Well-posedness for the Navier-Stokes equations in critical mixed-norm Lebesgue spaces
aa r X i v : . [ m a t h . A P ] A p r WELL-POSEDNESS FOR THE NAVIER-STOKES EQUATIONS IN CRITICALMIXED-NORM LEBESGUE SPACES
TUOC PHAN
Abstract.
We study the Cauchy problem in n -dimensional space for the system of Navier-Stokesequations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posednessof solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in themixed-norm Lebesgue spaces, both of the initial data and the class of solutions could be singularat certain points or decaying to zero at infinity with different rates in different spatial variabledirections. Some of these singular rates could be very strong and some of the decaying ratescould be significantly slow. Besides other interests, the results of the paper particularly show aninteresting phenomena on the persistence of the anisotropic behavior of the initial data under theevolution. To achieve the goals, fundamental analysis theory such as Young’s inequality, timedecaying of solutions for heat equations, the boundedness of the Helmholtz-Leray projection, andthe boundedness of the Riesz tranfroms are developed in mixed-norm Lebesgue spaces. Thesefundamental analysis results are independently topics of great interests and they are potentiallyuseful in other problems. April 16, 2019 1.
Introduction and main results
This paper establishes local and global well-posedness of the Cauchy problem for Navier-Stokesequations in critical mixed norm Lebesgue spaces. We consider the following initial value problemfor the system of Navier-Stokes equations of incompressible fluid in n -dimensional space u t − ∆ u + ( u · ∇ ) u + ∇ P = 0 in R n × (0 , T ) , div( u ) = 0 in R n × (0 , T ) ,u | t =0 = a on R n , (1.1)where u = ( u , u , · · · , u n ) : R n × (0 , T ) → R n is the unknown velocity of the considered fluid withsome T > n ≥
2. Moreover, P : R n × (0 , T ) → R is the unknown fluid pressure, and a is a given vector field initial data function which is assumed to be divergence-free. Global well-posedness of small solutions in critical mixed-norm Lebesgue spaces and local well-posenedness forlarge solutions in critical mixed-norm Lebesgue spaces are established. Being in the mixed-normLebesgue spaces, both of the initial data and the solutions obtained in the paper could possiblydecay to zero with different rates as | x | → ∞ in different directions. Similarly, they could also besingular at certain points in R n with different rates in different directions of the spatial x -variable.As a result, this paper demonstrates an important phenomenon on the persistence of the anisotropicproperties of the initial data under the evolution of the Navier-Stokes equations.To explain the ideas, motivation and to put our results in perspective, let us review and discussknown results concerning the Cauchy problem for the system of the Navier-Stokes equations (1.1)with possibly irregular initial data in critical spaces. In 1984, in the well-known work [17], T. Katoinitiated the study of (1.1) with initial data belonging to the space L n ( R n ) and he proved theglobal existence and uniqueness of solutions of (1.1) in a subspace of C ([0 , ∞ ) , L n ( R n )) provided Mathematics Subject Classification.
Key words and phrases.
Local well-posedness, global well-posedness, Navier-Stokes equations, mixed-normLebesgue spaces.T. Phan’s research is partially supported by the Simons Foundation, grant the norm k a k L n ( R n ) is sufficiently small. Similarly, local existence and uniqueness of solutions werealso obtained in [17] with initial data a ∈ L n ( R n ). As found in [15, 18, 20, 21, 31], in [5, 6, 28]and [1, Theorem 5.40, p. 234], this kind of global and local existence and uniqueness of solutionscontinues to hold with initial data in homogeneous Morrey spaces M q,q ( R n ) for 1 ≤ q ≤ n , andrespectively in homogeneous Besov spaces ˙ B − np p , ∞ ( R n ) for n ≤ p < ∞ . Here, for 1 ≤ q < ∞ and0 < λ ≤ n , we say that a L q -locally integrable function f : R n → R belongs to the Morrey space M q,λ ( R n ) provided that its norm k f k M q,λ ( R n ) = sup B ρ ( x ) ⊂ R n ( ρ λ − n Z B ρ ( x ) | f ( x ) | q dx ) q < ∞ , where B ρ ( x ) denotes the ball in R n of radius ρ > x ∈ R n . Also, for q ∈ [1 , ∞ ]and α >
0, ˙ B − αq, ∞ ( R n ) denotes the homogeneous Besov space consisting of distributions f whosenorm can be equivalently defined by k f k ˙ B − αq, ∞ ( R n ) ≈ sup t> t α k e ∆ t f ( · ) k L q ( R n ) < ∞ . (1.2)The significant breakthrough is due to the work [19] by H. Koch and D. Tataru in 2001. Inthis work, the authors established the global well-posedness of the Cauchy problem (1.1) for smallinitial data in the borderline BMO − ( R n ) space. Here, the space BMO − ( R n ) can be defined as thespace of all distributional divergences of BMO( R n ) vector fields. On the other hand, it should bealso noted that it has been shown recently by J. Bourgain and N. Pavlovi´c in [2] that the Cauchyproblem (1.1) is ill-posedness in a space even smaller than ˙ B − ∞ , ∞ ( R n ).Now, we would like to note that all of the spaces appear in the mentioned papers are invariantwith respect to the scaling f ( · ) → λf ( λ · ) , λ > f in some space E that we just mentioned, then k f ( · ) k E = k λf ( λ · ) k E , ∀ λ > . In other words, up to now, BMO − ( R n ) is the largest known space that is invariant under thescaling (1.3) on which the Cauchy problem for the system of the Navier-Stokes equations (1.1) isglobally well-posed for small initial data. Interested readers may find in [14, 16] for related results inbounded domains, and in [1, Chapter 5], [24, Chapters 7 - 9] and [32, Chapter 5] for further results,discussion, and more related references.Motivated by the mentioned work, this paper continues the study of the well-posedness of theCauchy problem (1.1) in critical spaces. We plan to refine and extend all the mentioned knownwork to a completely new and interesting direction. In this paper, we particularly focus on theLebesgue space setting. Unlike the mentioned results, we investigate the class of initial data andsolutions for the Cauchy problem (1.1) that possibly decay to zero with different rates as | x | → ∞ in different directions. Some of these rates could be extremely slow. Similarly, the class of initialdata and solutions investigated in this paper could also be singular at certain points in R n withdifferent singularity rates in different spatial directions, and some of which could be very strong. Asthe initial data and the solutions are in the same class of such functions, and besides other interests,the results of this paper particularly demonstrate the persistence of the anisotropic properties of theinitial data under the evolution of the Navier-Stokes equations. To the best of our knowledge, thisphenomenon is even not known for the heat equation. To achieve the goals, we follow the spirit ofKrylov in the work [23] and use mixed norm Lebesgue spaces to capture the features of those kindsof functions. Several important analysis inequalities and estimates in mixed norm Lebesgue spaceswill be also developed in this paper. See also [8, 9, 10, 30] for some other related work and [22] fora survey paper on some interesting features regarding mixed norm Lebesgue spaces. AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 3
