Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces
aa r X i v : . [ m a t h . A P ] M a r Existence and Weak* Stability for theNavier-Stokes System with Initial Values inCritical Besov Spaces
T Barker ∗ August 30, 2018
Abstract
In 2016, Seregin and ˘Sver´ak, conceived a notion of global in timesolution (as well as proving existence of them) to the three dimensionalNavier-Stokes equation with L solenoidal initial data called ’global L solutions’. A key feature of global L solutions is continuity withrespect to weak convergence of a sequence of solenoidal L initial data.The first aim of this paper is to show that a similar notion of ’ global˙ B − , ∞ solutions’ exists for solenoidal initial data in the wider criticalspace ˙ B − , ∞ and satisfies certain continuity properties with respect toweak* convergence of a sequence of solenoidal ˙ B − , ∞ initial data. Thisis the widest such critical space if one requires the solution to theNavier-Stokes equations minus the caloric extension of the initial datato be in the global energy class.For the case of initial values in the wider class of ˙ B − p p, ∞ initial data( p > < T < ∞ there exists a solution to theNavier-Stokes system on R × ]0 , T [ with this initial data. We discusshow properties of these solutions imply a new regularity criteria for3D weak Leray-Hopf solutions in terms of the norm k v ( · , t ) k ˙ B −
1+ 3 pp, ∞ (aswell as certain additional assumptions). ∗ OxPDE, Mathematical Institute, University of Oxford, Oxford, UK.
Email address: [email protected] ; he main new observation of this paper, that enables these results,regards the decomposition of homogeneous Besov spaces ˙ B − p p, ∞ . Thisdoes not appear to obviously follow from the known real interpolationtheory. In this paper, we consider the Cauchy problem for the Navier-Stokes systemin the space-time domain Q S = R × ]0 , S [ (0 < S ≤ ∞ ) for the vector-valuedfunction v = ( v , v , v ) = ( v i ) and scalar function q , satisfying the equations ∂ t v + v · ∇ v − ∆ v = −∇ q, div v = 0 (1.1)in Q S , the boundary conditions v ( x, t ) → | x | → ∞ for all t ∈ [0 , S [, and the initial conditions v ( · ,
0) = u ( · ) (1.3)In the recent paper [31], a notion of global in time solution to the Navier-Stokes equation was developed with L solenoidal initial data called ’global L solutions’. A key feature of global L solutions is as follows. Namely, if u ( n ) are global L solutions corresponding the the initial datum u ( n )0 , and u ( n )0 converge weakly in L ( R ) to u , then a suitable subsequence of u ( n ) convergesto global L solution u corresponding to the initial condition u . To explainthe notation of global weak L -solutions in [31] further, we introduce thenotation S ( t ) u ( x ) = Z R Γ( x − y, t ) u ( y ) dy, where Γ is the three dimensional heat kernel. Throughout this paper wewill often write V ( x, t ) := S ( t ) u ( x ). In [31], any global weak L -solution ofthe Navier-Stokes equation, with initial data u ∈ L ( R ), has the followingstructure. Namely, v ( x, t ) = V ( x, t ) + u ( x, t ) , (1.4)where u is globally in the energy class in R × ]0 , T [ for any finite T > u ( · , t ) k L + 2 t Z Z R |∇ u ( x, t ′ ) | dxdt ′ ≤ Z t Z R ( V ⊗ u + V ⊗ V ) : ∇ udxdt ′ . (1.5)In [31], the crucial estimate k u ( · , t ) k L + 2 t Z Z R |∇ u ( x, t ′ ) | dxdt ′ ≤ t C ( k u k L ( R ) ) . (1.6)is proven for u , with u and V as in (1.4)-(1.5). This estimate plays a centralrole in [31] in the proof of continuity of global weak L solutions, with respectto the weak convergence of the initial data.A natural question concerns whether an analogous notion of of global intime solutions (which we will refer to as ’global X solutions’ or sometimes N ( X )) with initial data u in other critical spaces X , that satisfy theproperties •
1) Global existence for any u ∈ X there exists a global in timesolution in N ( X ); •
2) Weak* stability u ( k )0 ∗ ⇀ u in X ⇒ u ( k ) ( · , u ( k )0 ) ∈ N ( X ) converges up to subsequence (in sense of distribu-tions) to u ( · , u ) ∈ N ( X );As mentioned in [3], such properties are useful if one wants to show thatcritical norms of the Navier-Stokes equations tend to infinity at a potentialblow up time. Let us now describe this in more detail. Suppose one wanted toprove that if v is a weak Leray-Hopf solution on R × ]0 , ∞ [, with sufficientlyregular initial data as well as a finite blow up time T , then necessarilylim t ↑ T k v ( · , t ) k X = ∞ . (1.7) We say X is a critical space, if u ∈ X ⇒ λu ( λ · ) ∈ X and k u λ k X = k u k X X is a critical space. One such strategy, given in [28] and subsequentlyused in [29],[5] and [1], for showing this is to assume for contradiction thatthere exists t n ↑ T with M := sup n k v ( · , t n ) k X < ∞ . (1.8)The next step is to perform the rescaling u ( n ) ( y, s ) = λ n v ( x, t ) , p ( n ) ( y, s ) = λ n q ( x, t ) , u ( n )0 ( y ) = λ n v ( λ n y, t n ) , (1.9) x = λ n y, t = T + λ n s, λ n = r T − t n u ( n ) , p ( n ) ) are solutions to the Navier-Stokes equations on R × ] − ,
0[ with sup n k u ( n )0 k X = k v ( · , t n ) k X = M. (1.11)The final part of the strategy is to obtain a non-trivial ancient solutions tothe Navier-Stokes equations and to then attempt to apply a Liouville typetheorem to it, based on backward uniqueness and unique continuation forparabolic operators developed in [12]. The motivation for considering ’global X solutions’, with properties
1) Global existence- 2) Weak* stability ,is that if such solutions exist then they are particularly useful in obtaining anon-trivial ancient solution in the above strategy.Unfortunately, the method for proving the existence of global L solutions,with properties
1) Global Existence- 2) Weak* stability , breaks downfor L , ∞ initial data. In [3]-[4], this difficulty was overcome. In particularin [3]- [4], a notion of ’global weak L , ∞ solutions’, satisfying
1) Globalexistence - 2) Weak* stability and having the structure (1.4)-(1.5), wasconceived. In [3]-[4] the key is establishing that for any global L , ∞ solution v , with solenoidal initial data u ∈ L , ∞ ( R ), we have k v ( · , t ) − S ( t ) u k L + 2 t Z Z R |∇ v ( x, t ′ ) − S ( t ) u | dxdt ′ ≤ t C ( k u k L , ∞ ( R ) ) . (1.12)A key part in showing this uses that for any N > u in L , ∞ ( R ) can be decomposed into two divergence free pieces¯ u N and f u N satisfying k f u N k L ≤ CN − k u k L , ∞ (1.13)4nd k ¯ u N k pL p ≤ CN p − k u k L , ∞ (1.14)for any 3 < p . This, together with appropriate decompositions of the Navier-Stokes equations (inspired by [8]), imply (1.12).Suppose one attempts constructs a global X solution to the Navier-Stokesequation, with X being a critical space, having the structure (1.4)-(1.5). Toensure the finiteness of the right hand side of the energy inequality (1.5), oneshould have that for any u ∈ X that S ( t ) u ∈ L ,loc (0 , ∞ ; L ( R )) . Hence, it is natural to consider X such that for any 0 < T < ∞ and 0 < ǫ
1) Global existence- 2) Weak* Stability andhaving the structure (1.4)-(1.5), X = ˙ B − , ∞ ( R ) is the widest critical spacefor such a possibility.The aim of this paper is to show that, for any divergence free initial data in˙ B − , ∞ , there exists ’global ˙ B − , ∞ solution’ to the Navier-Stokes equations (1.1)-(1.3), which satisfies the requirements
1) Global existence- 2) Weak*Stability . To the best of the author’s knowledge, there is currently no notionof solutions to the Navier-Stokes equations with arbitrary ˙ B − , ∞ solenodialinitial data. Note that under certain smallness conditions on the ˙ B − , ∞ data,5olutions were constructed by means of the Banach contraction principle,in [26] and [9]. Let us mention that the recent preprint [7] implies that if u ∈ ˙ B − , ∞ is discretely self similar , then there exists a discretely self similarsolution to the Navier-Stokes equation. These solutions will also belong toour class of global ˙ B − , ∞ solutions.Before giving the definition of ’global ˙ B − , ∞ ( R ) solutions’, we providesome relevant definitions and notation: J and ◦ J are the completion of the space C ∞ , ( R ) := { v ∈ C ∞ ( R ) : div v = 0 } with respect to L -norm and the Dirichlet integral (cid:16) Z R |∇ v | dx (cid:17) , correspondingly. Additionally, we define the space-time domains Q T := R × ]0 , T [ and Q ∞ := R × ]0 , ∞ [.For arbitrary vectors a = ( a i ) , b = ( b i ) in R n and for arbitrary matrices F = ( F ij ) , G = ( G ij ) in M n we put a · b = a i b i , | a | = √ a · a,a ⊗ b = ( a i b j ) ∈ M n ,F G = ( F ik G kj ) ∈ M n , F T = ( F ji ) ∈ M n ,F : G = F ij G ij , | F | = √ F : F .
