About some possible blow-up conditions for the 3-D Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] A p r About some possible blow-up conditions for the 3-DNavier-Stokes equations
Haroune Houamed ∗ April 30, 2019
Abstract
In this paper, we study some conditions related to the question of the possible blow-up ofregular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in aproof of a very recent result from [6], we prove that if one component of the velocity remainssmall enough in a sub-space of ˙ H ”almost” scaling invariant, then the 3D Navier Stokes isglobally wellposed. In a second time, we investigate the same question under some conditionson one component of the vorticity and unidirectional derivative of one component of the ve-locity in some critical Besov spaces of the form L pT ( ˙ B α, p − − α , ∞ ) or L pT ( ˙ B p + q − q, ∞ ). Keywords : Incompressible Navier-Stokes Equations, Anisotropic Littlewood-Paley The-ory, Blow-up criteria.AMS Subject Classification (2010): 35Q30, 76D03
In this work we are interested in the study of the possible blow-up for regular solutions tothe 3D incompressible Navier stokes equations(
N S ) ∂ t u + u · ∇ u − ∆ u + ∇ P = 0 , ( t, x ) ∈ R + × R div u = 0 u | t =0 = u where the unkowns of the equations u = ( u , u , u ), P are respectevely, the velocity and thepressure of the fluid. We recall that the set of the solutions to ( N S ) is invariant under thetransformation: u ,λ ( x ) def = λu ( λx ) , u λ ( t, x ) def = λu λ ( λ t, λx )That is if u ( t, x ) is a solution to ( N S ) on [0 , T ] × R associated to the initial data u , then,for all λ > u λ ( t, x ) is a solution to ( N S ) on [0 , λ − T ] × R associated to the initial data u ,λ .It is well known that system ( N S ) has a global weak solution with finite energy k u ( t ) k L + 2 Z t (cid:13)(cid:13) ∇ u ( t ′ ) (cid:13)(cid:13) L dt ′ ≤ k u k L (1) ∗ Universit´e Cˆote dAzur, CNRS, LJAD, France (email: [email protected]). his result was proved first by J.Leray in [18]. In dimension three, uniqueness for suchsolutions stands to be an open problem. J.Leray proved also in his famous paper [18] that, formore regular initial data, namely for u ∈ H ( R ), ( N S ) has a unique local smooth solution,that is there exists T ∗ > u in L ∞ T ∗ ( H ( R )) ∩ L T ∗ ( H ( R )).The question of the behaviour of this solution after T ∗ remains to be also an open problem.In order to give a ”formaly” large picture, let us define the set χ T def = (cid:18) ( L ∞ T L ∩ L T H ) · ( L ∞ T L ∩ L T H ) (cid:19) ′ (2)where( L ∞ T L ∩ L T H ) · ( L ∞ T L ∩ L T H ) def = (cid:8) uv : ( u, v ) ∈ ( L ∞ T L ∩ L T H ) × ( L ∞ T L ∩ L T H ) (cid:9) Multiplying (
N S ) by − ∆ u , and integrating by parts yield d dt k∇ u k L + k∇ u k H = − Z R d (cid:0) ∇ u · ∇ u (cid:12)(cid:12) ∇ u (cid:1) If we suppose that ∇ u is already bounded in G T some sub-space of χ T , then one may provethat ∇ u is bounded in L ∞ T L ∩ L T H . This is the case in dimension two where we get, for free,by the L -energy estimate (1) a uniform bound of ∇ u in L T L ⊂ (cid:0) ( L T L ) · L T L ) (cid:1) ′ ⊂ χ T .In the case of dimension three, several works have been done in this direction, establishinga global wellposedeness of ( N S ) under assumptions of the type ∇ u ∈ G T . We can setas an example of these results the well known Prodi-Serrin type criterions, saying that, if u ∈ L p ([0 , T ] , L q ( R )), with p + q = 1 and q ∈ ]3 , ∞ ], then ( N S ) is globally wellposed. Thelimit case where q = 3 was proved recently by L. Escauriaza, G. Seregin and V. Sver`ak in[10] proving that: if T ∗ def = T ∗ ( u ) denotes the life span of a regular solution u associated tothe initial data u then T ∗ < ∞ = ⇒ lim sup t → T ∗ k u ( t ) k L ( R ) = ∞ This was extended to the full limit in time in ˙ H ( R ) by G. Seregin in [24].In another hand, one may notice that the divergence free condition can provide us anothertype of conditions for the global regularity (let us say anisotropic ones) under conditionson some components of the velocity or its gradiant. Several works have been done in thisdirection, one may see for instance [19, 3, 4, 11, 13, 17, 20, 22, 23, 26] for examples in somescaling invariant spaces or not of Serrin-type regularity criterions, or equivalently provingthat, if T ∗ is finite then Z T ∗ (cid:13)(cid:13) u ( t, . ) (cid:13)(cid:13) pL q = ∞ or Z T ∗ (cid:13)(cid:13) ∂ j u ( t, . ) (cid:13)(cid:13) pL q = ∞ The first result in a scaling invariant space under only one component of the velocity hasbeen proved by J.-Y Chemin and P.Zhang in [7] for p ∈ ]4 ,
6[ and a little bit later by the sameauthors together with Z.Zhang in [9] for p ∈ ]4 , ∞ [. The case p = 2 has been treated veryrecently by J.-Y Chemin, I.Gallagher and P.Zhang in [6]. As mensionned in [6] such a resultin the case of p = ∞ , assuming it is true, seems to be the out of reach for the time being.however the authors in [6] proved some results for p = ∞ . Mainly they proved that if thereis a blow-up at some time T ∗ >
0, then it is not possible for one component of the velocityto tend to 0 too fast. More precisely they proved the following blow-up condition ∀ σ ∈ S , ∀ t < T ∗ , sup t ′ ∈ [ t,T ∗ [ (cid:13)(cid:13) u ( t ′ ) · σ (cid:13)(cid:13) ˙ H ≥ c log − (cid:18) e + k u ( t ) k L T ∗ − t (cid:19) he last result proved in their paper needs reinforcing slightly the ˙ H norm in some directions.Mainly, without loss of generality, their result can be stated as the following Theorem 1.
There exists a positive constant c such that if u is a maximal solution of ( N S ) in C ([0 , T ∗ [ , H ) , then for all positive real number E we have: T ∗ < ∞ = ⇒ lim sup y → T ∗ (cid:13)(cid:13) u (cid:13)(cid:13) ˙ H logh,E ≥ c . where k a k H logh,E def = Z R | ξ | log ( E | ξ h | + e ) | ˆ a ( ξ ) | dξ < ∞ Motivated by this result, we aim to show that, up to a small modification in the proofof Theorem 1, we can obtain the same blow-up condition in the case p = ∞ , by slightlyreinforcing the ˙ H norm in the vertical direction instead of the horizontal one. More precisely,we define Definition 1.
Let E be a positive real number. We define ˙ H log v ,E to be the sub space of ˙ H ( R ) such that: a ∈ ˙ H log v ,E ⇐⇒ k a k H logv,E def = Z R | ξ | log ( E | ξ v | + e ) | ˆ a ( ξ ) | dξ < ∞ We will prove
Theorem 2.
There exists a positive constant c such that if u is a maximal solution of ( N S ) in C ([0 , T ∗ [ , H ) , then for all positive real number E we have T ∗ < ∞ = ⇒ lim sup y → T ∗ (cid:13)(cid:13) u (cid:13)(cid:13) ˙ H logv,E ≥ c . Remark 1.
The blow-up condition stated in Theorem 2 above can be generalized to thefollowing one ∀ σ ∈ S , T ∗ < ∞ = ⇒ lim sup y → T ∗ k σ · u k ˙ H log ˜ σ,E ≥ c . where k a k H log ˜ σ,E def = Z R | ξ | log ( E | ξ ˜ σ | + e ) | ˆ a ( ξ ) | dξ < ∞ , with ξ ˜ σ def = ( ξ · σ ) σ The other two results that we will prove in this paper can be seen as some blow-up criterionsunder scaling invariant conditions on one component of the velocity and one component ofthe vorticity, whether in some anisotropic Besov spaces of the form L p (cid:0) ( B α , ∞ ) h ( B s p − α , ∞ ) v (cid:1) , for α ∈ [0 , s p ], or L p ( B q,p ), where s p def = 2 p − , and B q,p def = B q + p − q, ∞ . (3)We will prove Theorem 3.
Let u be a maximal solution of ( N S ) in C ([0 , T ∗ [; H ) . If T ∗ < ∞ , then ∀ p, m ∈ [2 , , ∀ α ∈ (cid:20) , p − (cid:21) , ∀ β ∈ (cid:20) , m − (cid:21) , we have: Z T ∗ (cid:13)(cid:13) ∂ u ( t ′ ) (cid:13)(cid:13) pB α,sp − α , ∞ dt ′ + Z T ∗ (cid:13)(cid:13) ω ( t ′ ) (cid:13)(cid:13) mB β,sm − β , ∞ dt ′ = ∞ heorem 4. Let u be a maximal solution of ( N S ) in C ([0 , T ∗ [; H ) . If T ∗ < ∞ , then for all q , q ∈ [3 , ∞ [ , for all p , p satisfying q i + 2 p i ∈ ]1 , , i ∈ { , } (4) we have Z T ∗ (cid:13)(cid:13) ∂ u ( t ′ ) (cid:13)(cid:13) p B q ,p dt ′ + Z T ∗ (cid:13)(cid:13) ω ( t ′ ) (cid:13)(cid:13) p B q ,p dt ′ = ∞ Remark 2.