For p , p , · · · , p n ∈ [1 , ∞ ), and for a given measurable function f : R n → R , we say that f belongs to the the mixed-norm Lebesgue space L p p ··· p n ( R n ) if its norm k f k L p p ··· pn ( R n ) = · · · (cid:18)Z R | f ( x , x , · · · x n ) | p dx (cid:19) p p dx · · · ! pn − pn − dx n − pnpn − dx n pn < ∞ . Similar definitions can be also formulated if some of the indices in { p , p , · · · , p n } are equal to ∞ .Note that it follows directly from the definition that if p = p = p = · · · = p n , then L p p ··· p n ( R n )is the same as the usual Lebesgue space L p ( R n ).To clearly explain our ideas as well as to understand the importance of the mixed norm Lebesguespaces, let us consider the following example about a function that is decaying to zero at differentrates as | x | → ∞ . Similar examples can be easily produced about different rates of singularity offunctions at some certain points. We consider a bounded measurable function f : R → satisfying | f ( x ) | ≤ N | x | − k | x ′ | k , x = ( x , x ′ ) ∈ R × R , | x | > , (1.4)with some given constant N > k ∈ N which could be very large. It can be seen that f ∈ L p p p ( R ) with p > k and p = p > k . However, if we consider the usual Lebesguespace, then f ∈ L p ( R ) only if p > k , which can be very large when we choose k sufficiently large.In other words, the very fast decaying directions in ( x , x )-variable of the function f is completelyinvisible in the usual unmixed Lebesgue spaces. As a consequence, in the unmixed spaces, the classof functions f as in (1.4) is viewed the same as the class of extremely slow decaying functions with | f ( x ) | ≤ N | x | − k , | x | > . Now, it is surprisingly interesting to note that for given numbers p , p , · · · , p n ∈ [1 , ∞ ], themixed-norm space L p p ··· p n ( R n ) is invariant under the scaling (1.3) if and only if1 p + 1 p + · · · + 1 p n = 1 . (1.5)The class of critical mixed-norm spaces L p p ··· p n ( R n ) such that (1.5) holds is the one we willestablish the well-posedness for solutions of the Navier-Stokes equations (1.1) in this paper. Notethat in the special case when p = p = · · · = p n and (1.5) holds, we have p = p = · · · = p n = n .On the other hand, we also note, as an example, that for the class of functions as in (1.4), it ispossible to choose p > k and p = p > p , p , p )satisfies the condition (1.5). Therefore, in some certain sense, this paper can be considered as anatural but completely non-trivial extension of the work [17].Before stating our results, let us introduce some notations used in the paper. For given p k ∈ [1 , ∞ ), we write P L p p ··· p n ( R n ) the space of all vector fields f ∈ L p p ··· p n ( R n ) such thatdiv( f ) = 0 in R n in the sense of distributions . Also, for given p = ( p , p , · · · , p n ) and q = ( q , q , · · · , q n ) such that p k ∈ (1 , ∞ ) and q k ∈ [ p k , ∞ )for all k = 1 , , . . . , n . Assume that (1.5) holds and1 q + 1 q + · · · + 1 q n = δ ∈ (0 , . (1.6)Then, with given T ∈ (0 , ∞ ], we denote X p,q,T the space consisting of all measurable vector fieldfunctions f : R n × [0 , T ) → R n such that for g ( x, t ) := t (1 − δ ) / f ( x, t ) , ˜ g ( x, t ) := t D x f ( x, t ) , for ( x, t ) ∈ R n × (0 , T ) T. PHAN then g ∈ C([0 , T ) , P L q q ··· q n ( R n )) , ˜ g ∈ C([0 , T ) , P L p p ··· p n ( R n ))and moreover g ( x,
0) = 0 , ˜ g ( x,
0) = 0 and the norm k f k X p,q,T = sup t ∈ (0 ,T ) h k g ( · , t ) k L q q ··· qn ( R n ) + k ˜ g k L p p ··· pn ( R n ) i < ∞ . (1.7)We also denote Y p,T the space consisting of all vector field functions f ∈ C ([0 , T ) , P L p p ··· p n ( R n ))such that t / D x f ∈ C ([0 , T ) , P L p p ··· p n ( R n )) and k f k Y p,T = sup t ∈ (0 ,T ) h k f ( t ) k L p p ··· pn ( R n ) + t / k D x f ( t ) k L p p ··· pn ( R n ) i < ∞ . (1.8)The following theorem on local and global well-posedness of the Cauchy problem (1.1) in thecritical mixed-norm Lebesgue spaces L p p ··· p n ( R n ) is the main result of the paper. Theorem 1.9.
Let p = ( p , p , · · · , p n ) and q = ( q , q , · · · , q n ) . Assume that p k ∈ (2 , ∞ ) and q k ∈ [ p k , ∞ ) for all k = 1 , , · · · , n . Assume also that (1.5) and (1.6) hold. Then, there exist asufficiently small constant λ > and a large number N > depending only on n and p, q suchthat the following assertions hold. (i) For every a ∈ L p p ··· p n ( R n ) n with ∇ · a = 0 , if k a k L p p ··· pn ( R n ) ≤ λ , then the Cauchyproblem (1.1) has unique global time solution u ∈ X p,q, ∞ ∩ Y p,q, ∞ with k u k X p,q, ∞ ≤ N k a k L p p ··· pn ( R n ) and k u k Y p,q, ∞ ≤ N k a k L p p ··· pn ( R n ) . (ii) For every a ∈ L p p ··· p n ( R n ) n with ∇ · a = 0 , there exists T > sufficiently smalldepending on n, p, q and a such that the Cauchy problem (1.1) has unique local time solution u ∈ X p,q,T ∩ Y p,q,T with k u k X p,q,T ≤ N k a k L p p ··· pn ( R n ) and k u k Y p,q,T ≤ N h k a k L p p ··· pn ( R n ) + k a k L p p ··· pn ( R n ) i . To the best of our knowledge, this is the first time that the kinds of solutions of Navier-Stokesequations in critical mixed-norm Lebesgue spaces are discovered. As demonstrated in the examplein (1.4) and the discussion after (1.5), it is possible to choose some of p , p , · · · , p n to be verylarge numbers so that the given numbers ( p , p , · · · , p n ) still satisfy the condition (1.5). Due tothis reason, in some directions, the class of the initial data and the solutions in Theorem 1.9 coulddecay significantly slow. Similarly, some of the singularity rates in some spatial directions could bevery strong. Hence, our class of solutions may not belong to L n ( R n ) nor L ( R n ), and the solutionsobtained in Theorem 1.9 may not belong to the classes of solutions found in the papers [17, 25, 26].Observe also that if p = p = · · · = p n and (1.5) holds, then p = p = · · · = p n = n . In this sense,this paper can be considered as a natural, but completely non-trivial extension of the work [17].Now, we summarize the above discussion with the following remarks regarding Theorem 1.9. Remark . The following interesting points are worth highlighting.(i) Under the condition (1.5), the initial data and the solutions obtained in Theorem 1.9 maydecay to zero very slow as | x | → ∞ . Similarly, they could also be strongly singular in somespatial directions. Therefore, the solutions obtained in Theorem 1.9 may not be in L n ( R n )nor L ( R n ). Consequently, these solutions may not be the same as the ones obtained in[17, 25, 26].(ii) Let p ∈ (max { p , p , · · · , p n , n } , ∞ ), where p , p , · · · , p n are as in Theorem 1.9. Then,if a ∈ L p p ··· p n ( R n ), it follows from the characterization of Besov spaces with negativeregularity (see Remark 2.21 below) that a ∈ ˙ B − np p , ∞ ( R n ) ⊂ BMO − ( R n ). In view of thisand the results obtained in [5, 6, 19, 28] and [1, Theorem 5.40, p. 234], Theorem 1.9 can be AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 5 seen as a refinement of these results regarding the persistence of the anisotropic propertiesof the initial data under the evolution of the Navier-Stokes equations. See also [3, Section3.3] for some different but related results.To prove Theorem 1.9, we follow the approach developed in [11, 12, 17] and in [14, 16, 19, 27].To implement the method, several important and fundamental analysis estimates in mixed normLebesgue spaces are developed. In Section 2, we develop and prove a version of Young’s inequalityin mixed norm Lebesgue spaces. We then use Young’s inequality in mixed norm Lebesgue spaces toestablish the time decaying estimates for solutions of heat equations in mixed norm Lebesgue spaces.The boundedness of the Riesz transform and the boundedness of the Helmholtz-Leray projection inmixed-norm Lebesgue spaces are also established and proved in Section 2. Clearly, these analysisinequalities and estimates are independently topics of great interests and they can be useful in manyother problems. To the best of our knowledge, this paper is the first time that those mixed-normanalysis estimates are developed. Therefore, besides the great interest and contribution of our studyin the Navier-Stokes equations, the contribution in real and harmonic analysis theory of this paperis also very significant. The paper concludes with Section 3 which provides the proof of Theorem1.9. 2.