Definition 1.1.
We say that v is a weak ˙ B − , ∞ -solution to Navier-Stokes IVPin Q T (with < T < ∞ ) if v = V + u, (1.19) with u ∈ L ∞ (0 , T ; J ) ∩ L (0 , T ; ◦ J ) and there exists an α > such that sup The role of the requirement (1.20) is to ensure that the righthand side of the energy inequality (1.1) is finite. See Lemma 4.1 for more de-tails. This therefore ensures that the function u satisfies the initial conditionin the strong L -sense, i.e., u ( · , t ) → in L . Next us state the main results of this paper.7 heorem 1.3. Let u ( k )0 ∗ ⇀ u in ˙ B − , ∞ and let v ( k ) be a sequence of a globalweak ˙ B − , ∞ -solutions to the Cauchy problem for the Navier-Stokes system withinitial data u ( k )0 . Then there exists a subsequence still denoted v ( k ) that con-verges to a global weak ˙ B − , ∞ -solution v to the Cauchy problem for the Navier-Stokes system with initial data u , in the sense of distributions. Corollary 1.4. There exists at least one global weak ˙ B − , ∞ -solution to theCauchy problem (1.1)-(1.3). Let us state our main observation, regarding decomposition of homoge-neous Besov spaces, that enables us to show the global existence of global˙ B − , ∞ solutions and that the property 2) Weak* stability holds true forthem. Note that from this point onwards, for p > 3, we will denote s p := − p < . Moreover, for 2 < α ≤ p > α , we define s p ,α := − α + 3 p < . Proposition 1.5. Suppose that < p < ∞ , g ∈ ˙ B s p p, ∞ ( R ) (1.25) and div g = 0 in sense of tempered distributions. Then the above assumptionsimply that there exists p < p < ∞ , < δ < − s p , γ := γ ( p ) > and γ := γ ( p ) > such that for any N > there exists divergence free tempereddistributions ¯ g N ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and e g N ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) withthe following properties. Namely, g = ¯ g N + e g N , (1.26) k ¯ g N k ˙ B sp δ p ,p ≤ N γ C ( p, k g k ˙ B spp, ∞ ) , (1.27) k e g N k L ≤ N − γ C ( p, k g k ˙ B spp, ∞ ) . (1.28) Furthermore, k ¯ g N k ˙ B spp, ∞ ≤ C ( p, k g k ˙ B spp, ∞ ) , (1.29) k e g N k ˙ B spp, ∞ ≤ C ( p, k g k ˙ B spp, ∞ ) . (1.30)8his refines previous decompositions for homogeneous Besov spaces, provenby the author in [6]. This is the main new observation of this paper. Un-like the decompositions in [6], it is not clear if related decompositions areobtainable by the known real interpolation theory of homogeneous Besovspaces.Once Proposition 1.5 is established, one can argue in a similar manner tothe case of global weak L , ∞ solutions to obtain improved decay propertiesof u ( x, t ) := v ( x, t ) − V ( x, t ) near the initial time, which we will state asa Lemma. Related properties have also been exploited by the author inestablishing weak strong uniqueness results for the Navier-Stokes equation,see [6]. Lemma 1.6. Let u , v and u be as in Definition 1.1. Let γ , γ , δ and p be as in Proposition 1.5. Then there exists a β ( γ , γ , δ ) > such that thefollowing estimate is valid for any < t < T : k u ( · , t ) k L + t Z Z R |∇ u | dxdt ′ ≤ C ( T, k u k ˙ B − , ∞ , δ ) t β . (1.31)Once this Lemma is established, Theorem 1.3 and Corollary 1.4 follow bysimilar arguments presented for global L , ∞ solutions in [3]-[4].We will also prove some conditional uniqueness and regularity statementsfor global weak ˙ B − , ∞ -solutions. Proposition 1.7. Let u ∈ ˙ B − , ∞ be divergence free in the sense of tempereddistributions. There exists an ε > such that if sup 1) Globalexistence-2) Weak* Stability . Proposition 1.5, together with ideas from[8] and a Lemma from [1], is sufficient to give a notion of solution of theNavier-Stokes equations with solenodial ˙ B s p p, ∞ initial data. To the best ofour knowledge, there is no previous notion of solution to the Navier-Stokesequations with arbitrary solenodial ˙ B s p p, ∞ initial data. Note that under certainsmallness conditions on the ˙ B s p p, ∞ data, solutions were constructed by meansof the Banach contraction principle, in [26] and [9]. We also mention anapplication of this solution in providing a new regularity criteria for weakLeray-Hopf solutions of the Navier-Stokes equations.Let 3 < p < ∞ and suppose u ∈ ˙ B s p p, ∞ is a divergence free tempereddistribution. Let u = ¯ u N + e u N denote the splitting from Proposition 1.5. Inparticular, p < p < ∞ , 0 < δ < − s p , γ := γ ( p ) > γ := γ ( p ) > N > u N ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and e u N ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) with u = ¯ u N + e u N , (1.34) k ¯ u N k ˙ B sp δ p ,p ≤ N γ C ( p, k u k ˙ B spp, ∞ ) , (1.35) k e u N k L ≤ N − γ C ( p, k u k ˙ B spp, ∞ ) . (1.36)Furthermore, k ¯ u N k ˙ B spp, ∞ ≤ C ( p, k u k ˙ B spp, ∞ ) , (1.37) k e u N k ˙ B spp, ∞ ≤ C ( p, k u k ˙ B spp, ∞ ) . (1.38) Theorem 1.8. Let u ∈ ˙ B s p p, ∞ ( R ) be divergence free, in the sense of tempereddistributions, and let < p < ∞ . For any finite T > there exists an N = N ( k u k ˙ B spp, ∞ , T ) > such that the following conclusions hold. In particular,there exists a solution to Navier-Stokes IVP in Q T such that v = W N + u N . (1.39) Here, W N is a mild solution to the Navier-Stokes equations, with initial data ¯ u N ∈ ˙ B s p + δ p ,p ( R ) , such that sup It can be shown that (1.40), (1.8) and the estimates for ¯ u N and e u N from Proposition 1.5, implies the following estimate for u N in theabove Theorem: sup Consider < S ≤ ∞ . Let u ∈ J ( R ) . (1.48) We say that v is a ’weak Leray-Hopf solution’ to the Navier-Stokes Cauchyproblem in Q S := R × ]0 , S [ if it satisfies the following properties: v ∈ L ( S ) := L ∞ (0 , S ; J ( R )) ∩ L (0 , S ; ◦ J ( R )) . (1.49) Additionally, for any w ∈ L ( R ) : t → Z R w ( x ) · v ( x, t ) dx (1.50) is a continuous function on [0 , S ] (the semi-open interval should be taken if S = ∞ ). The Navier-Stokes equations are satisfied by v in a weak sense: S Z Z R ( v · ∂ t w + v ⊗ v : ∇ w − ∇ v : ∇ w ) dxdt = 0 (1.51) for any divergent free test function w ∈ C ∞ , ( Q S ) := { ϕ ∈ C ∞ ( Q S ) : div ϕ = 0 } . The initial condition is satisfied strongly in the L ( R ) sense: lim t → + k v ( · , t ) − u k L ( R ) = 0 . (1.52) Finally, v satisfies the energy inequality: k v ( · , t ) k L ( R ) + 2 t Z Z R |∇ v ( x, t ′ ) | dxdt ′ ≤ k u k L ( R ) (1.53) for all t ∈ [0 , S ] (the semi-open interval should be taken if S = ∞ ). The corresponding global in time existence result, proven in [24], is asfollows. 