Regularity criteria with angular integrability for the Navier-Stokes equation
aa r X i v : . [ m a t h . A P ] J a n REGULARITY CRITERIA WITH ANGULAR INTEGRABILITYFOR THE NAVIER–STOKES EQUATION
RENATO LUC `A
Abstract.
We give new a priori assumptions on weak solutions of the Navier–Stokes equation so as to be able to conclude that they are smooth. Theregularity criteria are given in terms of mixed radial-angular weighted Lebesguespace norms. Introduction and main results
We consider the Cauchy problem on (0 , T ) × R n ∂ t u + ( u · ∇ ) u − ∆ u = −∇ P ∇ · u = 0 u ( x,
0) = u ( x ) . (1.1)It describes the motion of a viscous incompressible fluid in the absence of externalforces, where u is the velocity field and P is the pressure.The first equation is the Newton law while the second follows by the incompress-ibility of the fluid. In order to require incompressibility at time t = 0 it is necessaryto restrict to initial data u such that ∇ · u = 0.We shall use the same notation for the norm of scalar, vector or tensor quantities,for instance: k P k L := R P dx, k u k L := R P ni =1 u i dx, k∇ u k L := R P ni,j =1 ( ∂ i u j ) dx and we often write simply u ∈ L ( R n ) instead of u ∈ [ L ( R n )] n .The well-posedness of (1.1) is still open even if many partial results have beenobtained. In [12, 17] the authors proved global existence of weak solutions for initialdata in L but a satisfactory well-posedness theory is basically developed only inthe case of small initial data or data with a peculiar geometric structure.In this scenario it is useful to establish a priori conditions under which uniquenessand regularity of the weak solutions are guaranteed. Results of this kind are usuallycalled regularity criteria.In this paper we focus on some classical regularity criteria [2, 21, 22, 24] andtheir extension to the setting of weighted Lebesgue spaces [26]. In particular weshow how the results in [26] can be improved under the hypothesis of additionalangular integrability.The regularity is basically ensured by boundedness assumptions on quantitieslike u, ∇ u, ∇ × u in suitable critical spaces. A simple regularity criterion is forinstance k u k L sT L px := Z T (cid:18)Z R n | u ( t, x ) | p dx (cid:19) sp dt ! s < + ∞ , s + np ≤ . (1.2)Notice that in the endpoint case (1.2) is invariant with respect to u ( t, x ) → λu ( λ t, λx ) , (1.3) Date : July 24, 2018. that is the natural scaling of (1.1). In [21] smoothness in space variables hasbeen obtained in the case s + np <
1, while the endpoints have been fixed in[9, 11, 22, 24, 28]. We recall the following
Definition 1.1 ([2]) . We say that a point (¯ t, ¯ x ) ∈ (0 , T ) × R is regular for asolution u ( t, x ) of (1.1) if u is essentially bounded on a neighbourhood of (¯ t, ¯ x ). (Inthis case one can prove that u ( t, x ) is smooth near (¯ t, ¯ x ), see for instance [21]). Wesay that a set is regular if all its points are regular.Let us also recall that (0 , T ) × R n is regular provided that (1.2) is satisfied with2 /s + n/p = 1 (see for instance [22, 24]).Then we focus on the weighted norm approach: Theorem 1.2 ([26]) . Let n ≥ and u ∈ L ( R n ) be a divergence free vector field.Let then u be a weak solution of (1.1) and ¯ x ∈ R n such that k| x − ¯ x | − n u k L x < + ∞ , (1.4) k| x − ¯ x | α u ( x, t ) k L sT L px < + ∞ , (1.5) with s + np = 1 − α, − ≤ α < − α < s < + ∞ , n − α < p < + ∞ ; (1.6) or k| x − ¯ x | α u ( x, t ) k L − αT L ∞ x < + ∞ , − < α <
1; (1.7) or sup t ∈ (0 ,T ) k| x − ¯ x | α u ( x, t ) k L n − αx < ε, − ≤ α ≤
1; (1.8) with ε sufficiently small. Then (0 , T ) × { ¯ x } is a regular set.Remark . The condition s + np = 1 − α makes the norm k| x − ¯ x | α u ( x, t ) k L sT L px scaling invariant with respect to u ( t, x − ¯ x ) → λu ( λ t, λ ( x − ¯ x )) . Our goal is to point out the local aspect of Theorem 1.2: for each t ∈ (0 , T )there is a neighborhood Ω t, ¯ x of ¯ x such that u is smooth in { t } × Ω t, ¯ x .The restriction to a neighborhood of ¯ x can be heuristically explained in the case α <
0: the weight morally localizes the norm around ¯ x and a loss of information atinfinity occurs.We shall show how to recover this information by a suitable amount of angularregularity (if α <
0) and how to do the same in the case α > x = 0. All the followingresults are of course true provided with the norms and weights centered at ¯ x = 0.In order to quantify precisely our notion of angular regularity we define thenorms k f k L p | x | L e pθ := (cid:16)R + ∞ k f ( ρ · ) k pL e p ( S n − ) ρ n − dρ (cid:17) p , k f k L ∞| x | L e pθ := sup ρ> k f ( ρ · ) k L e p ( S n − ) . (1.9)If p = e p the norms reduce to the usual L p norms k u k L p | x | L pθ = k u k L p ( R n ) , We mean a neighborhood in the space variables for each fixed time.
NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 3 while for radial functions the value of e p is irrelevant u radial = ⇒ k u k L p L e p ≃ k u k L p ( R n ) ∀ p, e p ∈ [1 , ∞ ] . Notice also that the norms (ignoring the constants) are increasing in e p .The idea of distinguishing radial and angular directions is not new and has provedsuccessful in the context of Strichartz estimates and dispersive equations (see forinstance [1], [3], [10], [18], [19] [27]).We also notice that the mixed angular-radial norms have the same scaling oftheir classical counterparts, in fact k| x | α u ( t, x ) k L sT L p | x | L e pθ is invariant with respect to u ( t, x ) → λu ( λ t, λx ) , provided that s + np = 1 − α .We obtain new values e p G , e p L for the angular integrability such that global andlocal regularity are, respectively, achieved: e p L := n − p (2 α +1) p +2( n − if − ≤ α < n − pp +2( n − if 0 ≤ α < , (1.10) e p G := max (cid:16) n, ( n − pαp + n − (cid:17) if − n < α < ( n − pαp + n − if 0 ≤ α < . (1.11)Notice that neither the quantities are increasing in α , e p L < e p G , if α < / , e p L = e p G if α = 1 / e p L ≤ p < e p G , if α < , (1.12) e p L < e p G < p, if α >
0; (1.13)this is in fact consistent with the previous heuristic. For simplcity we state ourresults in the case of Schwartz initial data. In Section 4 we show how to refine thisassumption.
Theorem 1.3.
Let n ≥ and u be a divergence free vector field with each compo-nent in the Schwartz class. Let also u be a weak solution of (1.1) satisfying (2.6).Then (0 , T ) × R n is a regular set provided that α ∈ ((1 − n ) / , , max (cid:18) , n − α (cid:19) < p ≤ − nα , or p = 2 , (1.14) and k| x | α u k L sT L p | x | L e pθ < + ∞ , (1.15) with s + np = 1 − α, (1.16)max (cid:18) , − α (cid:19) < s < + ∞ , or s = 21 − α , (1.17) Here and in the following we mean global and local in space. Notice that e p L = p in the endpoint case α = − / NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 4 e p ≥ e p G := max (cid:18) n, ( n − pαp + n − (cid:19) ; (1.18) or α ∈ [0 , / , n < p ≤ + ∞ , (1.19) and k| x | α u k L sT L p | x | L e pθ < + ∞ , (1.20) with s + np = 1 − α, (1.21)21 − α ≤ s < + ∞ , (1.22) e p ≥ e p G := ( n − pαp + n − . (1.23) Remark . Let us point out again that the main information of the Theorem iscontained in the assumptions (1.18, 1.23), i.e. the angular integrability necessaryin order to get a global regularity result.It turns out by relations (1.12, 1.13) that in the case of negative weights addi-tional angular integrability ( e p G > p ) is necessary in order to get global regularity.On the other hand if we consider | x | α , α > e p G < p ). Remark . Notice that: • Our method misses the endpoint s = + ∞ ; • If n > α with respect to Theorem1.2. We have in fact − n < α instead of − ≤ α . This is also moresatisfactory because exhibits a dependence on the dimension. We have, onthe other hand, a loss in the positive values of α , i.e. α < instead of α < Theorem 1.4.
Let n ≥ and u be a divergence free vector field with each compo-nent in the Schwartz class. Let also u be a weak solution of (1.1) satisfying (2.6).Then (0 , T ) × { } is a regular set provided that α ∈ [ − / , , n < p ≤ + ∞ , (1.24) and k| x | α u k L sT L p | x | L e pθ < + ∞ , (1.25) with s + np = 1 − α, (1.26)max (cid:18) , − α (cid:19) < s < + ∞ , or s = 21 − α , (1.27) e p ≥ e p L := 2( n − p (2 α + 1) p + 2( n −
1) ; (1.28) or α ∈ [0 , , n − α < p ≤ + ∞ , (1.29) and k| x | α u k L sT L p | x | L e pθ < + ∞ , (1.30) with s + np = 1 − α, (1.31) NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 5 − α ≤ s < + ∞ , (1.32) e p > e p L := 2( n − pp + 2( n − . (1.33) Remark . Notice that: • Our main assumption is actually weaker that (1.5) because e p L < p ( e p L = p if α = − / • We have a loss in the negative values of α with respect to 1.2. We assumein fact − ≤ α instead of − ≤ α .It is interesting to compare this results with the regularity criteria obtained byworking in parabolic Morrey spaces [14, 15, 25]. Consider the norms k u k L pλ ((0 ,T ) × R n ) := sup ¯ t ∈ (0 ,T ) , ¯ x ∈ R n sup r> r λ/p k u k L p ( Q r (¯ t, ¯ x )) , where Q r (¯ t, ¯ x ) is the parabolic cylinder of radius r and centered in (¯ t, ¯ x ) Q r (¯ t, ¯ x ) := B r (¯ x ) × (¯ t − r , ¯ t + r )and focus on the formal corrispondence k u k L pλ ((0 ,T ) × R n ) ↔ sup ¯ x ∈ R n k| x − ¯ x | − λ/p u k L pT L px ;Since k| x − ¯ x | − λ/p u k L pT L px ≥ sup ¯ t ∈ (0 ,T ) sup r> r λ/p k u k L p ( Q r (¯ x, ¯ t ) it is clear thatboundedness assumptions in weighted spaces are stronger then their counterpart inMorrey spaces. This is heuristically because in the first case the weights provide aresidual information even for large | x | . As we have observed this information andangular integrability hypotesis provide a a more satisfactory ragularity theory.We exploit again this viewpoint through a really interesting example, i.e. theweighted counterpart of the following Theorem 1.5 ([2]) . Let n = 3 and u be a suitable weak solution of (1.1). There isan absolute constant ε such that if lim sup r → r Z Q ∗ r (¯ t, | u | + | p | / ≤ ε, (1.34) where Q ∗ r (¯ t,
0) := (cid:8) ( τ, y ) : | y | < r, ¯ t − / r < τ < ¯ t + 1 / r (cid:9) ; then (¯ t, is a regular point. We focus on the condition k| x | − / u k L T L | x | L θ < ∞ . (1.35)A little work is necessary in order to show that (1.35) is actually stronger than(1.34). We just sketch the argument that is classical in the context of the Navier–Stokes theory. At first notice that the pressure can be recovered by u through P = X i,j =1 R i R j u i u j , where R i is the Riesz transform in the i -th direction. So the second term in (1.34)can be bounded by using the Calderon-Zygmund inequality (see [23]) k| x | δ P k L r ( R ) ≤ C k| x | δ | u | k L r ( R ) , See also the next Section.
NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 6 r ∈ (1 , ∞ ) , − r < δ < − r , with the choice ( δ, r ) = ( − / , / T > ¯ t :lim sup r → r Z Q ∗ r (¯ t, | u | = lim sup r → r Z ¯ t +1 / r ¯ t − / r Z B (0 ,r ) | u | ≤ lim sup r → Z ¯ t +1 / r ¯ t − / r Z B (0 ,r ) | x | − | u | ≤ lim sup r → Z ¯ t +1 / r ¯ t − / r Z R n | x | − | u | = 0 . Then also notice that ( α, p, s ) = ( − / , ,
3) is an admissible choice of indexes inTheorem 1.2.Theorems 1.3, 1.4 suggest that it is possible to(1) get global regularity (in (0 , T ) × R n ) by a suitable amount of angular inte-grability in (1.35);(2) get regularity in (0 , T ) × { } even by weaker angular integrability in (1.35).The first point is achieved by applying Theorem 1.3 with( α, s, p, e p ) = ( − / , , , + ∞ ) , i.e. by assuming k| x | − / u k L T L | x | L ∞ θ < + ∞ ;notice that in this case the indexes satisfy the endpoint relation p = − nα so wehave to require L ∞ boundedness in the angular direction.Otherwise it is interesting to notice that Theorem 1.4 can not give a positiveanswer to the second point because the value α = − / L p | x | L e pθ spaces; in the fourthSection we prove the main Theorems.2. Integral formulation of the problem
We recall the integral formulation of the Navier–Stokes problem. By taking thedivergence of the first equation in (1.1) and by using the incompressibility: − ∆ P = n X i =1 ∂ i n X j =1 u j ∂ j u i (2.1)= n X i,j =1 ∂ i ∂ j ( u i u j ) , (2.2)so P can be, at least formally, recovered by u through P = − ∆ − P ni,j =1 ∂ i ∂ j ( u i u j ) . (2.3) NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 7
Thus (1.1) becomes (cid:26) u = e t ∆ u − R t e ( t − s )∆ P ∇ · ( u ⊗ u ) ds in [0 , T ) × R n ∇ · u = 0 in [0 , T ) × R n , (2.4)where ( u ⊗ u ) i,j := u i u j and P is formally defined by P f := f − ∇ ∆ − ( ∇ · f ) . (2.5)This operator is a really useful tool in the study of the Navier–Stokes problem. Itis actually a projection on the subspace of the divergence free vector fields ( P f = f ⇔ ∇ · f = 0). If f ∈ [ L ( R n )] n then P is rigorously defined by P f := f + R ⊗ R f, where R is the vector of the Riesz transformations. On the other hand P can bedefined on larger Banach spaces as a Calderon-Zygmund operator. Furthermorewe are basically interested in the operator P ( ∇· ) that, thanks to the differentia-tion, can be actually defined on [ L uloc ( R n )] n × n , i.e. the space of uniformly locallyintegrable functions (see [16] for further details).Now we focus on some properties of the Oseen kernel. At first we need thefollowing Lemma 2.1 ([16]) . Let ≤ i, j ≤ n . The operator ∆ − P nj =1 ∂ i ∂ j e t ∆ is a convo-lution operator P nj =1 O i,j ∗ f j with O i,j ( t, x ) := 1 t n o i,j (cid:18) x √ t (cid:19) and for each multi-index ηo i,j ∈ C ∞ ( R n ) , (1 + | x | ) n + | η | ∂ η o i,j ∈ L ∞ ( R n ) . This is the main technical tool necessary in order to study the properties of e t ∆ P ( ∇· ), it holds in fact the following Proposition 2.2 ([16]) . Let ≤ i, j, k ≤ n . The operator e t ∆ P ( ∇· ) is a convo-lution operator P nj,k =1 K i,j,k ( t ) ∗ f j,k with K i,j,k ( t, x ) := 1 t n +12 k i,j,k (cid:18) x √ t (cid:19) and for each multi-index ηk i,j,k ∈ C ∞ ( R n ) , (1 + | x | ) n +1+ | η | ∂ η k i,j,k ∈ L ∞ ( R n ) . We conclude the section with an useful equivalence result:
Theorem 2.3 ([16]) . Let u ∈ ∩ s We prove time decay estimates for the operators e t ∆ and e t ∆ P ( ∇· ). These esti-mates turn out to be fundamental in the study of the Navier–Stokes problem withsmall data since the pioneering work of Kato [13]. Following the same philosophywe take advantage of them in order to get regularity criteria. This is natural byworking with the integral formulation (2.4).We investigate the connection between homogeneous weights and angular reg-ularity by working in L p | x | L e pθ spaces. In particular we show that higher angularintegrability allows to consider a larger set of weights.As mentioned the idea of distinguish radial and angular integrability often occursin harmonic analysis and PDE’s. In particular we refer to [4] where this technologyhas been applied to recover in a more general setting the improvements to Sobolevembeddings and Caffarelli-Kohn-Nirenberg inequalities known in the radial case by[5, 6, 7, 8, 20].We need the following Lemma 3.1 ([4]) . Let n ≥ and ≤ p ≤ q ≤ ∞ , ≤ e p ≤ e q ≤ ∞ . Assume α, β, γ satisfy the set of conditions β > − nq , α < np ′ , α − β ≥ ( n − (cid:18) q − p + 1 e p − e q (cid:19) , (3.1) α − β + γ > n (cid:18) q − p (cid:19) . (3.2) Then k| x | β S γ φ k L q | x | L e qθ ≤ C k| x | α φ k L p | x | L e pθ , (3.3) where S γ φ := Z R n K ( x − y ) φ ( y ) dy, and the kernel K satisfies | K ( x ) | ≤ Const (1 + | x | ) γ/ . Remark . Let point out that: • The assumptions β > − nq , α < np ′ are necessary to ensure local integrability; • The assumption (3.2) is due to the smoothness of the kernel in the origin.It is less restrictive than its counterpart in the homogeneous case (see [4]) α − β + γ = n (cid:18) q − p (cid:19) , that follows by scaling; • The assumption α − β ≥ ( n − (cid:18) q − p + 1 e p − e q (cid:19) follows by testing the inequality under translations.It is useful to define the quantityΛ( α, p, e p ) := α + n − p − n − e p . (3.4)Notice that α − β ≥ ( n − (cid:18) q − p + 1 e p − e q (cid:19) ⇔ Λ( α, p, e p ) ≥ Λ( β, q, e q ) . NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 9 This notation is more convenient for our purposes; we use also simply Λ α when thevalues of p, e p will be clear by the context. Proposition 3.2. Let n ≥ , ≤ p ≤ q ≤ + ∞ and ≤ e p ≤ e q ≤ + ∞ . Assumefurther that α, β satisfy the set of conditions β > − nq , α < np ′ , Λ( α, p, e p ) ≥ Λ( β, q, e q ) . (3.5) Then for each multi-index η (1) k| x | β ∂ η e t ∆ u k L q | x | L e qθ ≤ c η t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ , t > , (3.6) provided that | η | + np − nq + α − β ≥ , (2) k| x | β ∂ η e t ∆ P ∇ · F k L q | x | L e qθ ≤ d η t (1+ | η | + np − nq + α − β ) / k| x | α F k L p | x | L e pθ , t > , (3.7) provided that | η | + np − nq + α − β > .Proof. The proof follows by Lemma (3.1) and scaling considerations. At first notice e t ∆ φ = S √ t e ∆ S / √ t φ, (3.8)where S λ is defined by ( S λ φ )( x ) = φ (cid:16) xλ (cid:17) . (3.9)Then k| x | β ∂ η S λ φ k L q | x | L e qθ = λ nq + β −| η | k| x | β φ k L q | x | L e qθ . (3.10)We get k| x | β ∂ η e t ∆ u k L q | x | L e qθ = k| x | β ∂ η S √ t e ∆ S / √ t u k L q | x | L e qθ = t ( nq + β −| η | ) / k| x | β ( ∂ η e ∆ ) S / √ t u k L q | x | L e qθ ≤ c η t ( − nq − β + | η | ) / k| x | α S / √ t u k L p | x | L e pθ = c η t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ , provided that Λ( α, p, e p ) ≥ Λ( β, q, e q ) . Notice that the third condition in (3.1) is trivially satisfied by the heat kernel. Toprove (3.7) we have to work with the operator e t ∆ P ( ∇· ) that is (see Lemma 2.2)a convolution operator with a kernel K such that K j,k,m ( t, x ) := k j,k,m (cid:18) x √ t (cid:19) , (3.11)and (1 + | x | ) n + | µ | ∂ µ k j,k,m ∈ L ∞ ( R n ) , (3.12)for each multi-index µ . By (3.11) follows K ( t ) ∗ φ = 1 √ t S √ t k ∗ S / √ t φ. (3.13) NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 10 So k| x | β ∂ η e t ∆ P ∇ · F k L q | x | L e qθ = k| x | β ∂ η K ( t ) ∗ F k L q | x | L e qθ = 1 √ t k| x | β ∂ η S √ t k ∗ S / √ t F k L q | x | L e qθ = 1 √ t t ( nq + β −| η | ) / k| x | β ( ∂ η k ) ∗ S / √ t F k L q | x | L e qθ ≤ d η t ( − nq − β +1+ | η | ) / k| x | α S / √ t F k L p | x | L e pθ = d η t (1+ | η | + np − nq + α − β ) / k| x | α F k L p | x | L e pθ , provided that Λ α ≥ Λ β . Notice that the optimal choice of γ allowed by (3.12) is γ = 1 + n + | η | that leads to α − β + 1 + n + | η | > n (cid:16) q − p (cid:17) ⇒ | η | + np − nq + α − β > . (cid:3) It’s remarkable that the restriction Λ α ≥ Λ β can be removed by localizing theestimate in the interior of a space-time parabola above the origin. The size ofthe parabola depends on the values of the difference Λ α − Λ β and increases asΛ α − Λ β → − . In the limit case Λ α = Λ β we recover in fact Proposition 3.2. Proposition 3.3. Let n ≥ , ≤ p ≤ q ≤ + ∞ and ≤ e p ≤ e q ≤ + ∞ . Assumefurther that α, β satisfy the set of conditions β > − nq , α < np ′ , Λ( α, p, e p ) < Λ( β, q, e q ) , (3.14) and define Λ α,β := Λ( α, p, e p ) − Λ( β, q, e q ) . Let then Π( R ) := (cid:26) ( t, x ) ∈ R + × R n : | x |√ t ≤ R (cid:27) , for each muti index η (1) k Π( R ) | x | β ∂ η e t ∆ u k L q | x | L e qθ ≤ c η R − Λ α,β t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ , t > , (3.15) provided that | η | + np − nq + α − β ≥ , Λ α,β < , (2) k Π( R ) | x | β ∂ η e t ∆ P ∇ · F k L q | x | L e qθ ≤ d η R − Λ α,β t (1+ | η | + np − nq + α − β ) / k| x | α F k L p | x | L e pθ , t > , (3.16) provided that | η | + np − nq + α − β > , Λ α,β < . Proof. Let us write simply Λ instead of Λ α,β . Of courseΛ < ⇒ R − Λ (cid:12)(cid:12)(cid:12) x √ t (cid:12)(cid:12)(cid:12) Λ ≥ , if ( t, x ) ∈ Π( R ) . NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 11 Then k Π( R ) | x | β ∂ η e t ∆ u k L q | x | L e qθ = k Π( R ) | x | β ∂ η S √ t e ∆ S / √ t u k L q | x | L e qθ ≤ R − Λ t Λ / k| x | β +Λ ∂ η S √ t e ∆ S / √ t u k L q | x | L e qθ = R − Λ t Λ / t ( nq + β +Λ −| η | ) / k| x | β +Λ ( ∂ η e ∆ ) S / √ t u k L q | x | L e qθ ≤ c η t ( − nq − β + | η | ) / k| x | α S / √ t u k L p | x | L e pθ = c η t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ , where the indexes relationships are consistent becauseΛ α ≥ Λ(Λ α,β + β, p, e p ) = Λ(Λ α − Λ β + β, p, e p ) = Λ α . The proof of (3.16) is analogous. (cid:3) Remark . We have observed observed that the inequalities hold with an addi-tional factor R − Λ after localization in the interior of a space-time parabola. Noticethat this factor goes to 1 as Λ → − . To get a constant independent on Λ it isinstead necessary to restrict the size of the parabola. If we chose the constant equalto K , we need to restrict to Π( K ) := (cid:26) | x |√ t ≤ K − (cid:27) . Notice that Π( K ) fills the whole space-time as Λ → − .Then integral estimates can be obtained by the time decay properties. Let usintroduce another useful notationΩ( α, p, s ) := α + np + 2 s . (3.17) Proposition 3.4. Let n ≥ , ≤ p ≤ q < np ( | η | + α − β ) p + n − , p < r < + ∞ and ≤ e p ≤ e q ≤ + ∞ . Assume further that α, β satisfy β > − nq , α < np ′ . (3.18) Then for each multi-index η k| x | β ∂ η e t ∆ u k L rt L q | x | L e qθ ≤ c η k| x | α u k L p | x | L e pθ , t > , (3.19) provided that | η | + Ω( α, p, ∞ ) = Ω( β, q, r ) , Λ( α, p, e p ) ≥ Λ( β, q, e q ); (3.20) and k Π( R ) | x | β ∂ η e t ∆ u k L rt L q | x | L e qθ ≤ c η R − Λ α,β k| x | α u k L p | x | L e pθ , t > , (3.21) provided that | η | + Ω( α, p, ∞ ) = Ω( β, q, r ) , Λ α,β := Λ( α, p, e p ) − Λ( β, q, e q ) < , (3.22) Proof. By the time decay k| x | β ∂ η e t ∆ u k L q | x | L e qθ ≤ c η t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ , NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 12 follows that ∂ η e t ∆ u is bounded in the Lorentz space L r, ∞ ( R + ; L q | x | βq d | x | L e qθ ) pro-vided that | η | + Ω( α, p, ∞ ) = Ω( β, q, r ). In fact kk| x | β ∂ η e t ∆ u k L q | x | L e qθ k L r, ∞ t ≤ c η (cid:13)(cid:13)(cid:13)(cid:13) t ( | η | + np − nq + α − β ) / k| x | α u k L p | x | L e pθ (cid:13)(cid:13)(cid:13)(cid:13) L r, ∞ t ≤ c η (cid:13)(cid:13)(cid:13)(cid:13) t ( | η | + np − nq + α − β ) / (cid:13)(cid:13)(cid:13)(cid:13) L r, ∞ k u k L p | x | L e pθ ≤ c η k u k L p | x | L e pθ , when ( | η | + np − nq + α − β ) / r ⇒ | η | + Ω( α, p, ∞ ) = Ω( β, q, r ) . Let now consider ( α , β , p , e p , q , e q , r ), ( α , β , p , e p , q , e q , r ) such that theassumptions of the Theorem are satisfied. We have the bounded operators ∂ η e t ∆ : L p | x | α p d | x | L e p θ −→ L r , ∞ t L q | x | β q d | x | L e q θ L p | x | α p d | x | L e p θ −→ L r , ∞ t L q | x | β q d | x | L e q θ . (3.23)and we can use real interpolation with parameters ( ξ, r ξ ) , ≤ ξ ≤ p ξ < r ξ , (3.24)1 p ξ = (1 − ξ ) 1 p + ξp , q ξ = (1 − ξ ) 1 q + ξq , r ξ = (1 − ξ ) 1 r + ξr , e p ξ = (1 − ξ ) 1 e p + ξ e p , e q ξ = (1 − ξ ) 1 e q + ξ e q ,α ξ = (1 − ξ ) α + ξα ,β ξ = (1 − ξ ) β + ξβ . Then since (cid:16) L r , ∞ t L q | x | β q d | x | L e q θ , L r , ∞ t L q | x | β q d | x | L e q θ (cid:17) ξ,r ξ = L r ξ t L q ξ | x | βξqξ d | x | L e q ξ θ , we get the bounded operators ∂ η e t ∆ u : L p ξ | x | αξpξ d | x | L e p ξ θ → L r ξ t L q ξ | x | βξqξ d | x | L e q ξ θ . It is now straightforward to check that the indexes satisfy (3.18, 3.20) and the otherassumptions. In particular (3.24) is ensured by q ξ < np ξ ( | η | + α ξ − β ξ ) p ξ + n − .Of course this method misses the endpoint r = 1. The estimates (3.21) can beproved in the same way by using the localized time decay. (cid:3) Then we bound the Duhamel term: NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 13 Proposition 3.5. Let n ≥ , ≤ p ≤ q ≤ + ∞ , < s < r < + ∞ and ≤ e p ≤ e q ≤ + ∞ . Assume further that α, β satisfy β > − nq , α < n − np , (3.25) then for each multi-index η (cid:13)(cid:13)(cid:13)(cid:13) | x | β ∂ η Z t e ( t − s )∆ P ∇ · ( u ⊗ u ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ d η k| x | α u k L st L p | x | L e pθ , t > , (3.26) provided that α, p, s ) = Ω( β, q, r ) + 1 − | η | , α, p, e p ) ≥ Λ( β, q, e q ); (3.27) in particular (for < r < ∞ ) (cid:13)(cid:13)(cid:13)(cid:13) | x | β Z t e ( t − s )∆ P ∇ · ( u ⊗ u ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ d η k| x | β u k L rt L q | x | L e qθ , t > , (3.28) provided that r + nq = 1 − β, Λ( β, q, e q ) ≥ . (3.29) Proof. By Minkowski inequality and (3.6) (cid:13)(cid:13)(cid:13)(cid:13) | x | β ∂ η Z t e ( t − s )∆ P ∇ · F ( x, s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z R + k| x | β ∂ η e ( t − s )∆ P ∇ · F k L q | x | L e qθ ds (cid:13)(cid:13)(cid:13)(cid:13) L rt ≤ d η (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z R + t − s ) (1+ | η | + np − nq + α − β ) / k| x | α F k L p | x | L e p θ ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L rt , provided that e p ≤ e q, p ≤ q | η | + np − nq + α − β > , Λ α ≥ Λ β . (3.30)Let then 1 + 1 r = 1 s + 1 k , (3.31)and use the Young inequality in Lorentz spaces k · k L r ≤ k · k L s k · k L k, ∞ , that is allowed if 1 < r, s , k < + ∞ . We get (cid:13)(cid:13)(cid:13)(cid:13) | x | β ∂ η Z t e ( t − s )∆ P ∇ · F ( x, s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ d η k| x | α F k L s t L p | x | L e p θ (cid:13)(cid:13)(cid:13)(cid:13)Z R + dtt (1+ | η | + np − nq + α − β ) / (cid:13)(cid:13)(cid:13)(cid:13) L k, ∞ t ≤ d η k| x | α F k L s t L p | x | L e p θ , provided that p ≤ q (1 + | η | + np − nq + α − β ) / k , Λ α ≥ Λ β , (3.32) We are not interested in the case k = + ∞ to which corresponds a singular behaviour. NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 14 since (cid:13)(cid:13)(cid:13)(cid:13)Z R + dtt /k (cid:13)(cid:13)(cid:13)(cid:13) L k, ∞ t = 1 . By (3.31) and the second in (3.32)Ω( α , p , s ) = 1 − | η | + Ω( β, q, r ) . (3.33)We now specify F = u ⊗ u (cid:13)(cid:13)(cid:13)(cid:13) | x | β ∂ η Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( x, s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ c η k| x | α | u | k L s t L p | x | L e p θ ≤ c η k| x | α / | u |k L s t L p | x | L e p θ (3.34) ≤ c η k| x | α u k L st L p | x | L e pθ , where we have set ( α / , s , p , e p ) = ( α, s, p, e p ) . (3.35)Notice that 2Ω( α, s, p ) = Ω( α , s , p ), 2Λ α = Λ α so (3.33), (3.35) and the last in(3.32) lead to 2Ω( α, p, s ) = Ω( β, q, r ) + 1 − | η | , α ≥ Λ β . Finally notice that (3.31) and (3.32) imply r > s = s/ , q ≥ p = p/ , e q ≥ e p = e p/ . These conditions are furthermore consistent with the choice ( α, s, p, e p ) = ( β, r, q, e q ),in such a way we recover inequality (3.28) (cid:13)(cid:13)(cid:13)(cid:13) | x | β ∂ η Z t e ( t − s )∆ P ∇ · ( u ⊗ u ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rt L q | x | L e qθ ≤ d η k| x | β u k L rt L q | x | L e qθ , provided that Ω( β, q, r ) = 1 − | η | , Λ( β, q, e q ) ≥ . (cid:3) Proof of the main results We refer to the relations2 s + np = 1 − α, α = 1 − np , s + np = 1 − β, as scaling assumptions.As mentioned Theorems 1.3, 1.4 actually hold under weaker assumptions on u ,we prove in fact: Theorem 4.1. Theorem 1.3 holds if u ∈ L ( R n ) is a divergence free vector fieldand k| x | α u k L p | x | L e p θ < + ∞ with α ∈ [(2 − n ) / , / (2 + n )) , α = 1 − np , e p ≤ e p G , (4.1) (cid:26) ≤ p ≤ e p G / if e p G ≤ n ≤ p ≤ e p G / , p < e p G e p G − n if e p G > n ; (4.2) or α ∈ [(2 − n ) / , / (2 + n )) , α = 1 − np , e p ≤ p , (4.3) NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 15 (cid:26) ≤ p ≤ p/ if p ≤ n ≤ p ≤ p/ , p < pp − n if p > n ; (4.4) while u has to satisfy (2.6) and (1.14, 1.15, 1.16, 1.17, 1.18), or (1.19, 1.20, 1.21,1.22, 1.23).Proof. Since we want to use the regularity condition (1.2) we need to show that k u k L rT L qx < + ∞ , with 2 r + nq = 1 . (4.5)Let’s start by the integral representation u = e t ∆ u − Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds and distinguish the cases α ∈ ((1 − n ) / , 0) and α ∈ [0 , / Case α ∈ ((1 − n ) / , . k u k L rT L qx ≤ k e t ∆ u k L rT L qx + (cid:13)(cid:13)(cid:13)(cid:13)Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rT L qx = I + II. By the scaling assumption and Proposition 3.4 I ≤ c k| x | α u k L p | x | L e p θ , (4.6)provided that p ≤ q < np p − , e p ≤ q, Λ( α , p , e p ) ≥ . (4.7)Actually the condition Λ( α , p , e p ) ≥ II ≤ d k| x | α u k L sT L p | x | L e pGθ . k| x | α u k L sT L p | x | L e pθ , provided that Λ( α, p, e p ) ≥ , (4.8)2 ≤ p ≤ + ∞ , < s < + ∞ , p/ , e p G / ≤ q, s/ < r. (4.9)Condition (4.8) is ensured by e p ≥ ( n − pαp + n − . (4.10)Notice also that (4.10), the scaling and α < n − α < p ≤ − nα , so the widestrange for p is attained as α → − . Then we need a couple ( r, q ) such that (4.9) isconsistent with r + nq = 1. We choose q = e p G / (cid:16) n, ( n − p αp +2( n − (cid:17) . This isallowed by (1 − n ) / < α , we have indeed2 r = 1 − nq = 1 − nαn − np ⇒ r − s = 1 − n − αn − , so (1 − n ) / < α ⇒ s/ < r ;Finally (4.7) becomes p ≤ e p G < np p − , that by a straightforward calculation leads to (4.2) and α ∈ [(2 − n ) / , / (2 + n )). NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 16 Case α ∈ [0 , / . The only difference is in the choice of ( r, q ). Here we set q = p/ α < / 2, in fact2 r = 1 − np ⇒ r − s = − α, so α < / ⇒ s/ < r. Notice that in this case we do not have the restriction p ≤ − nα . Then (4.7) becomes p ≤ q < np p − , that by a straightforward calculation leads to (4.4), α ∈ [(2 − n ) / , / (2 + n )).The choice q = p/ , β = 0 and the scaling assumptions force to be p > n .We show how the assumption Λ( α , p , e p ) ≥ α , p , e p ) and suppose Λ < 0. We can use the localizedestimate (3.21) to get the bound k Π( R ) u k L rT L qx ≤ R − Λ c k| x | α u k L p | x | L e p θ + d k| x | α u k L sT L p | x | L e pθ where Π( R ) := (cid:26) ( t, x ) ∈ R + × R n : | x |√ t ≤ R (cid:27) . So (0 , T ) × R n is a regular set by taking the limit R → + ∞ . (cid:3) Theorem 4.2. Theorem 1.4 holds if u ∈ H ∩ L | x | − n dx is a divergence free vectorfield such that k| x | α u k L p | x | L e p θ < + ∞ , Λ( α , p , e p ) ≥ , with α ∈ (cid:20) − n, − n n (cid:19) , α = 1 − np , e p ≤ p , (4.11) (cid:26) ≤ p ≤ p/ if p ≤ n ≤ p ≤ p/ , p < pp − n if p > n ; (4.12) or α ∈ (cid:20) − (1 − α ) n, − (1 − α ) 2 n n (cid:19) , α = 1 − np , e p ≤ p , (4.13)11 − α ≤ p ≤ p , p < p (1 − α ) p − n ; (4.14) while u has to satisfy (2.6) and (1.24, 1.25, 1.26, 1.27, 1.28), or (1.29, 1.30, 1.31,1.32, 1.33).Proof. Since we want to use directly Theorem 1.2 so we need to show that k| x | β u k L rT L qx < + ∞ , with 2 r + nq = 1 − β. (4.15)Let’s start by the integral representation u = e t ∆ u − Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds. and distinguish the cases α ∈ [ − / , 0) and α ∈ [0 , NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 17 Case α ∈ [ − / , . k| x | β u k L rT L qx ≤ k| x | β e t ∆ u k L rT L qx + (cid:13)(cid:13)(cid:13)(cid:13) | x | β Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L rT L qx = I + II. By the scaling assumption and Poposition 3.4 I ≤ c k| x | α u k L p | x | L e p θ (4.16)provided that p ≤ q < np ( α − β ) p + n − , e p ≤ q, Λ( α , p , e p ) ≥ . (4.17)We use Proposition 3.5 and scaling to bound II ≤ d k| x | α u k L sT L p | x | L e pθ , provided that 2Λ( α, p, e p ) ≥ β, (4.18)2 ≤ p ≤ + ∞ , < s < + ∞ , p/ , e p/ ≤ q, s/ < r. (4.19)Condition (4.18) is ensured by e p ≥ n − α − β ) p + 2( n − . (4.20)Then we need a triple ( β, r, q ) such that (4.19) is consistent with r + nq = 1 − β .We are using Theorem 1.2 so it is necessary to restrict to − ≤ β and, in order toget the lowest value for e p , we choose β = − 1. In such a way (4.20) becomes (1.28).By this choice we have e p ≤ p if − / ≤ α, that is in fact the range of α we have restricted on. Then we choose q = p/ r − s = 2 α − ≤ , that is consistent with s/ < r . Because of the choice q = p/ , β = − p > n . Then (4.17) becomes p ≤ q < np p − , that by a straightforward calculation leads to (4.12) and α ∈ h − n, − n n (cid:17) . Case α ∈ [0 , . The only difference is again in the choice of ( β, r, q ). Since α ≥ e p by setting 2 α − β = 1 − ε in (4.20), with ε > e p ≥ n − p (1 − ε ) p + 2( n − . Then we choose ( β, r, q ) = (cid:18) α − ε, s − εs , p (cid:19) . It is easy to check that this is consistent with the scaling relation. Notice also thatthe scaling assumptions force to be p > n/ (1 − α ). Finally, by (4.17) and scalingwe have p ≤ q < np (2 − α ) p − , NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 18 that by a straightforward calculation leads to (4.14) and α ∈ (cid:20) − (1 − α ) n, − (1 − α ) 2 n n (cid:19) . (cid:3) Outlooks and remarks In this paper we develop a technique that makes able to get new regularitycriteria for weak solutions of (1.1) from a given one. In principle this machinerycould be applied to many different criteria known in literature even if we basicallyfocus on (1.2) and on Theorem 1.2.The relations between the indexes in the main theorems are not the most generalpossible, for instance different choices are allowed than q = p/ q = e p G / Acknowledgements The author would like to thank professor Piero D’Ancona for constant help andsuggestions and professor Keith Rogers for useful discussions and reading the paper. References [1] F. Cacciafesta and P. D’Ancona. Endpoint estimates and global existence for the nonlinearDirac equation with potential. J. Diff. Eq. , 254(5):2233–2260, 2013.