbetweenTheorems1.3and 1.4: Theorem 1.5. There exists a constant δ > 0 such that the following holds. Let u be a suitable weak solution of Problem (1.1) with divergence free initial data u ∈L2(R3), and let p∈[2,4) and M >1. 0 If the norm [u0]pe of the initial data satisfies e θ1·[u0]pe≤δ, θ2·[u0]pe≤δe−4M2 (1.18) then the paraboloid |x|2 Π := (t,x)∈R+×R3 : t> (1.19) Mδ Mδ (cid:26) (cid:27) is a regular set for u(t,x). The result can be interpreted as follows. Since θ (p) → 0 as p → 4, we can 2 choose p=p as a function of M in such a way that M e e e4M2 ·θ (p )→0 as M →+∞. e e 2 M Of course we have p →4− as M →+∞. Then from the theorem it follows that, M e for all sufficiently large M, e [u0]peM ≤δ =⇒ ΠMδ is a regular set for u. In other words, if we take M → +∞ and the norm [u0]peM is less than δ, then the regular set invades the whole half space t > 0. Note that, as remarked above, the [u0]peM norm can be small even if the BMO−1 norm of u0 is large. Even in the special case p=2, which is coveredby Theorem D of [2], the result gives some new information on the regular set. Indeed, for p = 2 we have θ = 0, 1 θ =1, and we obtain: 2 e e Corollary 1.6. There exists a constant δ > 0 such that for any suitable weak solution u with divergence free initial data u ∈ L2(R3), and for every M > 1, if 0 the initial data satisfy k|x|−1/2u k ≤δe−4M2 0 L2 x then the paraboloid Π is a regular set for u. Mδ Thus, taking M → +∞, we see that if the weighted L2 norm of the data is sufficientlysmall,thentheregularsetinvadesthewholehalfspacet>0,asclaimed above. Therestofthepaperisorganizedasfollows. InSection2wecollectthenecessary tools,inparticularwerecallthefundamentalCaffarelli–Kohn–Nirenbergregularity criterion from [2]; in Section 3 we prove Theorem 1.4, and Section 4 is devoted to the proof of Theorem 1.5. ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 6 2. Preliminaries We recall some definitions from [2]. Definition 2.1. Let u ∈L2(R3). The couple (u,P) is a suitable weak solution of 0 Problem (1.1) if4 (1) (u,P) satisfies (1.1) in the sense of distributions; (2) u(t)→u weakly in L2 as t→0; 0 (3) for some constants E ,E 0 1 |u|2(t) dx≤E , 0 R3 Z for all t>0 and |∇u|2 dtdx≤E ; 1 Z ZR+×R3 (4) for all non negative φ∈C∞([0,∞)×R3) and for all t>0 c t |u|2φ(t)+2 |∇u|2φ (2.1) ZR3 Z0 ZR3 t t ≤ |u |2φ(0)+ |u|2(φ +∆φ)+ (|u|2+2P)u·∇φ. 0 t ZR3 Z0 ZR3 Z0 ZR3 Suitable weak solutions are known to exist for all L2 initial data, see [27] or the Appendix in [2]. Such solutions are also L2-weakly continous as functions of time (see [35], pp. 281–282),namely u(t,x)w(x) dx→ u(t′,x)w(x) dx (2.2) R3 R3 Z Z for all w ∈ L2(R3) as t → t′ (t,t′ ∈ [0,+∞)); thus it makes sense to impose the initial condition (2). Next we define the parabolic cylinder of radius r and top point (t,x) as Q (t,x):= (s,y): |x−y|<r, t−r2 <s<t r while the shifted parabolic c(cid:8)ylinder is (cid:9) Q∗(t,x):=Q (t+r2/8,x)≡ (s,y): |x−y|<r, t−7r2/8<s<t+r2/8 r r The crucial regularity result in(cid:8)[2] ensures that: (cid:9) Lemma 2.2. There exists an absolute constant ε∗ such that