13 heorem 1.11. Let u ∈ J ( R ) . Then, there exists at least one weak Leray-Hopf solution on Q ∞ . There are two big open problems concerning weak Leray-Hopf solutionsregarding uniqueness and regularity. Many regularity criteria exist for weakLeray-Hopf solutions. It was shown by Leray in [24] that if v is a weak Leray-Hopf solution with sufficiently regular initial data and finite blow up time T ,then there exists a C ( p ) > < p ≤ ∞ : k v ( · , t ) k L p ≥ c ( p )( T − t ) (1 − p ) . (1.54)The case p = 3 stood long open. It was shown in [12] that if v is a weakLeray-Hopf solution with sufficiently regular initial data and finite blow uptime T , then necessarily lim sup t ↑ T k v ( · , t ) k L ( R ) . (1.55)The proof in [12] is by a contradiction argument, involving a rescaling proce-dure producing a non trivial ancient solution and a Liouville theorem basedon backward uniqueness for parabolic equations. For an alternative approachto this type of regularity criteria, we refer to [20], [14] and [15].The criteria (1.55) was made more precise in [29], where it was shownthat if T is a finite blow up time then necessarilylim t ↑ T k v ( · , t ) k L ( R ) . (1.56)The proof in [29] uses ideas in [12], as well as the fact that the local en-ergy solutions of [25] on a fixed time interval, with L ( R ) initial data, arecontinuous with respect to weak convergence in L ( R ) of the initial data.Unfortunately, it is not known if the notion of local energy solutions in [25]carries over to the half space. Consequently, the proof of [29] doesn’t applyto the case of weak Leray-Hopf solutions on R × ]0 , ∞ [. This was overcomein [5]. In particular it was shown that the L ,q ( R ) of a weak Leray-Hopfsolution v on R × ]0 , ∞ [ must tend to infinity with 3 ≤ q < ∞ .A further refinement has been recently obtained in [1], who showed thatif T is a finite blow up time and 3 < p < ∞ , then necessarilylim t ↑ T k v ( · , t ) k ˙ B spp,p ( R ) . < p < ∞ L ( R ) ֒ → L ,p ( R ) ֒ → ˙ B s p p,p ( R ) ֒ → ˙ B s p p, ∞ ( R ) . Theorem 1.8 and the above Remark (specifically (1.46)-(1.47)), togetherwith ideas from [1] and [5], allow us to obtain a new of a regularity criteriafor weak Leray-Hopf solutions to the Navier- Stokes equations. Theorem 1.12. Let v be a global in time weak Leray-Hopf solution to theNavier-Stokes equations. Assume < T < ∞ is such that for any < ǫ < T v ∈ L ∞ ( ǫ, T − ǫ, L ∞ ( R )) . (1.57) Additionally assume that there exists an increasing sequence t k tending to T such that sup k k v ( · , t k ) k ˙ B − 1+ 3 pp, ∞ ( R ) = M < ∞ . (1.58) Furthermore, assume that for any ϕ ∈ C ∞ ( R )lim λ → λ Z R v ( y, T ) · ϕ ( y/λ ) dy = 0 . (1.59) The assumptions (1.57)-(1.59) then imply that for any < ǫ < Tv ∈ L ∞ ( ǫ, T ; L ∞ ( R )) . (1.60)Once we have Theorem 1.8 and Remark 1.9 (specifically (1.46)-(1.47))the proof the above refined regularity criteria can be completed by verbatimarguments of [1]. It should be noted that removing the assumption (1.59) in the aboveTheorem is a challenging open problem. A positive resolution would provideregularity of v at time T ifsup x ∈ R , We first introduce the frequency cut off operators of the Littlewood-Paleytheory. The definitions we use are contained in [2]. For a tempered distribu-tion f , let F ( f ) denote its Fourier transform. Let C be the annulus { ξ ∈ R : 3 / ≤ | ξ | ≤ / } . Let χ ∈ C ∞ ( B (4 / ϕ ∈ C ∞ ( C ) be such that ∀ ξ ∈ R , ≤ χ ( ξ ) , ϕ ( ξ ) ≤ , (2.1) ∀ ξ ∈ R , χ ( ξ ) + X j ≥ ϕ (2 − j ξ ) = 1 (2.2)and ∀ ξ ∈ R \ { } , X j ∈ Z ϕ (2 − j ξ ) = 1 . (2.3)For a being a tempered distribution, let us define for j ∈ Z :˙∆ j a := F − ( ϕ (2 − j ξ ) F ( a )) and ˙ S j a := F − ( χ (2 − j ξ ) F ( a )) . (2.4)Now we are in a position to define the homogeneous Besov spaces on R . Let s ∈ R and ( p, q ) ∈ [1 , ∞ ] × [1 , ∞ ]. Then ˙ B sp,q ( R ) is the subspace of tempereddistributions such that lim j →−∞ k ˙ S j u k L ∞ ( R ) = 0 , (2.5) k u k ˙ B sp,q ( R ) := (cid:16) X j ∈ Z jsq k ˙∆ j u k qL p ( R ) (cid:17) q . (2.6) Remark 2.1. This definition provides a Banach space if s < p , see [2]. Remark 2.2. It is known that if ≤ q ≤ q ≤ ∞ , ≤ p ≤ p ≤ ∞ and s ∈ R , then ˙ B sp ,q ( R ) ֒ → ˙ B s − p − p ) p ,q ( R ) . emark 2.3. It is known that for s = − s < and p, q ∈ [1 , ∞ ] , the normcan be characterised by the heat flow. Namely there exists a C > such thatfor all u ∈ ˙ B − s p,q ( R ) : C − k u k ˙ B − s p,q ( R ) ≤ kk t s S ( t ) u k L p ( R ) k L q ( dtt ) ≤ C k u k ˙ B − s p,q ( R ) . Here, S ( t ) u ( x ) := Γ( · , t ) ⋆ u where Γ( x, t ) is the kernel for the heat flow in R . We will also need the following Proposition, whose statement and proofcan be found in [2] (Proposition 2.22 there) for example. In the Propositionbelow we use the notation S ′ h := { tempered distributions u such that lim j →−∞ k S j u k L ∞ ( R ) = 0 } . (2.7) Proposition 2.4. A constant C exists with the following properties. If s and s are real numbers such that s < s and θ ∈ ]0 , , then we have, forany p ∈ [1 , ∞ ] and any u ∈ S ′ h , k u k ˙ B θs − θ ) s p, ( R ) ≤ Cs − s (cid:16) θ + 11 − θ (cid:17) k u k θ ˙ B s p, ∞ ( R ) k u k − θ ˙ B s p, ∞ ( R ) . (2.8) Proposition 2.5. Let u ∈ ˙ B − , ∞ be divergence free, in the sense of tempereddistributions. Then there exists a weakly divergence free sequence u ( k )0 ∈ L ( R ) such that u ( k )0 ∗ ⇀ u in ˙ B − , ∞ .Proof. Next, it is well known that for any u ∈ ˙ B − , ∞ ( R ) we have that u ,< | k | := P kj = − k ˙∆ j u converges to u in the sense of tempered distributions.Furthermore, we have that ˙∆ j ˙∆ j ′ u = 0 if | j − j ′ | > . Thus, k u ,< | k | k ˙ B − , ∞ ≤ C k u k ˙ B − , ∞ (2.9)and ˙∆ N u ,< | k | = 0 if | N | > k + 1 . (2.10)17t then follows from [2] that there exists a Schwartz function g ( k ) , whoseFourier transform is supported away from the origin, such that k g ( k ) − u ,< | k | k ˙ B − , ∞ < k . (2.11)Define u ( k )0 to be the Leray Projector P applied to g ( k ) , which is continuouson ˙ B − , ∞ ( R ). Obviously since g ( k ) is Schwartz, we have that u ( k )0 is a weaklydivergence free function in L ( R ). Since u is divergence free in the sense oftempered distributions, we have that P u ,< | k | = u ,< | k | . Thus, k u ( k )0 − u ,< | k | k ˙ B − , ∞ = k P ( g ( k )0 − u ,< | k | ) k ˙ B − , ∞ < Ck . (2.12)So for any Schwartz function ϕ we have | < u ( k )0 − u , ϕ > | ≤ | < u ( k )0 − u < | k | , ϕ > | + | < u − u < | k | , ϕ > | ≤≤ Ck k ϕ k ˙ B , + | < u − u < | k | , ϕ > | . Thus, u ( k )0 satisfies the desired properties of the Proposition. Proposition 2.6. Let u ∈ ˙ B − , ∞ ( R ) . Then we have sup Before proving Proposition 1.5, we take note of a useful Lemma presented in[2] (specifically, Lemma 2.23 and Remark 2.24 in [2]). Lemma 3.1. Let C ′ be an annulus and let ( u ( j ) ) j ∈ Z be a sequence of functionssuch that Supp F ( u ( j ) ) ⊂ j C ′ (3.1) and (cid:16) X j ∈ Z jsr k u ( j ) k rL p (cid:17) r < ∞ . (3.2) Moreover, assume in addition that s < p . (3.3) Then the following holds true. The series X j ∈ Z u ( j ) converges (in the sense of tempered distributions) to some u ∈ ˙ B sp,r ( R ) ,which satisfies the following estimate: k u k ˙ B sp,r ≤ C s (cid:16) X j ∈ Z jsr k u ( j ) k rL p (cid:17) r . (3.4)Let us state a useful Lemma, proven in [6], regarding decomposition ofhomogeneous Besov spaces. Proposition 3.2. For i = 1 , , let p i ∈ ]1 , ∞ [ , s i ∈ R and θ ∈ ]0 , be suchthat s < s < s and p < p < p . In addition, assume the followingrelations hold: s (1 − θ ) + θs = s , (3.5)1 − θp + θp = 1 p , (3.6) s i < p i . (3.7)19 uppose that u ∈ ˙ B s p ,p ( R ) . Then for all ǫ > there exists u ,ǫ ∈ ˙ B s p ,p ( R ) , u ,ǫ ∈ ˙ B s p ,p ( R ) such that u = u ,ǫ + u ,ǫ , (3.8) k u ,ǫ k p ˙ B s p ,p ≤ ǫ p − p k u k p ˙ B s p ,p , (3.9) k u ,ǫ k p ˙ B s p ,p ≤ C ( p , p , p , kF − ϕ k L ) ǫ p − p k u k p ˙ B s p ,p . (3.10)In order to prove Proposition 1.5, we must first state and prove twoLemmas. Here is the first. Lemma 3.3. Let < p < ∞ and suppose u ∈ ˙ B s p p, ∞ ( R ) . Then there exists p < p < ∞ and < δ < − s p such that for any N > there exists functions ¯ u ,N ∈ ˙ B s p + δp , ∞ ( R ) ∩ ˙ B s p p, ∞ ( R ) and ¯ u ,N ∈ ˙ B , ∞ ( R ) ∩ ˙ B s p p, ∞ ( R ) with u = u ,N + u ,N , (3.11) k u ,N k p ˙ B sp δp , ∞ ≤ N p − p k u k p ˙ B spp, ∞ , (3.12) k u ,N k B , ∞ ≤ C ( p, p , kF − ϕ k L ) N − p k u k p ˙ B spp, ∞ , (3.13) k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ (3.14) and k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ . (3.15) Proof. If p < p < ∞ , there exists a θ ∈ ]0 , 1[ such that1 − θp + θ p . (3.16)If we define δ := θ − θ ) > s p + δ )(1 − θ ) = s p . (3.18)Denote, f ( j ) := ˙∆ j u,f ( j ) M − := f ( j ) χ | f ( j ) |≤ M f ( j ) M + := f ( j ) (1 − χ | f ( j ) |≤ M ) . It is easily verified that the following holds: k f ( j ) M − k p L p ≤ M p − p k f ( j ) k pL p , k f ( j ) M − k L p ≤ k f ( j ) k L p , k f ( j ) M + k L ≤ M − p k f ( j ) k pL p and k f ( j ) M + k L p ≤ k f ( j ) k L p . Thus, we may write2 p ( s p + δ ) j k f ( j ) M − k p L p ≤ M p − p ( p ( s p + δ ) − ps p ) j ps p j k f ( j ) k pL p (3.19) k f ( j ) M + k L ≤ M − p − ps p j ps p j k f ( j ) k pL p . (3.20)With (3.19) in mind, we define M ( j, N, p, p , δ ) := N ( psp − p sp δ )) jp − p . For the sake of brevity we will write M ( j, N ). Using the relations of theBesov indices given by (3.16)-(3.17), we can infer that M ( j, N ) − p − ps p j = N p − p . The crucial point being that this is independent of j . Thus, we infer2 p ( s p + δ ) j k f ( j ) M ( j,N ) − k p L p ≤ N p − p ps p j k f ( j ) k pL p , (3.21)2 s p j k f ( j ) M ( j,N ) − k L p ≤ s p j k f ( j ) k L p , (3.22) k f ( j ) M ( j,N )+ k L ≤ N − p ps p j k f ( j ) k pL p (3.23)and 2 s p j k f ( j ) M ( j,N )+ k L p ≤ s p j k f ( j ) k L p . (3.24)Next, it is well known that for any u ∈ ˙ B s p p, ∞ ( R ) we have that P mj = − m ˙∆ j u converges to u in the sense of tempered distributions. Furthermore, we have21hat ˙∆ j ˙∆ j ′ u = 0 if | j − j ′ | > . Combing these two facts allows us to observethat˙∆ j u = X | m − j |≤ ˙∆ m f ( j ) = X | m − j |≤ ˙∆ m f ( j ) M ( j,N ) − + X | m − j |≤ ˙∆ m f ( j ) M ( j,N )+ . (3.25)Define u ,Nj := X | m − j |≤ ˙∆ m f ( j ) M ( j,N ) − , (3.26) u ,Nj := X | m − j |≤ ˙∆ m f ( j ) M ( j,N )+ (3.27)It is clear, that Supp F ( u ,Nj ) , Supp F ( u ,Nj ) ⊂ j C ′ . (3.28)Here, C ′ is the annulus defined by C ′ := { ξ ∈ R : 3 / ≤ | ξ | ≤ / } . Using,(3.21)-(3.24) we can obtain the following estimates:2 p ( s p + δ ) j k u ,Nj k p L p ≤ λ ( p , kF − ϕ k L )2 p ( s p + δ ) j k f ( j ) M ( j,N ) − k p L p ≤≤ λ ( p , kF − ϕ k L ) N p − p ps p j k f ( j ) k pL p , (3.29)2 s p j k u ,Nj k L p ≤ λ ( kF − ϕ k L )2 s p j k f ( j ) M ( j,N ) − k L p ≤≤ λ ( kF − ϕ k L )2 s p j k f ( j ) k L p , (3.30) k u ,Nj k L ≤ λ ( kF − ϕ k L ) k f ( j ) M ( j,N )+ k L ≤≤ λ ( kF − ϕ k L ) N − p ps p j k f ( j ) k pL p (3.31)and 2 s p j k u ,Nj k L p ≤ λ ( kF − ϕ k L )2 s p j k f ( j ) M ( j,N )+ k L p ≤≤ λ ( kF − ϕ k L )2 s p j k f ( j ) k L p . (3.32)It is then the case that (3.28)-(3) allow us to apply the results of Lemma 3.1.This allows us to achieve the desired decomposition with the choice u ,N = X j ∈ Z u ,Nj ,u ,N = X j ∈ Z u ,Nj . emma 3.4. Fix < α < . • For < α < , take p such that < p < α − α .For p and α satisfying these conditions, suppose that u ∈ ˙ B s p,α p,p ( R ) ∩ ˙ B s p p, ∞ ( R ) (3.33) Then the above assumptions imply that there exists p < p < ∞ and < δ < − s p such that for any ǫ > there exists functions U ,ǫ ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and U ,ǫ ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) with u = U ,ǫ + U ,ǫ , (3.34) k U ,ǫ k p ˙ B sp δ p ,p ≤ ǫ p − p k u k p ˙ B sp,αp,p , (3.35) k U ,ǫ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ , (3.36) k U ,ǫ k L ≤ C ( p, p , kF − ϕ k L ) ǫ − p k u k p ˙ B sp,αp,p (3.37) and k U ,ǫ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ . (3.38) Proof. Under this condition, we can find p < p < ∞ such that θ := p − p − p > α − . (3.39)Clearly, 0 < θ < − θp + θ p . (3.40)Define δ := 1 − α + θ − θ . (3.41)From (3.39), we see that δ > 0. One can also see we have the followingrelation: (1 − θ )( s p + δ ) = s p,α . (3.42)23he above relations allow us to apply Proposition 3.2 to obtain the fol-lowing decomposition: (we note that ˙ B , ( R ) coincides with L ( R ) withequivalent norms) u = U ,ǫ + U ,ǫ , (3.43) k U ,ǫ k p ˙ B sp δ p ,p ≤ ǫ p − p k u k p ˙ B sp,αp,p , (3.44) k U ,ǫ k L ≤ C ( p, p , kF − ϕ k L ) ǫ − p k u k p ˙ B sp,αp,p . (3.45)For j ∈ Z and m ∈ Z , it can be seen that k ˙∆ m (cid:16) ( ˙∆ j u ) χ | ˙∆ j u |≤ N ( j,ǫ ) (cid:17) k L p ≤ C ( kF − ϕ k L )) k ˙∆ j u k L p . (3.46)and k ˙∆ m (cid:16) ( ˙∆ j u ) χ | ˙∆ j u |≥ N ( j,ǫ ) (cid:17) k L p ≤ C ( kF − ϕ k L )) k ˙∆ j u k L p . (3.47)From [6], we see that the definitions of U ,ǫ and U ,ǫ used in Proposition 3.2are of the following form: U ,ǫ := X j X | m − j |≤ ˙∆ m (cid:16) ( ˙∆ j u ) χ | ˙∆ j u |≤ N ( j,ǫ ) (cid:17) and U ,ǫ := X j X | m − j |≤ ˙∆ m (cid:16) ( ˙∆ j u ) χ | ˙∆ j u |≥ N ( j,ǫ ) (cid:17) Using this, along with (3.46)-(3.47), we can infer that k U ,ǫ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ and k U ,ǫ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u k ˙ B spp, ∞ . Proof of Proposition 1.5 Proof. Applying Lemma 3.3, we see that there exists p < p < ∞ and 0 <δ < − s p such that for any N > u ,N ∈ ˙ B s p + δp , ∞ ( R ) ∩ ˙ B s p p, ∞ ( R ) and u ,N ∈ ˙ B , ∞ ( R ) ∩ ˙ B s p p, ∞ ( R ) with24 = u ,N + u ,N , (3.48) k u ,N k p ˙ B sp δp , ∞ ≤ N p − p k g k p ˙ B spp, ∞ , (3.49) k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ , (3.50) k u ,N k B , ∞ ≤ C ( p, p , kF − ϕ k L ) N − p k g k p ˙ B spp, ∞ (3.51)and k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ . (3.52)Since 3 < p < ∞ , it is clear that there exists an α := α ( p ) such that2 < α < < p < α − α . (3.53)With this p and α , we may apply Proposition 2.4 with s = − + p , s = − p and θ = 6 (cid:16) α − (cid:17) . In particular this gives for any f ∈ S ′ h : k f k ˙ B sp,αp, ≤ c ( p, α ) k f k α − )˙ B − 32 + 3 pp, ∞ k f k − α )˙ B − 1+ 3 pp, ∞ . (3.54)From Remark 2.2, we see that ˙ B s p,α p, ( R ) ֒ → ˙ B s p,α p,p ( R ) and ˙ B , ∞ ( R ) ֒ → ˙ B − + p p, ∞ ( R ) . Thus, we have the inclusion˙ B s p p, ∞ ( R ) ∩ ˙ B , ∞ ( R ) ⊂ ˙ B s p,α p,p ( R ) ∩ ˙ B s p p, ∞ ( R ) . (3.55)Now (3.54)-(3.55), together with (3.51)-(3.52), imply that u ,N ∈ ˙ B s p,α p,p ( R )and there exists β ( p ) > k u ,N k ˙ B sp,αp,p ≤ N − β ( p ) C ( p, p , kF − ϕ k L , k g k ˙ B spp, ∞ ) . (3.56)Noting (3.53), along with (3.52) and (3.56), we may now apply Lemma 3.4.In particular, there exists p < p < ∞ and 0 < δ < − s p such that forany λ > U ,λ ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and U ,λ ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) with u ,N = U ,λ + U ,λ , (3.57) k U ,λ k p ˙ B sp δ p ,p ≤ λ p − p k u ,N k p ˙ B sp,αp,p ≤ λ p − p N − β ( p ) p C ( p, p , kF − ϕ k L , k g k ˙ B spp, ∞ ) , (3.58) k U ,λ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ , (3.59) k U ,λ k L ≤ C ( p, p , kF − ϕ k L ) λ − p k u ,N k p ˙ B sp,αp,p ≤≤ λ − p N − β ( p ) p C ( p, p , p , kF − ϕ k L , k g k ˙ B spp, ∞ ) (3.60)and k U ,λ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ . (3.61)Taking λ = N κ gives that u = u ,N + U ,N κ + U ,N κ with u ,N ∈ ˙ B s p + δp , ∞ ( R ) ∩ ˙ B s p p, ∞ ( R ), U ,N κ ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and U ,N κ ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) . Furthermore, k u ,N k p ˙ B sp δp , ∞ ≤ N p − p k g k p ˙ B spp, ∞ , (3.62) k u ,N k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ , (3.63) k U ,N k k p ˙ B sp δ p ,p ≤ N κ ( p − p ) − β ( p ) p C ( p, p , kF − ϕ k L , k g k ˙ B spp, ∞ ) , (3.64) k U ,N κ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ , (3.65) k U ,N κ k L ≤ N κ (2 − p ) − β ( p ) p C ( p, p , p , kF − ϕ k L , k g k ˙ B spp, ∞ ) (3.66)and k U ,N κ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ . (3.67)Let p = 2 max( p , p ) and δ = min( δ,δ )2 . From Remark 2.2 and (3.62)-(3.63), we have that u ,N ∈ ˙ B s p p , ∞ ( R ) ∩ ˙ B s p + δp , ∞ ( R ) with estimates k u ,N k ˙ B sp δp , ∞ ≤ c ( p ) N p − pp k g k pp ˙ B spp, ∞ (3.68)and k u ,N k ˙ B sp p , ∞ ≤ C ( kF − ϕ k L , p ) k g k ˙ B spp, ∞ . (3.69)26e may apply Proposition 2.4 with s = s p , s = s p + δ and θ = 1 − δ δ ∈ ]0 , . In particular this gives for any f ∈ S ′ h : k f k ˙ B sp δ p , ≤ c ( p , δ, δ ) k f k − δ δ ˙ B sp p, ∞ k f k δ δ ˙ B sp δp, ∞ . (3.70)From Remark 2.2, we see that ˙ B s p + δ p , ( R ) ֒ → ˙ B s p + δ p ,p ( R ). This, and (3.68)-(3.70) imply that k u ,N k ˙ B sp δ p ,p ≤ N δ p − p ) δp C ( p , p, p , δ, δ , kF − ϕ k L , k g k ˙ B spp, ∞ ) . (3.71)Using identical reasoning, it can also be inferred that k U ,N k k ˙ B sp δ p ,p ≤ N δ κ ( p − p ) − β ( p ) p ) δ p C ( p, p , p , p , δ, δ , kF − ϕ k L , k g k ˙ B spp, ∞ ) . (3.72)The choice κ = 1 p − p (cid:16) pβ ( p ) + δ p ( p − p ) δp (cid:17) implies k u ,N + U ,N κ k ˙ B sp δ p ,p ≤ N δ p − p ) δp C ( p, p , p , p , δ, δ , δ , kF − ϕ k L , k g k ˙ B spp, ∞ ) . (3.73)It is also the case that k u ,N + U ,N κ k ˙ B spp, ∞ ≤ C ( kF − ϕ k L ) k g k ˙ B spp, ∞ . (3.74)To establish the decomposition of Theorem we define ¯ g N to be the Lerayprojector applied to u ,N + U ,N κ and e g N to be the Leray projector appliedto U ,N κ . Note that the Leray projector is a continuous linear operator onthe homogeneous Besov spaces under consideration.27 Weak* Stability of Global Weak ˙ B − , ∞ ( R ) -Solutions Let L s,l ( Q T ), W , s,l ( Q T ), W , s,l ( Q T ) be anisotropic (or parabolic) Lebesgueand Sobolev spaces with norms k u k L s,l ( Q T ) = (cid:16) T Z k u ( · , t ) k lL s dt (cid:17) l , k u k W , s,l ( Q T ) = k u k L s,l ( Q T ) + k∇ u k L s,l ( Q T ) , k u k W , s,l ( Q T ) = k u k L s,l ( Q T ) + k∇ u k L s,l ( Q T ) + k∇ u k L s,l ( Q T ) + k ∂ t u k L s,l ( Q T ) . Lemma 4.1. Assume that u ∈ L ∞ (0 , T ; J ) ∩ L (0 , T ; ◦ J ) and that thereexists an α > such that ess sup By the H¨older inequality and Proposition 2.6: k V · ∇ V k L ≤ k V k L k∇ V k L ≤ ct k u k B − , ∞ . From here, (4.2) is easily established. Again, by the H¨older inequality andProposition 2.6: k u · ∇ V k L ≤ k∇ V k L k u k L ≤ c k u k L , ∞ ( Q T ) t k u k ˙ B − , ∞ . u · ∇ V ∈ L , ( Q T ) . Using H¨older’s inequalityonce more, one can verify that T Z k V · ∇ u k L dt ≤ ( T Z k∇ u k L dt ) ( T Z k V k L dt ) . The desired conclusion is reached by noting that Proposition 2.6 gives: k V k L ≤ ct k u k B − , ∞ . The last estimate shows why there are difficulties to prove energy estimatefor u .By Proposition 2.6, (4.1) and the H¨older inequality : T Z Z R | V ⊗ u : ∇ u | dxdτ ≤ (cid:16) T Z Z R |∇ u | dxdτ (cid:17) (cid:16) T Z Z R | V ⊗ u | dxdτ (cid:17) ≤≤ k u k ˙ B − , ∞ (cid:16) T Z Z R |∇ u | dxdτ (cid:17) (cid:16) T Z Z R τ − α ess sup Lemma 4.2. Let v be a global weak ˙ B − , ∞ -solution with functions u and q asin Definition 1.1. Then ( u, q ) = X i =1 ( u i , p i ) (4.5) such that for any finite T : ( u i , ∇ p i ) ∈ W , s i ,l i ( Q T ) × L s i ,l i ( Q T ) (4.6) and ( s , l ) = (9 / , / , ( s , l ) = (2 , / , ( s , l ) = (3 / , / . (4.7)29 n addition ( u i , p i ) satisfy the following: ∂ t u − ∆ u + ∇ p = − u · ∇ u, (4.8) ∂ t u − ∆ u + ∇ p = − V · ∇ V (4.9) ∂ t u − ∆ u + ∇ p = − V · ∇ u − u · ∇ V (4.10) in Q ∞ , and div u i = 0 (4.11) in Q ∞ for i = 1 , , , u i ( · , 0) = 0 (4.12) for all x ∈ R and i = 1 , , . Before the next Lemma let us introduce some notation. Let u, v and u be as in Definition 1.1. Let u = ¯ u N + e u N denote the splitting fromProposition 1.5. In particular, 4 < p < ∞ , 0 < δ < − s p , γ > γ > N > u N ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B − , ∞ ( R ) and e u N ∈ L ( R ) ∩ ˙ B − , ∞ ( R ) with u = ¯ u N + e u N , (4.13) k ¯ u N k ˙ B sp δ p ,p ≤ N γ C ( k u k ˙ B − , ∞ ) , (4.14) k e u N k L ≤ N − γ C ( k u k ˙ B − , ∞ ) . (4.15)Furthermore, k ¯ u N k ˙ B − , ∞ ≤ C ( k u k ˙ B − , ∞ ) , (4.16) k e u N k ˙ B − , ∞ ≤ C ( k u k ˙ B − , ∞ ) . (4.17)Let us define the following:¯ V N ( · , t ) := S ( t )¯ u N ( · , t ) , (4.18)˜ V N ( · , t ) := S ( t )˜ u N ( · , t ) (4.19)and w N ( x, t ) := u ( x, t ) + ˜ V N ( x, t ) . (4.20)30 emma 4.3. In the above notation, we have the following global energy in-equality k w N ( · , t ) k L + 2 t Z Z R |∇ w N ( x, t ′ ) | dxdt ′ ≤≤ k ˜ u N k L + 2 Z t Z R ( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N dxdt ′ (4.21) that is valid for positive N and t .Proof. Let us mention that with Lemma 4.1 in hand, the proof of Lemma4.3 follows from very similar reasoning as presented in [3]-[4]. We provide allthe details here for the convenience of the reader.The first stage is showing that w N satisfies the local energy inequality.Let us briefly sketch how this can be done. Let ϕ ∈ C ∞ ( Q ∞ ) be a positivefunction. Observe that the assumptions in Definition 1.1 imply that thefollowing function t → Z Ω w N ( x, t ) · ¯ V N ( x, t ) ϕ ( x, t ) dx (4.22)is continuous for all t ≥ 0. It is not so difficult to show that this term hasthe following expression: Z R w N ( x, t ) · ¯ V N ( x, t ) ϕ ( x, t ) dx = t Z Z R ( w N · ¯ V N )(∆ ϕ + ∂ t ϕ ) dxdt ′ −− t Z Z R ∇ w N : ∇ ¯ V N ϕdxdt ′ + t Z Z R ¯ V N · ∇ ϕqdxdt ′ ++ 12 t Z Z R ( | v | − | w N | ) v · ∇ ϕdxdt ′ −− t Z Z R ( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N ϕdxdt ′ − t Z Z R ( ¯ V N ⊗ ¯ V N + ¯ V N ⊗ w N ) : ( w N ⊗ ∇ ϕ ) dxdt ′ . (4.23)It is also readily shown that Z R | ¯ V N ( x, t ) | ϕ ( x, t ) dx = t Z Z R | ¯ V N ( x, t ′ ) | (∆ ϕ ( x, t ′ ) + ∂ t ϕ ( x, t ′ )) dxdt ′ −− t Z Z R |∇ ¯ V N | ϕdxdt ′ . (4.24)Using (1.1), together with (4.1)-(4.1), we obtain that for all t ∈ ]0 , ∞ [ and forall non negative functions ϕ ∈ C ∞ ( Q ∞ ): Z R ϕ ( x, t ) | w N ( x, t ) | dx + 2 t Z Z R ϕ |∇ w N | dxdt ′ ≤≤ t Z Z R | w N | ( ∂ t ϕ + ∆ ϕ ) + 2 qw N · ∇ ϕ + | w N | v · ∇ ϕdxdt ′ ++2 t Z Z R ( ¯ V N ⊗ ¯ V N + ¯ V N ⊗ w N ) : ( ∇ w N ϕ + w N ⊗ ∇ ϕ ) dxdt ′ (4.25)In the next part of the proof, let ϕ ( x, t ) = ϕ ( t ) ϕ R ( x ). Here, ϕ ∈ C ∞ (0 , ∞ )and ϕ R ∈ C ∞ ( B (2 R )) are positive functions. Moreover, ϕ R = 1 on B ( R ),0 ≤ ϕ R ≤ |∇ ϕ R | ≤ c/R, |∇ ϕ R | ≤ c/R . Since e u N ∈ [ C ∞ , ( R )] L ( R ) , it is obvious that for ˜ V N ( · , t ) := S ( t ) e u N ( · , t ) wethe energy equality: k ˜ V N ( · , t ) k L + t Z Z R |∇ ˜ V N | dxdt ′ = k e u N k L . (4.26)32y semigroup estimates, we have for 2 ≤ p ≤ ∞ , 4 ≤ q ≤ ∞ : k ˜ V N ( · , t ) k L p ≤ C ( p ) t ( − p ) k e u N k L , (4.27) k ¯ V N ( · , t ) k L q ≤ C ( q ) t ( − q )+ k ¯ u N k ˙ B − , ∞ . (4.28)Hence, we have w N ∈ C w ([0 , T ]; J ) ∩ L (0 , T ; ◦ J ). Here, T is finite and C w ([0 , T ]; J ) denotes continuity with respect to the weak topology. UsingH¨older’s inequality and Sobolev embeddings, this implies that w N ∈ L p ( Q T ) for 2 ≤ p ≤ / . (4.29)Using these facts, it is obvious that the following limits hold:lim R →∞ Z R ϕ R ( x ) ϕ ( t ) | w N ( x, t ) | dx + 2 t Z Z R ϕ R ϕ |∇ w N | dxdt ′ == Z R ϕ ( t ) | w N ( x, t ) | dx + 2 t Z Z R ϕ |∇ w N | dxdt ′ , lim R →∞ t Z Z R ( | w N | ∂ t ϕ ϕ R + 2( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N ϕ ϕ R ) dxdt ′ == t Z Z R ( | w N | ∂ t ϕ + 2( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N ϕ ) dxdt ′ , lim R →∞ t Z Z R ( | w N | ϕ ∆ ϕ R + ϕ | w N | v · ∇ ϕ R ++2 ϕ ( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ( w N ⊗ ∇ ϕ R )) dxdt ′ = 0 . Let us focus on the term containing the pressure, namely t Z Z R qw N · ∇ ϕ R ϕ dxdt ′ . 33t is known that the pressure q can be represented as the composition ofRiesz transforms R . In particular, q = q + q , (4.30) q = R i R j ( u i u j ) (4.31)and q = R i R j ( u i V j + V i u j ) . (4.32)Since u ∈ C w ([0 , T ]; J ) ∩ L (0 , T ; ◦ J ), it follows from the H¨older inequalityand Sobolev embeddings that u ∈ L p ( Q T ) for 2 ≤ p ≤ / 3. Using this,Proposition 2.6 and continuity of the Riesz transforms on Lebesgue spaces,we infer that q ∈ L ( Q T ) and q ∈ L ( R × ] ǫ, T [) for any 0 < ǫ < T and T > . (4.33)From this and (4.29), we infer thatlim R →∞ t Z Z T ( R ) qw N · ∇ ϕ R ϕ dxdt ′ = 0 . Thus, putting everything together, we get for arbitrary positive function φ ∈ C ∞ (0 , ∞ ): Z R ϕ ( t ) | w N ( x, t ) | dx + 2 t Z Z R ϕ ( t ′ ) |∇ w N | dxdt ′ ≤≤ t Z Z R | w N | ∂ t ϕ + 2( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N ϕ dxdt ′ (4.34)From Remark 1.2, we see thatlim t → k w N ( · , t ) − ˜ u N ( · ) k L = 0 . (4.35)For ¯ V N , we have k ¯ V N ( · , t ) k L ≤ t k ¯ u N k ˙ B − , ∞ ≤ t C ( k u k ˙ B − , ∞ ) . (4.36)34sing that ˙ B s p + δ p ,p ( R ) ֒ → ˙ B − δ ∞ , ∞ ( R ), together with the heat flow charac-terisation of homogeneous Besov spaces with negative upper index, we inferthat k ¯ V N ( · , t ) k L ∞ ≤ t − δ k ¯ u N k ˙ B sp δ p ,p ≤ N γ t − δ C ( k u k ˙ B − , ∞ ) (4.37)Thus, we have the following estimates: t Z Z R | ¯ V N ⊗ w N : ∇ w N | dxdt ′ ≤≤ N γ C ( k u k ˙ B − , ∞ ) t Z Z R |∇ w N | dxdt ′ t Z k w N ( · , τ ) k L τ − δ dτ , (4.38) t Z Z R | ¯ V N ⊗ ¯ V N : ∇ w N | dxdt ′ ≤ Ct C ( k u k ˙ B − , ∞ ) t Z Z R |∇ w N | dxdt ′ . (4.39)Let ϕ ε ( s ) := ≤ s ≤ ε/ , s − ( ε/ /ε if ε/ ≤ s ≤ ε, ε ≤ s. Using (4.1)-(4.39) and by taking suitable approximations of ϕ ε , it can beshown that ϕ = ϕ ε is admissible in (4.1). From this we obtain that thefollowing inequality is valid for any ε > t > Z R | w N ( x, t ) | dx + 2 t Z Z R ϕ ε ( t ′ ) |∇ w N | dxdt ′ ≤≤ t Z Z R | w N | ∂ t ϕ ε + 2( ¯ V N ⊗ w N + ¯ V N ⊗ ¯ V N ) : ∇ w N ϕ ε dxdt ′ . (4.40)Using (4.35), we can obtain (4.3) by letting ε tend to zero in (4.1).35 roof of Lemma 1.6 Proof. First observe that u = w N − ˜ V N . Thus, using (4.26) we see that k u ( · , t ) k L + t Z Z R |∇ u | dxdt ′ ≤≤ k ˜ u N k L + 2 k w N ( · , t ) k L + 2 t Z Z R |∇ w N | dxdt ′ . By (4.15): k ˜ u N k L ≤ N − γ C ( k u k ˙ B − , ∞ ) . (4.41)From now on, denote y N ( t ) := k w N ( · , t ) k L . Using (4.3), estimates (4.1)-(4.39), (4.41) and the Young’s inequality obtainthat y N ( t ) + t Z Z R |∇ w N | dxdt ′ ≤ N γ C ( k u k ˙ B − , ∞ ) t Z y N ( τ ) τ − δ dτ ++( N − γ + t ) C ( k u k ˙ B − , ∞ ) . (4.42)By Gronwall’s Lemma we obtain y N ( t ) ≤ ( N − γ + t ) C ( k u k ˙ B − , ∞ ) ×× exp (cid:16) N γ C ( k u k ˙ B − , ∞ , δ ) t δ (cid:17) . (4.43)Hence, k u ( · , t ) k L + t Z Z R |∇ u | dxdt ′ ≤ ( N − γ + t ) C ( k u k ˙ B − , ∞ , δ ) ×× (cid:16) t δ N γ exp (cid:16) N γ C ( k u k ˙ B − , ∞ , δ ) t δ (cid:17) + 1 (cid:17) . (4.44)The conclusion is then easily reached by taking N = t − κ with0 < κ < δ / γ . .2 Proof of Weak* Stability and Existence of GlobalWeak ˙ B − , ∞ ( R ) -Solutions Once Lemma 1.6 and Lemma 4.1 are established, the proof of Theorem 1.3is along similar lines to arguments in [3]-[4]. We present the full details forcompleteness. Proof of Theorem 1.3 Proof. We have u ( k )0 ∗ ⇀ u in ˙ B − , ∞ and may assume that M := sup k k u ( k )0 k ˙ B − , ∞ < ∞ . Firstly, define V ( k ) ( · , t ) := S ( t ) u ( k )0 ( · , t ) , V ( · , t ) := S ( t ) u ( · , t ) . We see that V ( k ) converges to V on Q ∞ in the sense of distributions. ByProposition 2.6, we see that k V ( k ) ( · , t ) k L ≤ CMt , (4.45) k ∂ mt ∇ l V ( k ) ( · , t ) k L r ≤ CMt m + k + (1 − r ) . (4.46)Here r ∈ [4 , ∞ ]. For T < ∞ and l ∈ ]1 , ∞ [, we have the compact embedding W , l ( B ( n ) × ]0 , T [) ֒ → C ([0 , T ]; L l ( B ( n ))) . From this and (4.46) one immediately infers that for every n ∈ N and l ∈ ]1 , ∞ [: ∂ mt ∇ l V ( k ) → ∂ mt ∇ l V in C ([1 /n, n ]; L l ( B ( n ))) . (4.47)From Lemma 1.6 we have that for any 0 < T < ∞ :sup 32 ( QT ) ≤ f ( M, T, β, δ ) . (4.53)By the same reasoning as in Lemma 4.1, we obtain: k V ( k ) · ∇ V ( k ) k L , ( Q T ) ≤ f ( M, T ) , (4.54) k V ( k ) · ∇ u ( k ) + u ( k ) · ∇ V ( k ) k L , ( Q T ) ≤ f ( M, T, β, δ ) . (4.55)Split u ( k ) = P i =1 u i ( k ) according to Definition 1.1, namely (4.5). By coerciveestimates for the Stokes system, along with (4.53) obtain: k u k ) k W , , ( Q t ) + k∇ p ( k )1 k L , ( Q T ) ≤ Cf ( M, T, β, δ ) , (4.56) k u k ) k W , , ( Q T ) + k∇ p ( k )2 k L ( Q t ) ≤ Cf ( M, T ) , (4.57) k u k ) k W , , ( Q t ) + k∇ p ( k )3 k L , ( Q T ) ≤ Cf ( M, T, β, δ ) . (4.58)By the previously mentioned embeddings, we infer from (4.56)-(4.58) thatfor any n ∈ N we have the following convergence for a certain subsequence: u ( k ) → u in C ([0 , n ]; L ( B ( n )) . (4.59)Hence, using (4.49), we infer that for any s ∈ ]1 , / u ( k ) → u in L s ( B ( n ) × ]0 , n [) . (4.60)38t is also not so difficult to show that for any f ∈ L and for any n ∈ N : Z R u ( k ) ( x, t ) · f ( x ) dx → Z R u ( x, t ) · f ( x ) dx in C ([0 , n ]) . (4.61)Using (4.52) with (4.61), we establish thatlim t → k u ( · , t ) k L = 0 . (4.62)Using (4.52), along with the fact that u ∈ L , ∞ ( Q T ) for any 0 < T < ∞ , wesee that for any 0 < T < ∞ sup 39n ˙ B − , ∞ . It was shown in [31] that for any k there exists a global L -weaksolution v ( k ) , which satisfiessup 0) = 0 . Furthermore, assume that F ij ∈ C ∞ ( Q T ) . Then a formal solution to theabove initial boundary value problem has the form: v ( x, t ) = t Z Z R K ( x − y, t − s ) : F ( y, s ) dyds. The kernel K is derived with the help of the heat kernel Γ as follows:∆ y Φ( y, t ) = Γ( y, t ) ,K mjs ( y, t ) := δ mj ∂ Φ ∂y i ∂y i ∂y s ( y, t ) − ∂ Φ ∂y m ∂y j ∂y s ( y, t ) . Moreover, the following pointwise estimate is known: | K ( x, t ) | ≤ C ( | x | + t ) . (5.1)40efine G ( f ⊗ g )( x, t ) := t Z Z R K ( x − y, t − s ) : f ⊗ g ( y, s ) dyds. (5.2)It what follows, we will use the notation π u ⊗ u := R i R j ( u i u j ) , where R is aRiesz transform and the summation convention is adopted. Proposition 5.1. Suppose that u ∈ ˙ B − , ∞ is weakly divergence free. Thereexists a universal constant ε such that ifess sup Let us introduce the space X ( T ) := { f ∈ S ′ ( R × ]0 , T [) : ess sup 0) = 041n R . Using (5.1), it is easy to check that for solutions to the above linearproblem the following estimate is true k u ( k +1) k X ( T ) ≤ c k v ( k ) k X ( T ) , and thus we have k v ( k +1) k X ( T ) ≤ k V k X ( T ) + c k v ( k ) k X ( T ) for all k = 1 , , ... . Using arguments in [19], one can show that if k V k X ( T ) < ε < c , then we have k v ( k ) k X ( T ) < k V k X ( T ) (5.7)for all k = 1 , , ... .Furthermore, Kato’s arguments also give that there is a e v = V + e u suchthat k v ( k ) − e v k X ( T ) , k u ( k ) − e u k X ( T ) → , (5.8)We also can exploit our equation, together with the pressure equation, toderive the following estimate for the energy and pressure: k u ( k ) − u ( m ) k , ∞ ,Q T + k∇ u ( k ) − u ( m ) k ,Q T + k π v ( k ) ⊗ v ( k ) − π v ( m ) ⊗ v ( m ) k ,Q T ≤≤ c T Z Z R | v ( k ) ⊗ v ( k ) − v ( m ) ⊗ v ( m ) | dxdt. (5.9)Using (5.8), we immediately see the following u ( k ) → e u in W , ( Q T ) ∩ C ([0 , T ]; L ( R )) , (5.10) π v ( k ) ⊗ v ( k ) → π e v ⊗ e v in L ( Q T ) , (5.11) ∂ t e u − ∆ e u + ∇ π e v ⊗ e v = − d iv e v ⊗ e v, div e u = 0 (5.12)in Q T and e u ( · , 0) = 0 . (5.13)42urthermore, it is not difficult to show that for 0 < t < T k e u ( · , t ) k L + t Z Z R |∇ e u ( x, t ′ ) | dxdt ′ ≤ t Z Z R | e v ⊗ e v | dxdt ′ ≤ t k V k X ( T ) . (5.14)Thus sup Let p ∈ ]3 , ∞ ] and p + 2 r = 1 . (5.17) Suppose that w ∈ L p,r ( Q T ) , v ∈ L , ∞ ( Q T ) and ∇ v ∈ L ( Q T ) . Then for t ∈ ]0 , T [ : t Z Z R |∇ v || v || w | dxdt ′ ≤ C t Z k w k rL p k v k L dt ′ + 12 t Z Z R |∇ v | dxdt ′ . (5.18) Specifically Theorem 3.1 of [6]. .2 Proof of Proposition 1.7 Proof. The proof of Proposition 1.7 is based on ideas developed by the authorin [6].Suppose, sup 0a such that for 0 < t < T : k u ( · , t ) k L ≤ t β c ( T, k u k ˙ B − , ∞ , δ ) , (5.28)45 e u ( · , t ) k L ≤ t β c ( T, k u k ˙ B − , ∞ , δ ) (5.29)and k w ( · , t ) k L ≤ t β c ( T, k u k ˙ B − , ∞ , δ ) . (5.30)Hence, sup Proposition 6.1. Consider p and δ such that < p < ∞ , δ > and s p + δ < . Suppose that u ∈ ˙ B s p + δ p ,p ( R ) is a divergence free tempereddistribution. There exists a constant c = c ( p ) such that if < T < ∞ issuch that cT δ k u k ˙ B sp δ p ,p < , (6.1)47 hen there exists a w , which solves the Navier-Stokes equations (1.1)-(1.3) on Q T in the sense of distributions and satisfies the following properties. Thefirst property is that w satisfies the estimate sup Let u = U + V , with U ∈ ˙ B s p + δ p ,p ( R ) ( < δ < − s p )and V ∈ L ( R ) being divergence free tempered distributions. Moreover,assume < T < ∞ is such that cT δ k U k ˙ B sp δ p ,p < . (6.4) Then there exists a solution to Navier-Stokes IVP in Q T with the followingproperties. In particular, v = W + u. (6.5) Here, W is a mild solution to the Navier-Stokes equations from Proposition6.1, with initial data U ∈ ˙ B s p + δ p ,p ( R ) , such that sup Proof. Let u ∈ ˙ B s p p, ∞ be divergence free. Let u = ¯ u N + e u N denote thesplitting from Proposition 1.5. In particular, p < p < ∞ , 0 < δ < − s p , γ := γ ( p ) > γ := γ ( p ) > N > u N ∈ ˙ B s p + δ p ,p ( R ) ∩ ˙ B s p p, ∞ ( R ) and e u N ∈ L ( R ) ∩ ˙ B s p p, ∞ ( R ) with u = ¯ u N + e u N , (6.12) k ¯ u N k ˙ B sp δ p ,p ≤ N γ C ( p, k u k ˙ B spp, ∞ ) , (6.13) k e u N k L ≤ N − γ C ( p, k u k ˙ B spp, ∞ ) . (6.14)Furthermore, k ¯ u N k ˙ B spp, ∞ ≤ C ( p, k u k ˙ B spp, ∞ ) , (6.15) k e u N k ˙ B spp, ∞ ≤ C ( p, k u k ˙ B spp, ∞ ) . (6.16)49rom (6.13) we see that for every 0 < T < ∞ we can choose N ( T, p, δ , p , γ , k u k ˙ B spp, ∞ ) > c ( p ) T δ k ¯ u N k ˙ B sp δ p ,p < . The proof is now completed by means of Proposition 6.2.