[2] L. Caffarelli, R. Kohn and L. Nirenberg. Partial regularity of suitable weak solutions of theNavier–Stokes equations. Comm. Pure Appl. Math. , 35:771–831, 1982.[3] Y. Cho and T. Ozawa. Sobolev inequalities with symmetry. Commun. Contemp. Math. ,11(3):355–365, 2009.[4] P. D’Ancona and R. Luc`a. Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with higherangular integrability. J. Math. Anal. App. , 388(2):1061–1079, 2012.[5] P. L. De N´apoli, I. Drelichman and R. G. Dur´an. Radial solutions for Hamiltonian ellipticsystems with weights. Adv. Nonlinear Stud. , 9(3):579–593, 2009.[6] P. L. De N´apoli and I. Drelichman. Weighted convolution inequalities for radial functions. Ann. Mat. Pur. Appl. , to appear.[7] P. L. De N´apoli, I. Drelichman and R. G. Dur´an. Improved Caffarelli-Kohn-Nirenberg andtrace inequalities for radial functions. Comm. Pure Appl. Anal. , 11(5):1629–1642, 2012.[8] P. L. De N´apoli, I. Drelichman and R. G. Dur´an. On weighted inequalities for fractionalintegrals of radial functions. Illinois J. Math. , 55:575–587, 2011.[9] L. Escauriaza, G. Seregin and V. Sverak Backward uniqueness for parabolic equations. Arch.Ration. Mech. Anal. , 169:147–157, 2003.[10] D. Fang and C. Wang. Weighted Strichartz estimates with angular regularity and their ap-plications. Forum Math. , 23:181–205, 2011[11] Y. Giga. Solutions for semilinear parabolic equations in L p and regularity of weak solutionsof the Navier–Stokes system. J. Diff. Eq. , 62:186–212, 1986.[12] E. Hopf. Uber die Anfanqswertaufgabe f¨ur die hydrodynamischen Grundgleichungen. Math.Nachr. , 4:213–231, 1951.[13] T. Kato. Strong L p -solutions of the Navier–Stokes equation in R n , with applications to weaksolutions. Math. Z. , 187: 471–480, 1984.[14] I. Kukavica. On partial regularity for the Navier–Stokes equations. Discrete Contin. Dyn.Syst. , 21:717–728, 2008.[15] I. Kukavica. On regularity for the Navier–Stokes equations in Morrey spaces. Discrete Contin.Dyn. Syst. 26(4):1319–1328, 2010.[16] P. G. Lemari´e-Rieusset. Recent developments in the Navier–Stokes problem. CHAPMANAND HALL/CRC. Research Notes in Mathematics Series 431, 2002.[17] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. , 63:193–248, 1934. NGULAR INTEGRABILITY AND NAVIER–STOKES EQUATION 19 [18] S. Machihara, M. Nakamura, K. Nakanishi, and T. Ozawa. Endpoint Strichartz estimatesand global solutions for the nonlinear Dirac equation. J. Funct. Anal. , 219(1):1–20, 2005.[19] T. Ozawa and K. M. Rogers Sharp Morawetz estimates. J. Anal. Math. , 121:163–175, 2013.[20] B. Rubin. One-dimensional representation, inversion and certain properties of Riesz potentialsof radial functions. (russian). Mat. Zametki , 34(4):521—533, 1983. English translation: Math.Notes 34(3-4):751–757, 1983.[21] J. Serrin. On the interior regularity of weak solutions of the Navier–Stokes equations. Arch.Ration. Mech. Anal. , 9:187–195, 1962.[22] H. Sohr. Zur Regularit¨atstheorie der instationaren Gleichungen von Navier–Stokes. Math. Z. ,184:339–375, 1983.[23] E. M. Stein. Note on singular integrals. Proc. Am. Math. Soc. , 8:250–254, 1957.[24] M. Struwe. On partial regularity results for the Navier–Stokes equations. Commun. PureAppl. Math. Proc. Royal Soc. Edin. , 139A:661–671, 2009.[27] J. Sterbenz. Angular regularity and Strichartz estimates for the wave equation. Int. Math.Res. Not. , (4):187–231, 2005. With an appendix by Igor Rodnianski.[28] W. von Wahl. Regularity of weak solutions of the Navier–Stokes equations. In Proc. 1983Summer Inst. on Nonlinear Functional Analysis and Applications, Proceedings of Symposiain Pure Mathematics, vol. 45, pp. 497–503 (Providence, RI: American Mathematical Society,1989). Instituto de Ciencias Matematicas, Consejo Superior de Investigaciones Cientificas,Madrid, 28049, Spain. Supported by the ERC grant 277778 and MINECO grant SEV-2011-0087 (Spain). E-mail address ::