Preliminaries on Analysis inequalities in mixed-norm Lebesgue spaces
This section gives some main ingredients for the proof of the main theorems in the paper. Inparticular, we develop Young’s inequality in mixed-norm Lebesgue spaces, time decaying rate esti-mates for solutions of the Cauchy problem for the heat equation in mixed-norm Lebesgue spaces,and Helmholtz-Leray projection in mixed-norm Lebesgue spaces. These results are not only new,fundamental, but they are topics of independent interests and could be useful for many other pur-poses. For your convenience, we recall that for p , p , · · · p n ∈ [1 , ∞ ), and for a measurable function f : R n → R , we say that f is in the mixed-norm Lebesgue space L p p ··· p n ( R n ) if its norm k f k L p p ··· pn ( R n ) = · · · (cid:18)Z R | f ( x , x , · · · x n ) | p dx (cid:19) p p dx · · · ! pn − pn − dx n − pnpn − dx n pn < ∞ . Similar definitions can be also formulated if some of the indices { p , p , · · · , p n } are equal to ∞ .As we already discussed, the significant role of the the mixed-norm Lebesgue space L p p ··· p n ( R n )is that it captures very well the functions that are singular at certain points or decaying to zero as | x | → ∞ with different rates in different x -directions.2.1. Young’s inequality in mixed norm Lebesgue spaces.
This subsection establishes thefollowing new result on Young’s inequality in mixed norm Lebesgue spaces. The result will beuseful in the study of heat equations in mixed norm Lebesgue spaces. Our theorem can be statedas in the following.
Theorem 2.1 (Young’s inequality in mixed norm) . Let p k , r k and q k be given numbers in [1 , ∞ ] that satisfy p k + 1 = 1 q k + 1 r k , k = 1 , , · · · , n. Then k f ∗ g k L p p ··· pn ( R n ) ≤ k f k L q q ··· qn ( R n ) k g k L r r ··· rn ( R n ) (2.2) for every f ∈ L q q ··· q n ( R n ) and g ∈ L r r ··· r n ( R n ) . T. PHAN
Proof.
We use induction on n . Observe that when n = 1, the inequality (2.2) is the classical Young’sinequality. We now assume that the inequality holds true in ( n − n -dimension with n ≥
2. Let us denote p ′ k , q ′ k , r ′ k the H¨older’s conjugates of p k , q k , r k respectively. Bythe assumption, we see that1 r ′ k + 1 p k + 1 q ′ k = 1 , r k p k + r k q ′ k = 1 , and q k r ′ k + q k p k = 1 , k = 1 , , · · · , n. (2.3)We split the proof into three different cases. Case I . We assume that p < ∞ and q < ∞ . In this case, we also see that r < ∞ . For x, y ∈ R n , we write x = ( x , x ′ ) ∈ R × R n − and y = ( y , y ′ ) ∈ R × R n − . As q < ∞ , by using thelast two identities in (2.3), we have | ( f ∗ g )( x ) | ≤ Z R n | f ( y ) || g ( x − y ) | dy = Z R n − (cid:20)Z R | f ( y , y ′ ) || g ( x − y , x ′ − y ′ ) | dy (cid:21) dy ′ = Z R n − (cid:20)Z R | f ( y , y ′ ) | q r ′ (cid:16) | f ( y , y ′ ) | q p | g ( x − y , x ′ − y ′ ) | r p (cid:17) | g ( x − y , x ′ − y ′ ) | r q ′ dy (cid:21) dy ′ , Note that in the above inequality also holds when r ′ = ∞ with r ′ = 0. Now, by using the firstidentity in (2.3) and the H¨older’s inequality with respect to the integration in y -variable, we obtain | ( f ∗ g )( x ) |≤ Z R n − " | f ( y ′ ) | q r ′ | g ( x ′ − y ′ ) | r q ′ (cid:18)Z R | f ( y , y ′ ) | q | g ( x − y , x ′ − y ′ ) | r dy (cid:19) p dy ′ , (2.4)where we denote f ( y ′ ) = (cid:18)Z R | f ( y , y ′ ) | q dy (cid:19) q and g ( y ′ ) = (cid:18)Z R | g ( y , y ′ ) | r dy (cid:19) r , for a.e. y ′ ∈ R n − . (2.5)As p < ∞ , it follows from (2.4) that G ( x ′ ) := (cid:18)Z R | ( f ∗ g )( x , x ′ ) | p dx (cid:19) p ≤ ( Z R n − " | f ( y ′ ) | q r ′ | g ( x ′ − y ′ ) | r q ′ (cid:18)Z R | f ( y , y ′ ) | q | g ( x − y , x ′ − y ′ ) | r dy (cid:19) p dy ′ ! p dx ) p . From this, and by using the Minskowski’s inequality, we see that (cid:18)Z R | ( f ∗ g )( x , x ′ ) | p dx (cid:19) p ≤ Z R n − (cid:18) | f ( y ′ ) | q r ′ | g ( x ′ − y ′ ) | r q ′ I ( x ′ , y ′ ) (cid:19) dy ′ , where I ( x ′ , y ′ ) = (cid:18)Z R (cid:18)Z R | f ( y , y ′ ) | q | g ( x − y , x ′ − y ′ ) | r dy (cid:19) dx (cid:19) p . AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 7
By the Fubini’s theorem, we see that I ( x ′ , y ′ ) = (cid:18)Z R (cid:18)Z R | f ( y , y ′ ) | q | g ( x − y , x ′ − y ′ ) | r dx (cid:19) dy (cid:19) p = k g ( · , x ′ − y ′ ) k r p L r ( R ) k f ( · , y ′ ) k q p L q ( R ) = | g ( x ′ − y ′ ) | r p | f ( y ′ ) | q p . Therefore, G ( x ′ ) ≤ Z R n − (cid:18) | f ( y ′ ) | q r ′ + q p | g ( x ′ − y ′ ) | r q ′ + r p (cid:19) dy ′ . From this, and by using the last two identities in (2.3), we obtain G ( x ′ ) = ( f ∗ g )( x ′ ) . Then, by induction hypothesis, we see that k f ∗ g k L p p ··· pn ( R n ) = k G k L p p ··· pn ( R n − ) ≤ k g k L r r ··· rn ( R n − ) k f k L q q ··· qn ( R n − ) = k g k L r r ··· rn ( R n − ) k f k L q q ··· qn ( R n − ) . This proves the desired estimate for the case q < ∞ and p < ∞ . Case II . We assume that p = ∞ and q < ∞ . In this case, we observe that r ′ = q ∈ [1 , ∞ ). Inthis case, we write | f ∗ g ( x ) | ≤ Z R n − (cid:20)Z R | f ( y , y ′ ) || g ( x − y , x ′ − y ′ ) | dy (cid:21) dy ′ . If r < ∞ , as q + r = 1, we apply the H¨older’s inequality for the integration with respect to y toobtain | ( f ∗ g )( x ) | ≤ Z R n − f ( y ′ ) g ( x ′ − y ′ ) dy ′ = ( f ∗ g )( x ′ ) , for a.e. x = ( x , x ′ ) ∈ R × R n − , where f , g are defined as in (2.5). Observe also that the similar estimate can be also done when r = ∞ . From this, the desired inequality follows by the induction hypothesis as in Case I . Theproof is of this case therefore completed.