1. All the spaces stated in Theorems 3 and Theorem 4 above are scaling in-variant spaces under the natural 3-D Navier-Stokes scaling.2. The regularity of the spaces stated in the blow-up conditions in Theorem 4 is negative,more precisely under assumption (4) , q i + p i − ∈ ] − , . Moreover, the integrabilityasked for in the associated Besov spaces is always higher than , which make these spaceslarger than L pT (cid:0) H p − (cid:1)
3. Taking in mind the embedding L pT (cid:0) H p − (cid:1) ֒ → L pT (cid:0) B α, p − − α , ∞ (cid:1) , for all α ∈ [0 , p − ] (seelemma A.2.3), it is obvious then that the blow-up conditions stated in Theorem 3 impliesthe ones in L pT (cid:0) H p − (cid:1) .4. In the case p = 4 (resp. m = 4 ) in Theorem 3, α (resp. β ) is necessary zero, this meansthat the anisotropic space abobe is nothing but L T ( ˙ B , ∞ ) , which is still larger than L T ( L ) . The proof in this case can be done without any use of anisotropic technics. The structure of the paper is the following: In section 2, we reduce the proof of the Theoremsto the proofs of three lemmas. In Section 3, we should present the proofs of these threelemmas, where we will use some results which will be recalled/proved in Section 4 ”Appendix”together with the definition and the properties of the functional spaces used in this work.
Notations If A and B are two real quantities, the notaion A . B means A ≤ CB for someuniversal constant C which is independent on varying parameters of the problem.( c q ) q ∈ Z (resp. ( d q ) q ∈ Z ) will be a sequence satisfying X q ∈ Z c q ≤ X q ∈ Z d q ≤ L rT ( L ph L qv ) def = L r ((0 , T ); L p (( R h ); L q ( R v ))) , ˙ H sh ( ˙ H tv ) def = ˙ H s,t ( R ) k·k ˙ H sh ( ˙ H tv ) def = k·k ˙ H s,t ( R ) , k·k ˙ B sp,q def = k·k ˙ B sp,q ( R ) Denoting by: J iℓ ( u, u ) def = Z R ∂ i u ∂ u ℓ ∂ i u ℓ The proof of Theorem 2 is then based on the following lemma
Lemma 1.
There exists
C > such that, for any E > , we have: (cid:12)(cid:12) J iℓ ( u, u ) (cid:12)(cid:12) ≤ (cid:18)
110 + C (cid:13)(cid:13) u (cid:13)(cid:13) H logv,E (cid:19) k∇ h u k H + C (cid:13)(cid:13) u (cid:13)(cid:13) H (cid:13)(cid:13) ∂ u h (cid:13)(cid:13) L E hile, the proofs of Theorem 3 and Theorem 4 are essentialy based on the following ones Lemma 2.
For all q ∈ ]2 , , for all α ∈ ]0 , s p [ , where s p = q − , we have: (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) ≤ k h k H ( R ) + C k g k pB α,sp − α , ∞ k g k L ( R ) Lemma 3.
For any p, q ∈ [1 , ∞ ] satisfying q + p ∈ ]1 , we have (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) ≤ k g k H ( R ) + C k f k p B q,p k g k L ( R ) As mentionned above, let us assume that lemmas 1, 2 and 3 hold true, which we will provein the next section, and let us prove Theorems 2, 3 and 4.