if (u,P) is a suitable weak solution of (1.1) and 1 limsup |∇u|2 ≤ε∗, (2.3) r r→0 Z ZQ∗r(t,x) then (t,x) is a regular point. Weshallmakefrequentuseofthefollowinginterpolationinequalityfrom[1](see also [9, 10] for extensions of the inequality): Lemma 2.3. Assume that (1) r≥0, 0<a≤1, γ <3/r, α<3/2, β <3/2; (2) −γ+3/r=a(−α+1/2)+(1−a)(−β+3/2); (3) aα+(1−a)β ≤γ; (4) when −γ+3/r=−α+1/2, assume also that γ ≤a(α+1)+(1−a)β. 4This definition of suitable weak solutions is appropriate to work with the initial datum u0. FormoredetailscomparetheSections2and7of[2]. ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 7 Then kσνγukLr(R3) ≤Ckσνα∇ukaL2(R3)kσνβuk1L−2(aR3), (2.4) where σ :=(ν+|x|2)−1/2, ν ≥0, with a constant C independent of ν. ν Akeyroleinthefollowingwillbeplayedbytime-decayestimatesforconvolutions with the heat and Oseen kernels. It is convenient to introduce the notation 2 2 Λ(α,p,p):=α+ − . p p Proposition 2.4 ([22]). Let 1≤p≤eq ≤∞, 1≤p≤q ≤∞ and e 3 3 β >− , α<3− , Λ(α,pe,p)e≥Λ(β,q,q). (2.5) q p For every multiindex η, e e (1) if |η|+ 3 − 3 +α−β ≥0, then p q 1 k|x|β∂ηet∆u0kLq|x|Lqθe . t(|η|+p3−q3+α−β)/2k|x|αu0kLp|x|Lpθe, t>0; (2.6) (2) if 1+|η|+ 3 − 3 +α−β >0, then p q 1 k|x|β∂ηet∆P∇·FkLq|x|Lqθe . t(1+|η|+p3−3q+α−β)/2k|x|αFkLp|x|Lpθe, t>0. (2.7) An easy consequence of Proposition (2.4) is the embedding Lp Lpe ֒→B−1+3/q if α=1− 3, p≥ 2p , q ≥max(p,p), |x|αpd|x| dθ q,∞ p p−1 whichisnotneededinthefollowing,butallowstoceompareTheorem1.4witheearlier results;recallalsothatB−1+3/q ֒→BMO−1forq >3. Indeed,usingestimate(2.6), q,∞ we can write ket∆φkLq(R3) ≤Ct−(3/p−3/q+α)/2k|x|αφkLp Lpe ≡Ct−(1−3/q)/2k|x|αφkLp Lpe |x| θ |x| θ andthentheembeddingfollowsimmediatelyfromthefollowing’caloric’defininition of Besov spaces (see e.g. [19]): Definition 2.5. Adistributionφ∈S′ belongstoB−1+3/q(R3)(q >3)ifandonly q,∞ if ket∆φkLq(R3) ≤Ct−(1−3/q)/2 for 0<t≤1. (2.8) The best constant C in (2.8) is equivalent to the norm kφk . Bq−,∞1+3/q(R3) We conclude this sectionwith anestimate for singularintegralsin mixedradial- angular norms. Let K ∈C1(S2) with zero mean value and K(y) y Tf(x):=PV f(x−y) dy, y = . (2.9) R3 |y|n |y| Z b Theorem 2.6. Let 1<p<∞, 1<p<∞. Then b kTfkLp|xe|Lpθe .kfkLp|x|Lpθe. (2.10) The inequality (2.10) has been recently provedby A. C´ordoba in the case p=2 ([6], Theorem 2.1); essentially the same argument gives also the other cases. e ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 8 Proof. Consider first the case p>p. Let 1/q+p/p=1 and denote by X the set of all g ∈S(R) with +∞gq(ρ)ρ2dρ=1. Then we can write 0 kTfRkpe = +∞e |Tf(ρ,eθ)|pedS ppe ρ2 dρ ppe Lp Lpe 0 S2 θ |x| θ =(cid:16)suRp +(cid:0)∞R |Tf(ρ,θ)|peg(ρ(cid:1))ρ2 dS d(cid:17)ρ 0 S2 θ g∈X =supR |TRf(x)|peg(|x|) dx. R3 g∈X R Write I(f,g):= |Tf(x)|peg(|x|)dx. By Proposition 1 in [7] we have R3 R I(f,g).s R3|f(x)|pe(Mgs(x))1s dx, for all 1 < s < ∞, where M iRs the Hardy–Littelwood maximal operator and gs(x)=(g(|x|))s. Since Mgs is radially symmetric, this can be written I(f,g).s 0+∞ S2|f(ρ,θ)|pe(Mgs(ρ))1s ρ2dSθdρ. Now, for s<q = p−ppe, Ho¨ldRer’s inRequality with exponents p/p, q gives I(f,g). 