Case III . We are left to consider the case that q = ∞ . In this case, it follows that p = ∞ and r = 1. By defining G ( x ′ ) = sup x ∈ R | ( f ∗ g )( x , x ′ ) | , f ( x ′ ) = sup x ∈ R | f ( x , x ′ ) | , g ( x ′ ) = Z R | g ( x , x ′ ) | dx we see that G ( x ′ ) ≤ ( f ∗ g )( x ′ ) . Then, we also obtain the same desired estimate. The proof is then completed. (cid:4)
Remark . Theorem 2.1 gives the classical unmixed-norm Young’s inequality when p = p = · · · = p n and q = q = · · · = q n . T. PHAN
Heat equations in mixed norm Lebesgue spaces.
This subsection develops estimates oftime decaying rates for solutions of heat equations in mixed norm Lebesgue spaces. We considerthe Cauchy problem for the heat equation (cid:26) u t − ∆ u = 0 in R n × (0 , ∞ ) ,u | t =0 = u on R n . (2.7)Under some suitable conditions on the initial data u , it is well known that u ( x, t ) = e ∆ t u ( x ) = ( G t ∗ u )( x ) , ( x, t ) ∈ R n × (0 , ∞ ) , (2.8)is a solution of (2.7), where G t ( x ) = 1(4 πt ) n e − | x | t , ( x, t ) ∈ R n × (0 , ∞ ) . The following new and fundamental result on the time decaying rates of the solutions (2.8) of theheat equation (2.7) in mixed norm Lebesgue spaces is the main result of this subsection.
Theorem 2.9 (Time decaying of solutions for heat equation in mixed-norm) . Let ≤ q k ≤ p k ≤ ∞ .There exists a positive constant N depending only on p , p , · · · , p n , q , q , · · · , q n such that for everysolution u ( x, t ) = e ∆ t u ( x ) defined in (2.8) of the Cauchy problem (2.7) with u ∈ L q q ··· q ( R n ) ,then for t > k u ( · , t ) k L p p ··· pn ( R n ) ≤ N t − P nk =1 ( qk − pk ) k u k L q q ··· qn ( R n ) . (2.10) Moreover, for every l = 1 , , · · · and for t > k D lx u ( · , t ) k L p p ··· pn ( R n ) ≤ N t − l − P nk =1 ( qk − pk ) k u k L q q ··· qn ( R n ) , (2.11) where D lx denotes the l th -derivative in x -variable.Proof. We begin with the proof of (2.10). For each k = 1 , , · · · , n , by the assumption that q k ≤ p k ,we can find r k ∈ [1 , ∞ ] such that 1 p k + 1 = 1 r k + 1 q k . (2.12)Then, because u ( x, t ) = ( G t ∗ u )( x ), we can use the mixed-norm Young’s inequality in Theorem2.1 to see that k u ( · , t ) k L p p ··· pn ( R n ) ≤ k G t ( · ) k L r r ··· rn ( R n ) k u k L q q ··· qn ( R n ) . (2.13)We now note that we can write G t as G t ( x ) = g t ( x ) g t ( x ) · · · g t ( x n ) for x = ( x , x , · · · , x n ) ∈ R n , t > g t is the heat kernel in R : g t ( s ) = 1 √ πt e − s t , s ∈ R , t > . (2.14)Note also that for each r ∈ [1 , ∞ ) we have k g t ( · ) k L r ( R ) = 1 √ πt (cid:18)Z R e − rs t (cid:19) r = 1 √ πt r tr ! r (cid:18)Z R e − z dz (cid:19) r = N ( r ) t − (1 − r ) . On the other hand, we also see that k g t ( · ) k L ∞ ( R ) ≤ N t − , t > . Therefore, we conclude that for every r ∈ [1 , ∞ ] k g t ( · ) k L ∞ ( R ) ≤ N ( r ) t − (1 − r ) , t > . (2.15) AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 9
From this, we infer that k G t ( · ) k L r r ··· rn ( R n ) = k g t ( · ) k L r ( R ) k g t ( · ) k L r ( R ) · · · k g t ( · ) k L rn ( R ) = N ( r , r , · · · , r n ) t − ( n − P ni =1 1 ri ) = N ( p , p , · · · p n , q , q , · · · q n ) t − P ni =1 ( qi − pi ) , t > , where we have used (2.12) in the last estimate. This last estimate together with (2.13) implies(2.10).Next, we prove (2.11). We only demonstrate the proof of (2.11) with l = 1 as the general casecan be done in a similar way. We observe that for each i = 1 , , · · · , nD x i u ( x, t ) = ([ D x i G t ] ∗ u )( x ) , ( x, t ) ∈ R n × (0 , ∞ ) . Then, by the mixed norm Young’s inequality in Theorem 2.1, we have k D x i u ( · , t ) k L p p ··· pn ( R n ) ≤ k D x i G t ( · ) k L r r ··· rn ( R n ) k u ( · ) k L q q ··· qn ( R n ) . (2.16)It remains to estimate the mixed norm k D x i G t ( · ) k L r r · rn ( R n ) . Note that D x i G t ( x ) = − πt ) n x i t e − | x | t , x = ( x , x , · · · , x n ) ∈ R n t > . Consequently, D x i G t ( x ) = h t ( x i ) Y k = i g t ( x k ) , x = ( x , x , · · · , x n ) ∈ R n t > , where g t is defined as in (2.14) and h t ( s ) = − √ πt s t e − s t , s ∈ R and t > . We observe that | h t ( s ) | ≤ Nt | s |√ t e − s t = Nt | z | e −| z | , where z = s √ t . As | z | e − z is a bounded function for z ∈ R , we conclude that k h t k L ∞ ( R ) ≤ Nt , t > . On the other hand, if r i ∈ [1 , ∞ ), we see that k h t k L ri ( R ) = N ( r i ) t (cid:20)Z R (cid:16) √ r i | s |√ t (cid:17) r i e − ris t ds (cid:21) ri = N ( r i ) t − − (1 − ri ) (cid:20)Z R | z | r i e − z dz (cid:21) ri = N ( r i ) t − − (1 − ri ) . Therefore, we conclude that for every r i ∈ [1 , ∞ ] k h t k L ri ( R ) = N ( r i ) t − − (1 − ri ) , t > . From this estimate and (2.15), we see that k D x i G t k L r r ··· rn ( R n ) = k h t k L ri ( R ) Y k = i k g t k L rk ( R ) ≤ N ( r , r , · · · , r n ) t − − ( n − P nk =1 1 rk ) . From this, and by using (2.12), we infer that k D x i G t k L r r ··· rn ( R n ) ≤ N ( r , r , · · · , r n ) t − − P nk =1 ( qk − pk ) . This last estimate and (2.16) imply (2.11) with l = 1. The proof of the lemma is complete. (cid:4) Next, we introduce and prove the following simple lemma on the continuity property of thesolutions of the heat equation (2.7) in mixed norm spaces. The result will be useful in the paper.