Proof of Theorem 2
Following the idea of [6] we begin by establishing a bound of ∇ h u in L ∞ T ( L ) ∩ L T ( ˙ H ),then we use this estimate to prove a bound of ∂ u in L ∞ T ( L ) ∩ L T ( ˙ H ). To do so we multiply( N S ) by − ∆ h u , usual calculation leads then to: d dt k∇ h u k L + k∇ h u k H = X i =1 E i ( u ) with: (5) E ( u ) def = − X i =1 (cid:0) ∂ i u h · ∇ h u h (cid:12)(cid:12) ∂ i u h (cid:1) L E ( u ) def = − X i =1 (cid:0) ∂ i u h · ∇ h u (cid:12)(cid:12) ∂ i u (cid:1) L E ( u ) def = − X i =1 (cid:0) ∂ i u ∂ u h (cid:12)(cid:12) ∂ i u h (cid:1) L E ( u ) def = − X i =1 (cid:0) ∂ i u ∂ u h (cid:12)(cid:12) ∂ i u h (cid:1) L A direct computation shows that E ( u ) , E ( u ) and E ( u ) can be expressed as a sum of termsof the form I ( u ) def = Z R ∂ i u ∂ j u k ∂ ℓ u m where: ( j, ℓ ) ∈ { , } and ( i, k, m ) ∈ { , , } .Next, by duality, product rules and then interpolation, for any p ∈ [1 , + ∞ ], one may easlyshow that I ( u ) . (cid:13)(cid:13) ∇ h u (cid:13)(cid:13) ˙ H p − (cid:13)(cid:13)(cid:13) ∂ j u k ∂ ℓ u m (cid:13)(cid:13)(cid:13) ˙ H − p . (cid:13)(cid:13) u (cid:13)(cid:13) ˙ H p + 12 k∇ h u k H − p . (cid:13)(cid:13) u (cid:13)(cid:13) ˙ H p + 12 k∇ h u k p L k∇ h u k − p ˙ H Notice that I ( u ) provides a globale bound if u ∈ L p ( ˙ H p + ) for some p ∈ [1 , ∞ ]. It is in fact the term E ( u )which poses a problem, and this is why this methode doesn’t give a complet answer to the regularity criteria underone component only in the case p = 2 as mentionned in [6]. n particular for p = ∞ we have: I ( u ) . (cid:13)(cid:13) u (cid:13)(cid:13) H k∇ h u k H (6)The term E ( u ), can be estimated by using lemma 1, to obtain E ( u ) ≤ (cid:18)
110 + C (cid:13)(cid:13) u (cid:13)(cid:13) H logv,E (cid:19) k∇ h u k H + C (cid:13)(cid:13) u (cid:13)(cid:13) H (cid:13)(cid:13) ∂ u h (cid:13)(cid:13) L E We define then T ∗ def = sup (cid:26) T ∈ [0 , T ∗ [ / sup t ∈ [0 ,T ] (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) H logv,E ≤ C (cid:27) Therefore, for all t ≤ T ∗ , relation (5) together with estimate (6), lemma 1 and the classical L − energy estimate lead to k∇ h u ( t ) k L + Z t k∇ h u ( s ) k H ds ≤ k∇ h u k L + k u k L E (7)In the other hand, as explained in [6], multiplying ( N S ) by − ∂ u , integrating over R ,integration by parts together with the divergence free condition lead to d dt k ∂ u k L + k ∂ u k H . k ∂ u k L k∇ h u k L k ∂ u k L . k ∂ u k H + C k∇ h u k L k∇ h u k ˙ H k ∂ u k L (7) above leads then to a bound for u in L ∞ T ∗ ( ˙ H ).Thus, by contraposition, if the quantity k u ( t ) k ˙ H blows-up at a finite time T ∗ >
0, then ∀ t ∈ [0 , T ∗ [: sup s ∈ [0 ,t ] (cid:13)(cid:13) u ( s ) (cid:13)(cid:13) ˙ H > c = 14 C which gives the desired result by passing to the limit t → T ∗ .Theorem 2 is proved. ✷ Proof of Theorem 3
Following for example an idea from [27], we multiply (
N S ) by − ∆ u and we integrate inspace to obtain d dt k∇ u k L + k ∆ u k L = Z R (cid:0) u · ∇ u (cid:1) · ∆ u = − Z R X i,j,k =1 ∂ k u j ∂ j u i ∂ k u i For the time being, we don’t know how to deal with the tri-linear term on the right hand-sideabove in order to obtain a global-estimate of u in L ∞ T ˙ H x ∩ L T ˙ H x , so to close the estimatesthe idea is similar to the one in Theorems 1 and 2, and it consists in looking at this term asa bilinear operator acting on (cid:0) L ∞ T ˙ H x ∩ L T ˙ H x (cid:1) after assuming a condition which allows tocontrole some components of the matrix ∂ i u j . et us recall the Biot-Savart law identity which allows to write the so-called div-curl decom-position of u h as u h = ∇ ⊥ h ∆ − h ( ω ) − ∇ h ∆ − h ( ∂ u ) (8)Identity (8) inssures that, for ( i, j ) ∈ { , } , ∂ i u j can be writing in terms of ω and ∂ u ,modulo some anisotropic Fourier-multiplyers of order zero, more presisely we have, for ( i, j ) ∈{ , } ∂ i u j = R i,j ω + ˜ R i,j ∂ u where ˜ R i,j and R i,j are zero-order Fourier multiplyers bounded from L q into L q for all q in]1 , ∞ [. In the other hand, the quantity ∂ k u j ∂ j u i ∂ k u i contains always, at least, one term ofthe form ∂ i u j with ( i, j ) ∈ { , } or i = j = 3, we infer that d dt k∇ u k L + k ∆ u k L . X ( l,k,m,n ) ∈{ , , } ( i,j ) ∈{ , } (cid:12)(cid:12)(cid:12)(cid:12) Z R (cid:0) R i,j ω + ˜ R i,j ∂ u (cid:1) ∂ k u l ∂ m u n (cid:12)(cid:12)(cid:12)(cid:12) (9)Lemma 3 gives then d dt k∇ u k L + k∇ u k H ≤ k∇ u k H + C (cid:18) k ∂ u k p ˙ B α,sp − α , ∞ + (cid:13)(cid:13) ω (cid:13)(cid:13) m ˙ B β,sm − β , ∞ (cid:19) k∇ u k L Gronwall lemma leads then to k∇ u ( t ) k L + Z t (cid:13)(cid:13) ∇ u ( t ′ ) (cid:13)(cid:13) H dt ′ . k∇ u k L exp (cid:20) C Z t (cid:18) (cid:13)(cid:13) ∂ u ( t ′ ) (cid:13)(cid:13) p ˙ B α,sp − α , ∞ + (cid:13)(cid:13) ω ( t ′ ) (cid:13)(cid:13) m ˙ B β,sm − β , ∞ (cid:19) dt ′ (cid:21) (10)That is if, for some α, β, p, m satisfying the hypothesis of Theorem 3, the quantity in theright hand side of (10) is finite, then u is bounded in L ∞ T ( ˙ H ). By contraposition, if there isa blow-up of the ˙ H norm at some finite T ∗ then, for all α, β, p, m Z T ∗ (cid:18) (cid:13)(cid:13) ∂ u ( t ′ ) (cid:13)(cid:13) p ˙ B α,sp − α , ∞ + (cid:13)(cid:13) ω ( t ′ ) (cid:13)(cid:13) m ˙ B β,sm − β , ∞ (cid:19) dt ′ = ∞ Theorem 3 is proved. ✷ Proof of Theorem 4
The proof of Theorem 4 doesn’t differ a lot from the previous one. We restart from (9),applying lemma 2 gives d dt k∇ u k L + k∇ u k H ≤ k∇ u k H + C (cid:18) k ∂ u k p B q ,p + (cid:13)(cid:13) ω (cid:13)(cid:13) p B q ,p (cid:19) k∇ u k L Next, integrating in time interval [0 , t ], and applying Gronwall lemma gives k∇ u ( t ) k L + Z t (cid:13)(cid:13) ∇ u ( t ′ ) (cid:13)(cid:13) H dt ′ . k∇ u k L exp (cid:20) C Z t (cid:18) k ∂ u k p B q ,p + (cid:13)(cid:13) ω (cid:13)(cid:13) p B q ,p (cid:19) dt ′ (cid:21) Same arguments as in the conclusion of the previous theorem lead to the desired result.Theorem 4 is Proved. ✷ Note that the case q i = ∞ is included in the estimates proved in Lemma 3, however we did not say anythingabout this case in Theorem 4 due to the lack of continuity of Riesz operators R i,j and ˜ R i,j from L ∞ into L ∞ . Proof of the three lemmas
Proof of lemma 1
Let us recall a definition from [6]. For E ∈ R + and a ∈ S ′ ( R ): a ♭,E − def = F − ( B h (0 ,E − ) ˆ a ) , a ♯,E − def = F − ( B ch (0 ,E − ) ˆ a )Based on this decomposition, we write J iℓ ( u, u ) = J ♭E + J ♯E where J ♭E def = Z R ∂ i u ∂ u ℓ♭,E − ∂ i u ℓ , and J ♯E def = Z R ∂ i u ∂ u ℓ♯,E − ∂ i u ℓ The main point consists in estimating J ♯E . Using Bony’s decomposition with respect to thehorizontal variables, to write J ♯E = J ♯, E + J ♯, E with J ♯, E def = Z R v (cid:18) Z R h ∂ i u ℓ ( x h , x ) ˜ T h∂ i u ( x h ,x ) ∂ u ℓ♯,E − ( x h , x ) dx h (cid:19) dx J ♯, E def = Z R v (cid:18) X k ∈ Z Z R h ∆ hk ∂ i u ℓ ( x h , x ) ˜∆ hk T h∂ u ℓ♯,E − ( x h ,x ) ∂ i u ( x h , x ) dx h (cid:19) dx .J ♯, E can be estimated by duality then by using some product laws (lemma A.2.2), we obtain J ♯, E . k∇ h u k L ∞ v ( ˙ H h ) (cid:13)(cid:13)(cid:13) ˜ T h∂ i u ∂ u ℓ♯,E − (cid:13)(cid:13)(cid:13) L v ( ˙ H − h ) . k∇ h u k L ∞ v ( ˙ H h ) (cid:13)(cid:13) ∇ h u (cid:13)(cid:13) L v ( ˙ H − h ) k ∂ u k L v ( ˙ H h ) Using then the inequality: k∇ h u k L ∞ v ( ˙ H h ) . k∇ h u k ˙ H ( R ) (see lemma A.2.4), we infer that J ♯, E . (cid:13)(cid:13) u (cid:13)(cid:13) ˙ H ( R ) k∇ h u k H ( R ) (11)In order to estimate J ♯, E we split it into a sum of a good term J ♯, , G E and a bad one J ♯, , B E based on the dominated frequencies of ∂ u ℓ ∂ u ℓ♯,E − = ∂ u ℓ, G ♯,E − + ∂ u ℓ, B ♯,E − with ∂ u ℓ, G ♯,E − def = X q ≤ k ∆ hk ∆ vq ∂ u ℓ♯,E − and ∂ u ℓ, B ♯,E − def = X k Let p ∈ [2 , 4] and α ∈ (cid:2) , p − (cid:3) . We define q and θ such that2 q def = 1 − p (19) θ def = 12 + α − p (20)One may check that q ∈ ]2 , ∞ [ and θ ∈ (cid:20) , q (cid:21) ∩ (cid:20) , (cid:21) which allow us to use the following embedding, due to lemmas A.2.3 and A.2.5 L ∞ T ( L ( R )) ∩ L T ( ˙ H ( R )) ֒ → L qT ( ˙ B q , ( R )) ֒ → L qT (cid:0) ( ˙ B q − θ , ) h ( ˙ B θ , ) v (cid:1) (21)Thus, by using lemma A.2.2, if g ∈ ( ˙ B q − θ , ) h ( ˙ B θ , ) v then g · g ∈ ( ˙ B q − θ − , ) h ( ˙ B θ − , ) v .By virtue of (19), (20) and embedding (21), we infer that k g · g k ( ˙ B − α , ) h ( ˙ B − p + 12 + α , ) v . k g k 2( ˙ B q − θ , ) h ( ˙ B θ , ) v which gives by duality, embedding (21) and lemma A.2.5 (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) . k f k ( ˙ B α , ∞ ) h ( ˙ B p − − α , ∞ ) v k g · g k ( ˙ B − α , ) h ( ˙ B − p + 12 + α , ) v . k f k ( ˙ B α , ∞ ) h ( ˙ B sp − α , ∞ ) v k g k B q , . k f k ( ˙ B α , ∞ ) h ( ˙ B sp − α , ∞ ) v k g k p L k g k − p )˙ H Finally we obtain (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) ≤ k g k H + C k f k ( ˙ B α , ∞ ) h ( ˙ B sp − α , ∞ ) v k g k L Lemma 2 is proved. ✷ Proof of lemma 3 According to lemma A.2.5 in Appendix, in particular inequality (26) gives k g ( t, . ) k ˙ B m , ( R )) . k g ( t, . ) k m ˙ H ( R ) k g ( t, . ) k − m L ( R ) , ∀ m ∈ ]2 , ∞ [ (22)We use then the Bony’s decomposition to study the product g · g .Let ( q, p ) ∈ [1 , ∞ ] satisfying q ∈ [3 , ∞ ] and 3 q + 2 p ∈ ]1 , m , m ) be in [2 , ∞ ] × ]2 , ∞ [ , given by23 m def = 1 q and 2 (cid:0) − m (cid:1) def = 3 q + 2 p ∈ ]1 , ⇐⇒ m ∈ ]2 , ∞ [ (23) et us defined the real number N m associated to the embedding ˙ H m ( R ) in L N m ( R )1 N m def = 12 − m ∈ (cid:20) , (cid:21) Let us also define r to be the conjugate of q , that is1 r def = 1 − q ∈ (cid:20) , (cid:21) We wirte ∆ j ( g · g ) = 2∆ j T g ( g ) + ∆ j R ( g, g )where T and R are the operators associated to the Bony’s decomposition, defined in theAppendix.We turn now to estimate the two parts of ∆ j ( g · g ). We have k ∆ j T g ( g ) k L r . k S j − g k L Nm k ∆ j g k L . k g k L Nm d j − j m k g k ˙ B m , using then the embedding k g k L Nm . k g k ˙ H m (24)together with the interpolation inequality (22) gives k ∆ j T g ( g ) k ˙ B m r, . − j m d j k g k m + m ˙ H k g k − m + m L (25)For the reminder term, we proceed almost similary