0+∞ S2|f(ρ,θ)|pedSθ ppe ρ2 dρ ppe 0+∞(Mges(ρ))qs ρ2dρ 1q .(cid:16)kfRkpe (cid:0)RkMgsk1/s (cid:1).kfkpe(cid:17) (cid:16)kRgsk1/s (cid:17) Lp Lpe Lq/s(R3) Lp Lpe Lq/s(R3) |x| θ |x| θ 1 ≃kfkpe +∞gq(ρ)ρ2 dρ q =kfkpe Lp Lpe 0 Lp Lpe |x| θ |x| θ (cid:16)R (cid:17) and taking the supremum over all g ∈ X we get the claim in the case p > p. The case p=p is classical, and the case p<p follows by duality. (cid:3) e Using the continuity of T in weighted Lebesgue spaces (see Stein [31]) e e k|x|βTfkLp(R3) .k|x|βfkLp(R3) for 1<p<∞, −p3 <β <3− p3 (2.11) we can also obtain weighted versions of (2.10). In particular, by interpolation of (2.10) in the case (α ,p ,p )=(0,2,10) 0 0 0 (2.12) (2.11) in the case (α ,p ,p )=(−4/3,2,2), 1 1 1 with θ =3/8 (⇒(α ,p ,p )=(−1/2,2,4)),weeget θ θ θ e k|x|−1/2Tfk .k|x|−1/2fk . (2.13) L2 L4 L2 L4 e |x| θ |x| θ Remark 2.1. We denote with R the Riesz transform in the direction of the j- j th coordinate and R := (R ,R ,R ). By (2.11, 2.13) the boundedness of R in 1 2 3 j L2(R3,|x|−1dx) and L2 L4(R3,|x|−1dx) follows, and so that of P≡Id+R⊗R. |x| θ 3. Proof of Theorem 1.4 We first need two technical lemmas. By standard machinery, integral estimates for the heat flow and for the bilinear operator appearing in the Duhamel represen- tation (1.2) can be deduced by the time-decay estimates of Proposition 2.4. Lemma 3.1 ([22]). Let β >−3/q, α<3−3/p, 1≤p≤q ≤∞, 1<r <∞ and 1≤p≤q ≤∞ if (|η|+α−β)p+1<0, (3.1) 1≤p≤q < 3p if (|η|+α−e β)ep+1≥0. (|η|+α−β)p+1 Assume further that 3 3 2 |η|+α+ =β+ + , Λ(α,p,p)≥Λ(β,q,q). (3.2) p q r e e ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 9 Then for every multiindex η we have k|x|β∂ηet∆u0kLrLq Lqe .k|x|αu0kLp Lpe. (3.3) t |x| θ |x| θ Remark 3.1. Oncewehaveassumedthescalingrelationin(3.2),itisstraighforward to check that the assumption (3.1) is equivalent to p<r. Proof. The family of estimates (3.3) follows by the family of estimates (2.6) and by the Marcinkiewickz interpolation theorem. The condition p < r, which as re- markedaboveturnsoutto beequivalentto(3.1),isnecessaryinordertoapplythe Marcinkiewickz theorem (see Proposition3.4 in [22] for details). (cid:3) Lemma 3.2. Let 3<q <∞, 2<r <∞ satisfying 2/r+3/q=1. Then t (cid:13)(cid:13)Z0 e(t−s)∆P∇·(u⊗v)(s) ds(cid:13)(cid:13)LrtLqx .kukLrtLqxkvkLrtLqx. (3.4) (cid:13) (cid:13) The inequa(cid:13)lity (3.4) is well known, see fo(cid:13)r instance Theorem 3.1(i) in [12]. The LrLq Lebesgue spaces have been extensively used in the context of Navier–Stokes t x equation since [12, 15]. Using the previous estimates, it is a simple matter to prove Theorem 1.4. We follow the scheme of the proof of Theorem 20.1(B) in [20] and we take advance of the inequalities (2.6, 3.3). Proof of Theorem 1.4. Let p :=2p/(p−1). We show that the space G X :=(cid:26)u : keukLrtLqx <∞, st>up0t1/2kukL∞x (t)<∞(cid:27), epqauthipsppeadcewiwthiththaednaoprtmedksp·kaXce:X=k:·=kLLrtpLqx+suLptpe>G0.t1/2k·kL∞x (t), is an admissible 0 |x|αpd|x| θ The estimate ket∆fk . kfk follows indeed by the inequalities (2.6, 3.3); it X X0 is straightforward