Lemma 2.17.
For each k = 1 , , · · · , n , let p k ∈ [1 , ∞ ) . Assume that u ∈ L p p ··· p n ( R n ) . Let u ( x, t ) = e ∆ t u be the solution of the heat equation (2.7) defined in (2.8) . Then, u : C ([0 , ∞ ) , L p p ··· p n ( R n )) and lim t → + k u ( · , t ) − u k L p p ··· pn ( R n ) = 0 . (2.18) Proof.
We only need to prove (2.18), as the proof of the continuity of u at t > ǫ >
0, by using the truncation and a multiplication by a suitable cut-off function, wecan find a bounded compactly support function ˜ u such that k u − ˜ u k L p p ··· pn ( R n ) ≤ ǫ N , where N = N ( n, p , p , · · · , p n ) > k e ∆ t ( u − ˜ u ) k L p p ··· pn ( R n ) ≤ N k u − ˜ u k L p p ··· pn ( R n ) ≤ ǫ . From the previous two estimates, we see that k e ∆ t ( u − ˜ u ) − ( u − ˜ u ) k L p p ··· pn ( R n ) ≤ ǫ ǫ N ≤ ǫ . (2.19)Our next goal is to show that lim t → + k e ∆ t ˜ u − ˜ u k L p p ··· pn ( R n ) = 0 . Take p > max { p , p , · · · , p n } and choose the numbers q k ∈ ( p k , ∞ ) such that1 q k = 1 p k − p , k = 1 , , · · · , n. Then, by applying the H¨older’s inequality repeatedly for each integration with respect to eachvariable x k , we see that k e ∆ t ˜ u − ˜ u k L p p ··· pn ( R n ) ≤ k e ∆ t ˜ u − ˜ u k L p ( R n ) k e ∆ t ˜ u − ˜ u k L q q ··· qn ( R n ) ≤ k e ∆ t ˜ u − ˜ u k L p ( R n ) h k e ∆ t ˜ u k L q q ··· qn ( R n ) + k ˜ u k L q q ··· qn ( R n ) i ≤ N k e ∆ t ˜ u − ˜ u k L p ( R n ) k ˜ u k L q q ··· qn ( R n ) . Observe that as ˜ u is bounded and compactly supported, k ˜ u k L q q ··· qn ( R n ) < ∞ . Therefore, k e ∆ t ˜ u − ˜ u k L p p ··· pn ( R n ) ≤ ˜ N k e ∆ t ˜ u − ˜ u k L p ( R n ) → t → + , where in the last assertion, we used the classical result of the continuity of the heat flow in L p ( R n )and the fact that ˜ u ∈ L p ( R n ). From this and (2.19), we conclude that there is δ = δ ( ǫ ) > k e ∆ t u − u k L p p ··· pn ( R n ) ≤ ǫ, ∀ t ∈ (0 , δ ) . This proves (2.18) as desired. (cid:4)
AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 11
Remark . It is interesting to note that Theorem 2.9 shows the persistence of the anisotropicproperties of the initial data under the evolution of the heat equation. This phenomenon seems tobe new. Theorem 2.9 and Lemma 2.17 recover the classical results when p = p = · · · = p n and q = q = · · · = q n . Remark . For given numbers p , p , · · · , p n ∈ (1 , ∞ ) that satisfy (1.5), if u ∈ L p p ··· p n ( R n ),by Theorem 2.9, we see that t (1 − np ) k e ∆ t u k L p ( R n ) ≤ N k u k L p p ··· pn ( R n ) , for p ∈ (max { p , p , · · · , p n , n } , ∞ ) and for N = N ( n, p , p , p , · · · , p n ). Then, it follows from thecharacterization of Besov spaces with negative regularity (see [1, Theorem 2.34, p. 72], or [24, eqn(8.6), p. 177], and also [4, 5]) that u ∈ ˙ B − np p , ∞ ( R n ) with its norm is defined as in (1.2) k u k ˙ B − np p , ∞ ( R n ) ≈ sup t> t (1 − np ) k e ∆ t u k L p ( R n ) < ∞ . In particular, it follows from this and [19, eqn (23)] that u ∈ BMO − ( R n ).2.3. Helmholtz-Leray projection in mixed-norm Lebesgue spaces.
Let P = Id − ∇ ∆ − ∇· be the Helmholtz-Leray projection onto the divergence-free vector fields. This subsection provesthat k P ( f ) k L p p ··· pn ( R n ) ≤ N k f k L p p ··· pn ( R n ) , for every f ∈ L p p ··· p n ( R n ) n and for p , p , · · · , p n ∈ (1 , ∞ ). This estimate is an important in-gredient in our paper. To achieve it, we need to recall the following definition of Muckenhoupt A q ( R n )-class of weights, which is needed for the proof of Theorem 2.23 below. For each q ∈ (1 , ∞ ),a non-negative, locally integrable function ω : R n → R is said to be in the Muckenhoupt A q ( R n )-classof weights if[ ω ] A q := sup R> ,x ∈ R n | B R ( x ) | Z B R ( x ) ω ( x ) dx ! B R ( x ) Z B R ( x ) ω ( x ) − q − dx ! q − < ∞ , where B R ( x ) denotes the ball in R n of radius R centered at x ∈ R n . In the following, for eachgiven p ∈ [1 , ∞ ) and each given weight ω : R n → R , a measurable function f : R n → R is said to bein the weighted Lebesgue space L p ( R n , ω ) if its norm k f k L p ( R n ,ω ) = (cid:18)Z R n | f ( x ) | p ω ( x ) dx (cid:19) p < ∞ . We also recall the following amazing result from [22, Theorem 6.2], which is a beautiful applicationof the Rubio De Francia extrapolation theory (see [7] for instance).
