Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces

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1 Commun. Math. Phys. 263, (25) Digital Obect Ientifier (DOI) 1.17/s Communications in Mathematical Physics Lower Bouns for an Integral Involving Fractional Laplacians an the Generalize Navier-Stokes Euations in Besov Spaces Jiahong Wu Department of Mathematics, Oklahoma State University, Stillwater, OK 7478, USA. Receive: 24 March 25 / Accepte: 25 July 25 Publishe online: 29 November 25 Springer-Verlag 25 Abstract: When estimating solutions of issipative partial ifferential euations in L p -relate spaces, we often nee lower bouns for an integral involving the issipative term. If the issipative term is given by the usual Laplacian, lower bouns can be erive through integration by parts an embeing ineualities. However, when the Laplacian is replace by the fractional Laplacian ( ) α, the approach of integration by parts no longer applies. In this paper, we obtain lower bouns for the integral involving ( ) α by combining pointwise ineualities for ( ) α with Bernstein s ineualities for fractional erivatives. As an application of these lower bouns, we establish the existence an uniueness of solutions to the generalize Navier-Stokes euations in Besov spaces. The generalize Navier-Stokes euations are the euations resulting from replacing in the Navier-Stokes euations by ( ) α. 1. Introuction This paper is concerne with the generalize incompressible Navier-Stokes (GNS) euations t u + u u + P = ν( ) α u, u =, (1.1) where ν> an α> are real parameters, an the fractional Laplacian ( ) α is efine in terms of the Fourier transform ( ) α u(ξ) = (2π ξ ) 2α û(ξ). We accomplish two maor goals. First, we obtain lower bouns for the integral D(f ) f p 2 f ( ) α fx, (1.2) R

2 84 J. Wu where p 2 an α. Secon, we apply these lower bouns to establish the existence an uniueness of solutions to the GNS euations in homogeneous Besov spaces. We shall now explain in some etail our maor results together with backgroun information necessary for unerstaning these results. When α = 1, the GNS euations (1.1) reuce to the usual Navier-Stokes euations. One avantage of working with the GNS euations is that they allow simultaneous consieration of their solutions corresponing to a range of α s. For example, the 3-D GNS euations with any α 4 5 always possess global classical solutions [15]. For the general -D GNS euations, we have shown in [17] that α guarantees global regularity. In this paper, we consier the GNS euations with a general fraction α an one of the ifficulties is how to obtain a lower boun for D(f ) efine in (1.2). The uantity D(f ) arises very naturally in the process of bouning solutions of the GNS euations in L p -relate spaces. In the special case when α = 1, lower bouns for D(f ) are often erive through integrating by parts ([1, 2, 16]). However, for a general fraction α, ( ) α is a nonlocal operator an this approach fails. In this paper, we establish lower bouns for D(f ) with a general fraction α by combining the pointwise ineualities for ( ) α an Bernstein s ineualities for fractional erivatives. In [1] an [11], A. Córoba an D. Córoba showe that for any α 1 an any f C 2 (R ) that ecays sufficiently fast at infinity, the pointwise ineuality 2 f(x)( ) α f(x) ( ) α f 2 (x), x R (1.3) hols. By moifying the proof in [1], N. Ju prove in [14] that if p an f is, in aition, nonnegative, then (p + 1)f p (x) ( ) α f(x) ( ) α f p+1 (x). We obtain here the ineuality in the general form (p 1 + p 2 )f p 1 (x) ( ) α f p 2 (x) p 2 ( ) α f p 1+p 2 (x), (1.4) where p 1 = k 1 l1 an p 2 = k 2 l2 with l 1 an l 2 being o an k 1 l 2 + k 2 l 1 being even (see Proposition 3.2 for more etails). Another type of generalization of (1.3) was consiere by P. Constantin, who establishe an ientity for ( ) α acting on the prouct of two functions. This ientity allowe him to obtain a calculus ineuality involving fractional erivatives [7]. In this paper, we combine suitable pointwise ineualities with Bernstein s ineualities for fractional erivatives to erive several lower bouns for D(f ). In particular, we have which is vali for any f C 2 (R ) satisfying D(f ) C 2 2α f p L p, (1.5) supp f {ξ R : K 1 2 ξ K 2 2 }. The precise statement of this result is provie in Theorem 3.4. As an application of these lower bouns, we stuy the solutions of the GNS euations in the homogeneous Besov space B r (R ) with general inices an establish several existence an uniueness results. In particular, it is shown that the GNS euations possess a uniue global solution in B r (R ) with r = 1 2α + p for any initial atum u that is comparable to ν in B r (R ). This result hols for any 1 an for