to check that (3.1) and p,p ≤ q are equivalent5 to (1.6) and G that the last assumption in (3.2) and in (2.5) is satisfied because Λ(α,p,p ) = G Λ(0,q,q) = Λ(0,∞,∞) = 0. Notice also theat the set of q for which the third inequality in (1.6) is satisfied is not empty provided p<5. e It remains to show that kB(u,v)kX . kukXkvkX. The bound kB(u,v)kLrtLqx . kukLrtLqxkvkLrtLqx followsbyLemma3.2. Inordertooestimatesupt>0t1/2kB(u,v)kL∞(t), we split this quantity as supt1/2kB(u,v)k (t)≤I+II L∞ x t>0 where I =sup t1/2 t/2e(t−s)∆P∇·(u⊗v)(s) ds t>0 0 L∞ (cid:13) (cid:13) x (cid:13)R (cid:13) (cid:13) (cid:13) II =sup t1/2 t e(t−s)∆P∇·(u⊗v)(s) ds , t>0 t/2 L∞ (cid:13) (cid:13) x (cid:13)R (cid:13) (cid:13) (cid:13) 5Exceptthatthevalueq=pisnotallowedin(1.6). ANGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 10 andwe use Minkowskiinequalityandthe time-decayestimate (2.7). For I we have t/2 1 I . stu>p0t1/2Z0 (t−s)(1+q/32)/2kukLqxkvkLqx(s) ds t/2 . supt−3/q kukLqxkvkLqx(s) ds t>0 Z0 1−2 r . stu>p0t−3/qkukLrtLqxkvkLrtLqx(cid:18)Z χ[0,t/2](s) ds(cid:19) . kukLrtLqxkvkLrtLqxt−3/q−2/r+1 =kukLrtLqxkvkLrtLqx while for II we have t 1 1 II . supt1/2 s1/2kuk (s) s1/2kvk (s) ds (t−s)1/2s L∞x L∞x t>0 Zt/2 (cid:16) (cid:17)(cid:16) (cid:17) t 1 . supt1/2kuk supt1/2kvk supt−1/2 ds L∞x L∞x (t−s)1/2 (cid:18)t>0 (cid:19)(cid:18)t>0 (cid:19)t>0 Zt/2 t/2 . supt1/2kuk supt1/2kvk supt−1/2 (t−s)1/2 L∞ L∞ (cid:18)t>0 x (cid:19)(cid:18)t>0 x (cid:19)t>0 h it . supt1/2kuk supt1/2kvk . L∞ L∞ x x (cid:18)t>0 (cid:19)(cid:18)t>0 (cid:19) Summing up we obtain kB(u,v)kX .kukLrtLqxkukLrtLqx+ supt>0t1/2kukL∞x supt>0t1/2kvkL∞x .kukXkukX. The existence of a unique solutio(cid:0)n u to Problem (1(cid:1).2(cid:0)) satisfying (cid:1) kukLrtLqx +st>up0t1/2kukL∞x (t).k|x|αu0kLp|x|Lpθe (3.5) follows by Proposition 1.1 and by the obvious inequality k|x|αu0kLp LpeG .k|x|αu0kLp Lpe. |x| θ |x| θ Finally,inequality (3.5)impliesthe boundednessofthe solutionuin(δ,∞)×R3 for all δ > 0, and this implies smoothness of the solution (see Theorem 3.4 in [12] or [11, 15, 28, 30, 32, 36]). (cid:3) We denote with BC([0,∞);L2) the Banach space of bounded continuous func- tions u:[0,∞)→L2 equipped with the norm sup ku(t)k . t≥0 L2 Corollary 3.3. Assume all the hypotheses of Theorem 1.4 are satisfied, and in ad- dition assume u ∈L2(R3). Then the solution u(t) belongs to BC([0,∞);L2(R3)). 0 In particular u is a strongsolution of (1.1), u(t)→u strongly in L2(R3) as t→0, 0 and the energy identity ku(t)k2 +2k∇uk2 =ku k2 holds for all t>0. L2 L2L2 0 L2 x t x Proof. Let X,X be the same admissible and adapted spaces used in the proof of 0 Theorem 1.4. As in that proof, we shall show that the space X ∩BC([0,∞);L2) x equippedwiththenormk·k +k·k isanadmissiblepathspacewithadapted X L∞L2 t x space X ∩L2 equipped with the norm k·k +k·k . 0 x X0 L2x The estimate ket∆fk . kfk again follows by (2.6, 3.3). X∩BC([0,∞);L2x) X0∩L2x Since we have already proved kB(u,v)k . kuk kvk , it remains to show that X X X kB(u,v)k . kuk kvk . By Minkowski inequality L∞L2 X∩BC([0,∞);L2) X∩BC([0,∞);L2) t x x x