Theorem 2.22.
Let p k ∈ (1 , ∞ ) for all k = 1 , , · · · , n . Then, there exists a constant K = K ( n, p , p , · · · , p n ) ≥ such that the following holds true. For a pair of given measurable functions f, g : R n → R such that if k f k L p ( R n ,ω ) ≤ k g k L p ( R n ,ω ) for every ω ∈ A p with [ ω ] A p ≤ K , then we have k f k L p p ··· pn ( R n ) ≤ n k g k L p p ...pn ( R n ) . Now, we begin with the following important result on the boundedness of the Riesz transform inmixed-norm Lebesgue spaces. Interested readers may find [8, Corollary 2.7] and [30, Lemma 2.1]for other interesting related results in mixed-norm spaces.
Theorem 2.23.
For any j = 1 , , · · · , n and any p , p , · · · , p n ∈ (1 , ∞ ) , there exists a positiveconstant N = N ( p , p , · · · , p n , n ) such that k R j ( f ) k L p p ··· pn ( R n ) ≤ N k f k L p p ··· pn ( R n ) for every f ∈ L p p ··· p n ( R n ) , where R j is the j th -Riesz transform defined by R j ( f ) = ∂ x j ( − ∆) − f .Proof. We plan to apply Theorem 2.22. For given p , p , · · · , p n ∈ (1 , ∞ ), let K be as in Theorem2.22. By using the truncation and a multiplication with suitable cut-off functions, we can approx-imate f ∈ L p p ··· p n ( R n ) by a sequence of bounded compactly supported functions. Therefore, wemay assume that f is bounded and compactly supported in R n . Without loss of generality, we canalso assume that p = min { p , p , · · · , p n } . Under these assumptions, we see that f ∈ L p ( R n , ω )for every weight ω ∈ A p . Then, since p ∈ (1 , ∞ ), by the classical Calder´on-Zygmund theory (see[7, 13] for instance), there exists a constant N = N ( p , n, K ) such that k R j ( f ) k L p ( R n ,ω ) ≤ N k f k L p ( R n ,ω ) , (2.24)for every ω ∈ A p with [ ω ] A p ≤ K . From (2.24) and Theorem 2.22, we infer that k R j ( f ) k L p p ··· pn ( R n ) ≤ n N k f k L p p ··· pn ( R n ) . This is the desired estimate and the proof is therefore completed. (cid:4)
The following consequence of Theorem 2.23 gives the boundedness of the Helmholtz-Leray pro-jection in mixed norm Lebesgue spaces, which is an important ingredient in the paper.
Corollary 2.25.
Let P = Id − ∇ ∆ − ∇· be the Helmholtz-Leray projection onto the divergence-freevector fields. Let p , p , · · · , p n ∈ (1 , ∞ ) . Then, one has k P ( f ) k L p p ··· pn ( R n ) ≤ N k f k L p p ··· pn ( R n ) , for every f ∈ L p p ··· p n ( R n ) n , where N = N ( p , p , · · · , p n , n ) is a positive constant.Proof. Note that with f = ( f , f , · · · f n ) ∈ L p p ··· p n ( R n ) n , we have P ( f ) = ( P ( f ) , P ( f ) , · · · , P ( f ) n )with P ( f ) k = f k + R k n X j =1 R j f j , k = 1 , · · · , n, where R j is the j th -Riesz transform. Therefore, it follows from Theorem 2.23 that k P ( f ) k L p p ··· pn ( R n ) ≤ N k f k L p p ··· pn ( R n ) which is our desired estimate. (cid:4) Navier-Stokes equations in critical mixed-norm Lebesgue spaces
This section provides the proof of Theorem 1.9. We follow the approach introduced in [11, 12, 17]and in [19, 27]. Recall that P denotes the Helmholtz-Leray projection which is defined in Corollary2.25. By applying P on the system (1.1), we see that the system (1.1) is recasted in the abstractway as the following (cid:26) u t + A u + F ( u, u ) = 0 in R n × (0 , ∞ ) ,u ( · ,
0) = a ( · ) on R n , (3.1)where A = − P ∆ = − ∆ P and F ( u, v ) = P (( u · ∇ ) v ) . (3.2)By the Duhamel’s principle, the system (3.1) is then converted to the following integral equation u = u + G ( u, u ) , (3.3) AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 13 where u ( t ) = e −A t a , and G ( u, v )( t ) = − Z t e − ( t − s ) A F ( u ( s ) , v ( s )) ds. (3.4)To proceed, we need several estimates. We begin with the following lemma on the time decayingproperties for the semi-group e −A t in mixed norm Lebesgue spaces. Lemma 3.5.
For each k = 1 , , · · · , n , let < p k ≤ q k < ∞ be given numbers. Also, let σ ≥ bedefined by σ = n X k =1 h p k − q k i . (i) There exists a number N depending only on n, p , p , · · · , p n and q , q , · · · , q n such that k e −A t P f k L q q ··· qn ( R n ) ≤ N t − σ k f k L p p ··· pn ( R n ) , k D x e −A t P f k L q q ··· qn ( R n ) ≤ N t − (1+ σ ) k f k L p p ··· pn ( R n ) , (3.6) for every f ∈ L p p ··· p n ( R n ) n . (ii) For each f ∈ L p p ··· p n ( R n ) n , the following assertions hold lim t → + t σ k e −A t P f k L q q ··· qn ( R n ) = 0 if σ > and also lim t → + k [ e −A t P f ] − P f k L p p ··· pn ( R n ) = 0 , and lim t → + t − (1+ σ ) k D x e −A t P f k L q q ··· qn ( R n ) = 0 . (3.7) Proof.