k ∆ j R ( g, g ) k L r . X j ′ ≥ j − (cid:13)(cid:13)(cid:13) ˜∆ j ′ g (cid:13)(cid:13)(cid:13) L Nm (cid:13)(cid:13) ∆ ′ j g (cid:13)(cid:13) L . − j m X j ′ ≥ j − (cid:0) d j ′ − ( j ′ − j ) m (cid:1) k g k L Nm k g k ˙ B m , Where ˜∆ ′ j def = X i ∈{− , , } ∆ j ′ + i . By convolution inequatlity, interpolation inequality (22) andthe embedding one (24), we get k ∆ j R ( g, g ) k L r . − j m d j k g k m + m ˙ H k g k − m + m L which gives, together with (25) k ∆ j ( g · g ) k L r . − j m d j k g k m + m ˙ H k g k − m + m L In the other hand, by duality, we get (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) . X j ∈ Z k ∆ j f k L q k ∆ j ( g · g ) k L r . X j ∈ Z (cid:0) − j m k ∆ j f k L q d j (cid:1) k g k m + m ˙ H k g k − m + m L . k f k ˙ B − m q, ∞ k g k m + m ˙ H k g k − m + m L y virtue of (23) we have1 − (cid:18) m + 1 m (cid:19) = 1 p and − m = 3 q + 2 p − (cid:12)(cid:12)(cid:0) f g | g (cid:1) L (cid:12)(cid:12) ≤ k g k H + C k f k p B q,p k g k L Lemma 3 is proved. ✷ A Appendix A.1 Functional framwork In this part we recall some notions/definitions used in the previous sections.Let us first recall some notions of the Littlewood-Paley theory, the anisotropic Besov spacesused in this paper and some of their properties. For more details one may see for instance[2]. Let ( ψ, ϕ ) be a couple of smooth functions with value in [0 , 1] satisfying:Supp ψ ⊂ { ξ ∈ R : | ξ | ≤ } , Supp ϕ ⊂ { ξ ∈ R : 34 ≤ | ξ | ≤ } ψ ( ξ ) + X q ∈ N ϕ (2 − q ξ ) = 1 ∀ ξ ∈ R , X q ∈ Z ϕ (2 − q ξ ) = 1 ∀ ξ ∈ R \{ } . Let a be a tempered distribution, ˆ a = F ( a ) its Fourier transform and F − denotes the inverseof F . We define the homogeneous dyadic blocks ∆ q by setting∆ vq a def = F − (cid:0) ϕ (2 − q | ξ | ˆ a ) (cid:1) , ∀ q ∈ Z , ∆ hj a def = F − (cid:0) ϕ (2 − j | ξ h | ˆ a ) (cid:1) , ∀ j ∈ Z ,S vq def = X q ′ In this part we present seven lemmas used in the previous section, we will prove the threelast ones and give references for the four first ones.We Start by recalling a Bernstein type lemma from [7] Lemma A.2.1. Let B h (resp. B v ) be a ball of R h (resp. R v ) and C h (resp. C v ) a ring of R h (resp. R v ). Let also a be a tempered distribution and ˆ a its Fourier transform. Then for ≤ p ≤ p ≤ ∞ and ≤ q ≤ q ≤ ∞ we have:Supp ˆ a ⊂ k B h = ⇒ (cid:13)(cid:13) ∂ αx h a (cid:13)(cid:13) L p h ( L q v ) . k (cid:0) | α | +2 (cid:0) p − p (cid:1)(cid:1) k a k L p h ( L q v ) Supp ˆ a ⊂ l B v = ⇒ (cid:13)(cid:13)(cid:13) ∂ βx a (cid:13)(cid:13)(cid:13) L p h ( L q v ) . l (cid:0) β + (cid:0) q − q (cid:1)(cid:1) k a k L p h ( L q v ) Supp ˆ a ⊂ k C h = ⇒ k a k L p h ( L q v ) . − kN sup | α | = N (cid:13)(cid:13) ∂ αx h a (cid:13)(cid:13) L p h ( L q v ) Supp ˆ a ⊂ l C v = ⇒ k a k L p h ( L q v ) . − lN (cid:13)(cid:13) ∂ Nx a (cid:13)(cid:13) L p h ( L q v ) Let us also recall an anisotropic version of the usual product laws in Besov spaces (see lemma4.5 from [7]) Lemma A.2.2. Let q ≥ , p ≥ p ≥ with p + p ≤ , and s < p , s < p (resp. s ≤ p , s ≤ p if q = 1 ) with s + s > . Let σ < p , σ < p (resp. σ ≤ p , σ ≤ p if q = 1 ) with σ + σ > . Then for a in B s ,σ p ,q and b in B s ,σ p ,q , the product ab belongs to B s + s − p ,σ + σ − p p ,q and we have k ab k B s s − p ,σ σ − p p ,q . k a k B s ,σ p ,q k b k B s ,σ p ,q A very useful lemma in the anisotropic context (lemma 4.3 from [7]), is the following Lemma A.2.3. For any s positive, for all ( p, q ) ∈ [1 , ∞ ] and any θ ∈ ]0 , s [ , we have k f k ( ˙ B s − θp,q ) h ( ˙ B θp, ) v . k f k ˙ B sp,q Finally, we recall lemma A.2 from [6] Lemma A.2.4. For any function a in the space ˙ H + s ( R ) with ≤ s < , there holds k a k L ∞ v ( ˙ H sh ) ≤ √ k a k ˙ H 12 + s ( R ) ext, we will prove an interpolation version in space-time spaces Lemma A.2.5. For all p ∈ ]2 , ∞ [ , there exists a constant c p > , such that for all u in L ∞ T ( L ( R )) ∩ L T ( ˙ H ( R )) we have k u k L pT ( ˙ B p , ( R )) ≤ c p k u k p L T ( ˙ H ( R )) k u k − p L ∞ T ( L ( R )) Proof The prood is classical, we proceed as the follwing:Let N ( t ) > X j ∈ Z p k ∆ j u ( t, . ) k L = X j ≤ N ( t ) p k ∆ j u ( t, . ) k L + X j>N ( t ) p − k ∆ j ∇ u ( t, . ) k L ≤ N ( t ) k u ( t, . ) k L + 2 N ( t ) − k∇ u ( t, . ) k L The choice of N ( t ) such that 2 N ( t ) def = (cid:18) − p p (cid:19) k∇ u ( t, . ) k L k u ( t, . ) k L gives k u ( t, . ) k ˙ B p , ( R )) ≤ c p k u ( t, . ) k p ˙ H ( R ) k u ( t, . ) k − p L ( R ) (26)The lemma follows by taking the L p norm in time. ✷ The following lemma can be used when the horizontal frequencies controle the vertical ones Lemma A.2.6. Let s, t be two real numbers, let f be a regular function, we define f G as f G def = X q ≤ k ∆ hk ∆ vq f. Then we have: (cid:13)(cid:13) ∂ f G (cid:13)(cid:13) ˙ H s,t . k∇ h f k ˙ H s,t Proof Let us use Plancherel-Parseval identity to write: (cid:13)(cid:13) ∂ f G (cid:13)(cid:13) H s,t ≈ Z R | ξ h | s | ξ v | t (cid:12)(cid:12)(cid:12)(cid:12) X q ≤ k | ξ v | ϕ hk ( ξ ) ϕ vq ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) | ˆ f ( ξ ) | dξ (27)where: ϕ hk def = ϕ (2 − k | ξ h | ), ϕ vq def = ϕ (2 − q | ξ v | ), and ϕ is the function defined at the begining ofthe Appendix part. Thus, using the support properties of ϕ hk , ϕ vq , and the condition q < k ,we infer that, for all ξ = ( ξ h , ξ v ) ∈ Supp ( ϕ hk ) × supp ( ϕ vq ) | ξ v | . q ≤ k . | ξ h | (28)plugging (28) into (27) concludes the proof of the lemma. ✷ The last lemma that we will prove is usful to estimate some parts of the anisotropic Bony’s de-composition for functions having a dominated vertical frequencies compared to the horizontalones, and which are supported away from zero horizontaly in Fourier side. emma A.2.7. Let f be regular function, and E > . We define f B ♯,E − as f B ♯,E − def = X k According to the support properties we have∆ vq S hj − ( f B ♯,E − ) = (cid:18) ∆ vq S hj − X i ∈{− , , } S hq − i ∆ vq + i (cid:19) f ♯,E − therefore, Bernstein’s inequality, we can write (cid:13)(cid:13)(cid:13) ∆ vq S hj − ( f B ♯,E − ) (cid:13)(cid:13)(cid:13) L v ( L ∞ h ) . X E − . k . q k (cid:13)(cid:13)(cid:13) ∆ hk ∆ vq f (cid:13)(cid:13)(cid:13) L ( R ) . (cid:18) X E − . k . q c k (cid:19) c q k∇ h f k L ( R ) . (cid:0) log ( E q + e ) (cid:1) c q k∇ h f k L ( R ) Thus the first inequality is proved. For the second, one we first write (cid:13)(cid:13)(cid:13) S vq − S hj − ( f B ♯,E − ) (cid:13)(cid:13)(cid:13) L ∞ v ( L ∞ h ) . X m ≤ q m (cid:13)(cid:13)(cid:13) ∆ vm S hj − ( f B ♯,E − ) (cid:13)(cid:13)(cid:13) L v ( L ∞ h ) inequality (29) gives then2 − q (cid:13)(cid:13)(cid:13) S vq − S hj − ( f B ♯,E − ) (cid:13)(cid:13)(cid:13) L ∞ v ( L ∞ h ) . 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