3 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 85 either 2 1 <αan p = 2or 2 1 <α 1 an 2 <p<. We efer the exact statement to Theorem 6.1. The proof of this theorem is base on the contraction mapping principle an two maor a priori ineualities. The first one states that any solution u of the GNS euations satisfies t u B r + Cν u B r+ 2α C u B 1 2α+ p u B r+ 2α, (1.6) where <α 1, r R an 1 <. The ineuality for the case = is slightly ifferent (see Theorem 4.1 for etails). The secon ineuality ( ) t F B +Cν F B C v B w B + w B v B r r+ 2α 1 2α+ p r+ 2α 1 2α+ p r+ 2α (1.7) bouns solutions of the euation t F + ν( ) α F = P(v w), (1.8) where P is the matrix operator proecting onto the ivergence free vector fiels. For arbitrarily large initial atum u B r (R ), the local existence an uniueness of solutions in B r (R ) is establishe. The proof of the local existence result reuires ifferent a priori bouns an they are provie in Theorems 5.2 an 5.4 of Sect. 5. These results for the GNS euations together with those in [4, 8, 9, 17 2] contribute significantly to unerstaning how the general fractional issipation effects the regularity of solutions to issipative partial ifferential euations. The results of this paper have two important special conseuences. First, the 3-D Navier-Stokes euations have a uniue global solution for any initial atum u comparable to ν in B 1+ p 3 (R 3 ), an a uniue local solution for any large atum in this space, where 1 p< an 1. Solutions of the 3-D Navier-Stokes euations in B 1+ p 3 (R 3 ) have previously been stuie in [5] an [13]. Secon, these existence an uniueness results also hol for solutions of the GNS euations in the usual Sobolev spaces W r,p (R ) with r = 1 2α + p. This is because B r reuces to W r,p when p =. The rest of this paper is ivie into five sections. Section 2 provies the efinition of the homogeneous Besov spaces an states Bernstein s ineualities for both integer an fractional erivatives. Section 3 presents the general form of the pointwise ineuality (1.4) an the lower boun (1.5). Section 4 erives the a priori bouns (1.6) an (1.7). Section 5 establishes a priori bouns for the GNS euations an for E. (1.8) in L ((,T); B r ) an L ((,T); B r ). Section 6 proves the existence an uniueness results. 2. Besov Spaces In this section, we provie the efinition of the homogeneous Besov space an state the Bernstein ineualities for integer an fractional erivatives. We first fix some notation. Let S be the usual Schwarz class an S the space of tempere istributions. The Fourier transform f of a L 1 -function f is given by

4 86 J. Wu f(ξ)= f(x)e 2π ix ξ x. R (2.1) For f S, the Fourier transform of f is obtaine by ( f,g)= (f, ĝ) for any g S. The Fourier transform is a boune linear biection from S to S whose inverse is also boune. For this reason, the fractional Laplacian ( ) α with α R can be efine through its Fourier transform, namely, ( ) α f(ξ)= (2π ξ ) 2α f(ξ). For notational convenience, we sometimes write for ( ) 2 1. We use S to enote the following subset of S, { } S = φ S, φ(x)x γ x =, γ =, 1, 2,. R Its ual S is given by S = S /S = S /P, where P is the space of multinomials. In other wors, two istributions in S are ientifie as the same in S if their ifference is a multinomial. We now introuce a yaic partition of R. For each Z, we efine Now, we choose φ S(R ) such that Then, we set an efine S by A ={ξ R :2 1 < ξ < 2 +1 }. supp φ ={ξ :2 1 ξ 2} an φ > ona. φ (ξ) = φ (2 ξ) (ξ) = φ (ξ) φ (ξ). It is clear that an satisfy Furthermore, (ξ) = (2 ξ), supp A, (x) = 2 (2 x). k= k (ξ) = { 1ifξ R \{}, ifξ =.