We begin with the proof of (i). As A = − P ∆ = − ∆ P , we see that A = − ∆ when actingon the class of divergenge free vector fields. Therefore, e −A t P = e ∆ t P . Then, by using the decayestimate for the heat equation in mixed norm developed in Theorem 2.9, we see that k e −A t P f k L q q ··· qn ( R n ) ≤ N t − σ k P ( f ) k L p p ··· pn ( R n ) . On the other hand, from Corollary 2.25, we see that the Helmholtz-Leray projection P : L p p ··· p n ( R n ) n → L p p ··· p n ( R n ) n is bounded. From this and the last estimate, we obtain the first estimate in (3.6). The secondestimate in (3.6) can be proved in the same way.Next, we prove (ii). We assume that σ > f is bounded and compactly supported if needed. Let ǫ >
0. Then, by usingapproximation, we can find g ∈ L p p ··· p n ( R n ) n ∩ L q q ··· q n ( R n ) n such that k f − g k L p p ··· pn ( R n ) ≤ ǫ N where N > t σ k e −A t P ( f − g ) k L q q ··· qn ( R n ) ≤ N k f − g k L p p ··· pn ( R n ) ≤ ǫ . On the other hand, using the first assertion in (i) again, we also obtain t σ k e −A t P g k L q q ··· qn ( R n ) ≤ N t σ k g k L q q ··· qn ( R n ) → t → + . Now, combine the last two estimates, we infer that there is small number δ = δ ( ǫ ) > t σ k e −A t P f k L q q ··· qn ( R n ) ≤ ǫ, ∀ t ∈ (0 , δ ) . This implies that lim t → + t σ k e −A t P f k L q q ··· qn ( R n ) = 0 , and the first assertion in (3.7) is proved. Observe also that the last assertion in (ii) can be done in asimilar way. Meanwhile, the second assertion of (3.7) is due to the continuity of the heat semi-group in Lemma 2.17 and the continuity of the Helmholtz-Leray in the mixed norm L p p ··· p n ( R n ) n as fromCorollary 2.25. The proof of the lemma is therefore completed. (cid:4) Our next lemma gives some important estimates in mixed norm for the bilinear term G ( u, v )defined in (3.4). Lemma 3.8.
Let p k ∈ (1 , ∞ ) and α k , β k , γ k ∈ (0 , be given numbers satisfying γ k ≤ α k + β k < p k , k = 1 , , · · · , n. Let α = n X k =1 α k p k , β = n X k =1 β k p k , and γ = n X k =1 γ k p k . Then, k G ( u, v )( t ) k L p γ p γ ··· pnγn ( R n ) ≤ N Z t ( t − s ) − α + β − γ k u ( s ) k L p α p α ··· pnαn ( R n ) k D x v ( s ) k L p β p β ··· pnβn ( R n ) ds, k D x G ( u, v )( t ) k L p γ p γ ··· pnγn ( R n ) ≤ N Z t ( t − s ) − α + β − γ k u ( s ) k L p α p α ··· pnαn ( R n ) k D x v ( s ) k L p β p β ··· pnβn ( R n ) ds, where N is a positive number depending only on n , p k , α l , β k , α k for k = 1 , , · · · , n .Proof. We only prove the first assertion in the lemma as the proof of the second one can be donesimilarly. By applying the first estimate in (3.6), we see that k G ( u, v )( t ) k L p γ p γ ··· pnγn ( R n ) ≤ N Z t ( t − s ) − α + β − γ k F ( u ( s ) , v ( s )) k L p α β p α β ··· pnαn + βn ( R n ) ds, where the bilinear function F is defined in (3.2). From this and the boundedness of the Helmholtz-Leray projection P as stated in Corollary 2.25, we see that k G ( u, v )( t ) k L p γ p γ ··· pnγn ( R n ) ≤ N Z t ( t − s ) − α + β − γ k ( u · ∇ ) v k L p α β p α β ··· pnαn + βn ( R n ) ds. Then, as α k + β k p k = α k p k + β k p k , for all k = 1 , , · · · , n we can repeatedly apply the H¨older’s inequality for each integration with respect to each variable x k to find that k ( u · ∇ ) v k L p α β p α β ··· pnαn + βn ( R n ) ≤ k u ( s ) k L p α p α ··· pnαn ( R n ) k D x v ( s ) k L p β p β ··· pnβn ( R n ) . The desired estimate then follows and the proof is complete. (cid:4)
To prove Theorem 1.9, our goal is to show that the abstract equation (3.3) has unique fixed pointsin suitable spaces. For this purpose, let us recall the following abstract lemma which is useful inthe study of initial value problem for Navier-Stokes equations, see [29, Lemma 3.1] and also [27].
Lemma 3.9.
Let X be a Banach space with norm k·k X . Let G : X × X → X be a bilinear mapsuch that there is N > so that k G ( u, v ) k X ≤ N k u k X k v k X , ∀ u, v ∈ X. Then, for every u ∈ X with N k u k X < , the equation u = u + G ( u, u ) AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 15 has unique solution u ∈ X with k u k X ≤ k u k X . We are now ready to prove Theorem 1.9.
Proof of Theorem 1.9.
Let p = ( p , p , · · · , p n ) , q = ( q , q , · · · q n ) with p k ∈ (2 , ∞ ) , q k ∈ [ p k , ∞ ) for k = 1 , , · · · , n . Assume that (1.5) and (1.6) hold. Let a ∈ L p p ··· p n ( R n ) n with ∇ · a = 0 andrecall that δ = 1 q + 1 q + · · · + 1 q n ∈ (0 , . (3.10)We now prove (i). Recall the definitions of X p,q, ∞ and Y p,q, ∞ in (1.7) and (1.8). We plan to provethe existence of solution u ∈ X p,q, ∞ of (3.3), and then prove that the solution u ∈ Y p,q, ∞ . Our goalis to apply Lemma 3.9 to obtain the existence and uniqueness of solution of (3.3) in X p,q, ∞ . To thisend, we begin with the proof that u ∈ X p,q, ∞ . From (i) of Lemma 3.5 and the definition of u in(3.4), we have k u ( t ) k L q q ··· qn ( R n ) ≤ N t − − δ k a k L p p ··· pn ( R n ) and k D x u ( t ) k L p p ··· pn ( R n ) ≤ N t − k a k L p p ··· pn ( R n ) , ∀ t > , where N > n, p and q . Moreover, it follows from (ii)of Lemma 3.5 that t (1 − δ ) / e −A t P is uniformly bounded from L p p ··· p n ( R n ) n to P L q q ··· q n ( R n ) andtends to zero as t → + , we see that t (1 − δ ) / u vanishes as t = 0. Similarly, as t / D x e −A t P isuniformly bounded from L p p ··· p n ( R n ) n to P L p p ··· p n ( R n ) n and tends to zero as t → + , we alsohave t / D x u equals to zero as t → + . In conclusion, we have shown that u ∈ X p,q, ∞ and k u k X p,q, ∞ ≤ N k a k L p p ··· pn ( R n ) . (3.11)It now remains to prove that the bilinear form G : X p,q, ∞ × X p,q, ∞ → X p,q, ∞ is bounded. By (3.10)and (1.5), we apply the first assertion in Lemma 3.8 with β k = 1 and γ k = α k = p k q k ∈ (0 ,
1] to findthat k G ( u, v )( t ) k L q q ··· qn ( R n ) ≤ N Z t ( t − s ) − k u ( s ) k L q q ··· qn ( R n ) k D x v ( s ) k L p p ··· pn ( R n ) ds ≤ N k u k X p,q, ∞ k v k X p,q, ∞ Z t ( t − s ) − s − δ ds. To control the integration in the last estimate, we split it into two time intervals (0 , t/
2) and ( t/ , t ).We then obtain k G ( u, v )( t ) k L q q ··· qn ( R n ) ≤ N k u k X p,q, ∞ k v k X p,q, ∞ "Z t/ ( t − s ) − s − δ ds + Z t/ t/ ( t − s ) − s − δ ds ≤ N k u k X p,q, ∞ k v k X p,q, ∞ " t − Z t/ s − δ ds + t − δ Z t/ t/ ( t − s ) − ds ≤ N t − − δ k u k X p,q, ∞ k v k X p,q, ∞ . Similarly, By using (3.10) and (1.5), and applying the second assertion in Lemma 3.8 with γ k = 1, β k = 1 and α k = p k q k ∈ (0 , k D x G ( u, v )( t ) k L p p ··· pn ( R n ) ≤ N Z t ( t − s ) − δ k u ( s ) k L q q ··· qn ( R n ) k D x u ( s ) k L p p ··· pn ( R n ) ds ≤ N k u k X p,q, ∞ k v k X p,q, ∞ Z t ( t − s ) − δ s − δ ds = N k u k X p,q, ∞ k v k X p,q, ∞ "Z t/ ( t − s ) − δ s − δ ds + Z tt/ ( t − s ) − δ s − δ ds = N k u k X p,q, ∞ k v k X p,q, ∞ " t − δ Z t/ s − δ ds + t − δ Z tt/ ( t − s ) − δ ds ≤ N t − / k u k X p,q, ∞ k v k X p,q, ∞ . (3.12)From the last two estimates and the definition of G ( u, v ) and Lemma 3.5, it follows that t (1 − δ ) / G ( u, v ) :[0 , ∞ ) → P L q q ··· q n ( R n ) is continuous and vanishes at t = 0. Similarly, we can also prove that t / D x G ( u, v ) : [0 , ∞ ) → P L p p ··· p n ( R n ) is continuous and vanishes at t = 0. Therefore, weconclude that G ( u, v ) ∈ X p,q, ∞ and k G ( u, v ) k X p,q, ∞ ≤ N k u k X p,q, ∞ k v k X p,q, ∞ , ∀ u, v ∈ X p,q, ∞ , (3.13)where N is a constant depending only on n, p and q . In other words, the bilinear form G : X p,q, ∞ × X p,q, ∞ → X p,q, ∞ is bounded.Next, let us choose λ > N N λ < , (3.14)where N is defined in (3.11), and N is defined in (3.13). Note that both of these numbers dependonly on p, q and n . Now, if k a k L p p ··· pn ( R n ) ≤ λ , then it follows from (3.11) that4 N k u k X p,q, ∞ ≤ N N k a k L p p ··· pn ( R n ) ≤ N N λ < . From this and by applying Lemma 3.9, we can find a unique solution u ∈ X p,q, ∞ of the equation(3.3) such that k u k X p,q, ∞ ≤ k u k X ∞ ≤ N k a k L p p ··· pn ( R n ) . (3.15)Now, to complete the proof (i), we need to show that u ∈ Y p,q, ∞ . We recall that the definition of Y p,q, ∞ is given in (1.8). Since u ( t ) = u ( t ) + G ( u, u )( t ) , we have k u ( t ) k L p p ··· pn ( R n ) ≤ k u ( t ) k L p p ··· pn ( R n ) + k G ( u, u )( t ) k L p p ··· pn ( R n ) , and k D x u ( t ) k L p p ··· pn ( R n ) ≤ k D x u ( t ) k L p p ··· pn ( R n ) + k D x G ( u, u )( t ) k L p p ··· pn ( R n ) . (3.16)Then, by applying Lemma 3.5, we see that k u ( t ) k L p p ··· pn ( R n ) ≤ N k a k L p p ··· pn ( R n ) , and k D x u ( t ) k L p p ··· pn ( R n ) ≤ N t − / k a k L p p ··· pn ( R n ) . (3.17) AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 17
On the other hand, by (3.10) and (1.5), we can apply the first assertion in Lemma 3.8 with γ k = 1, α k = p k q k ∈ (0 ,
1] and β k = 1 to infer that k G ( u, u )( t ) k L p p ··· pn ( R n ) ≤ N Z t ( t − s ) − δ k u ( s ) k L q q ··· qn ( R n ) k D x u ( s ) k L p p ··· pn ( R n ) ds ≤ N k u k X p,q, ∞ Z t ( t − s ) − δ s − (1 − δ ) ds = N k u k X p,q, ∞ "Z t/ ( t − s ) − δ s − (1 − δ ) ds + Z tt/ ( t − s ) − δ s − (1 − δ ) ds = N k u k X p,q, ∞ " t − δ Z t/ s − (1 − δ ) ds + t − (1 − δ ) Z tt/ ( t − s ) − δ ds ≤ N k a k L p p ··· pn ( R n ) , (3.18)where in the last estimate, we used (3.15). Also, by (3.12) and (3.15), it follows that k D x G ( u, u ) k L p p ··· pn ( R n ) ≤ N t − / k u k X p,q, ∞ ≤ N t − / k a k L p p ··· pn ( R n ) . (3.19)Then, from the estimates (3.16), (3.17), (3.18), (3.19) and the fact that k a k L p p ··· pn ( R n ) is suffi-ciently small that, we see that k u k Y p,q, ∞ ≤ N k a k L p p ··· pn ( R n ) . The proof of (i) is therefore complete.Now, we turn to prove (ii). As in the proof of (3.11), we see that u ∈ X p,q, ∞ . From thedefinition of the norm of the space X p,q, ∞ in (1.7), the continuity and the vanishes of t (1 − δ ) / u andof t / D x u at t = 0, we can choose a sufficiently small number T > n, p, q and a so that k u k X p,q,T ≤ λ , where λ is defined as in (3.14). Moreover, by following the proof of (3.13), we can also see that thebilinear form G : X p,q,T × X p,q,T → X p,q,T is bounded with k G ( u, v ) k X p,q,T ≤ N k u k X p,q,T k v k X p,q,T , ∀ u, v ∈ X p,q,T . Then, applying Lemma 3.9 again, we can find a unique local time solution u ∈ X p,q,T of (3.3)satisfying k u k X p,q,T ≤ N k a k L p p ··· pn ( R n ) . Now, we only need to prove that the solution u that we found is indeed in Y p,q,T . However, thiscan be done exactly the same as in the proof that u ∈ Y p,q, ∞ in (i), and we skip it. The proof ofthe theorem is then complete. (cid:4) Remark . The pressure P in (1.1) can be solved from the solution u = ( u , u , · · · , u n ) as P = n X i,j =1 R i R j ( u i u j ) , where R j is the i th Riesz transform, which is defined in Theorem 2.23. Since p k > k = 1 , , · · · , n , we can apply Theorem 2.23 to obtain k P ( · , t ) k L p p ··· pn ( R n ) ≤ N k u ( · , t ) k L p p ··· pn ( R n ) . Acknowledgement.
The author would like to thanks professor Lorenzo Brandolese (InstitutCamille Jordan, Universit´e Lyon 1) and professor Nam Le (Indiana University) for their valuablecomments.
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AVIER-STOKES EQUATIONS IN CRITICAL MIXED-NORM LEBESGUE SPACES 19 (T. Phan)
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