5 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 87 Thus, for a general function ψ S,wehave But, if ψ S, then That is, for ψ S, an hence k= k= k (ξ) ψ(ξ) = ψ(ξ) for ξ R \{}. k (ξ) ψ(ξ) = ψ(ξ) for all ξ R. k= k= k ψ = ψ in the weak* topology of S for any f S. To efine the homogeneous Besov space, we set k f = f (2.2) f = f, =, ±1, ±2,. (2.3) For s R an p, [1, ], we say that f B s if f S an B s B s = ( 2 s f L p) < for <, sup 2 s f L p < for =. << is a Banach space when euippe with the norm f B s f B s = 1/ ( ) 2 s f L p sup 2 s f L p for =. << for <, with this norm will be referre to as the homogeneous Besov space. The usual homogeneous Sobolev space W s,p efine by W s,p = s L p is a special type of the homogenous Besov space. That is, B s p,p = W s,p. The homogenous Besov spaces obey the inclusion relations state in the following proposition (see [3]). Part 2) of this proposition will be referre to as the Besov embeing.

6 88 J. Wu Proposition 2.1. Assume that β R an p, [1, ]. 1) If 1 1 2, then B β 1 (R ) B β 2 (R ). 2) If 1 p 1 p 2 an β 1 = β 2 + ( 1 p 1 1 p 2 ), then B β 1 p 1,(R ) B β 2 p 2,(R ). We now turn to Bernstein s ineualities. When the Fourier transform of a function is supporte on a ball or an annulus, the L p -norms of the erivatives of the function can be boune in terms of the L p -norms of the function itself. Ineualities of this nature are referre to as Bernstein s ineualities. The classical Bernstein s ineualities only allow integer erivatives. Proposition 2.2. Let k be an integer an 1 p. 1) If supp f {ξ R : ξ K2 } for some K>an integer, then sup D γ f L C 2 k+( p 1 1 ) f L p. γ =k 2) If supp f {ξ R : K 1 2 ξ K 2 2 } for some K 1,K 2 > an integer, then C 2 k+( p 1 1 ) f L p sup D γ f L C 2 k+( p 1 1 ) f L p, γ =k where C s an C are constants inepenent of. Bernstein s ineualities can actually be extene to involve fractional erivatives. Proposition 2.3. Let α an 1 p. 1) If supp f {ξ R : ξ K2 } for some K>an integer, then α f L C 2 α+( 1 p 1 ) f L p. 2) If supp f {ξ R : K 1 2 ξ K 2 2 } for some K 1,K 2 > an integer, then C 2 α+( 1 p 1 ) f L p α f L C 2 α+( 1 p 1 ) f L p. Proposition 2.3 is a simple extension of Proposition 2.2. I am inebte to Davi Ullrich who communicate Proposition 2.3 to me. We remark that a lemma in Danchin work [12, p. 632] implies a special case of 2) of Proposition 2.3, which states that for s> an for any even integer p, f satisfying implies supp f {ξ R : K 1 2 ξ K 2 2 } s (f p 2 ) L 2 C 2 s f p 2 L 2. We now point out several simple facts concerning the operators : k =, if k 2; (2.4)

7 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 89 S 1 k= k I, as ; (2.5) (S k 1 f k f)=, if k 3. (2.6) I in (2.5) enotes the ientity operator an (2.5) is simply another way of writing (2.2). Finally, we provie here two elementary ineualities involving summations over infinite terms: Höler ineuality an Minkowski s ineuality. Lemma 2.4. Let p [1, ] an p = 1.If{a k} l p an {b k } l, then [ ] 1 [ p ] 1 a k b k a k p b k. (2.7) k=1 k=1 Lemma 2.5. Let p [1, ].Ifa k an b k for k = 1, 2, 3,...,then [ ] 1 [ p ] 1 [ p ] 1 (a k + b k ) p a p p k + b p k. k=1 k=1 k=1 k=1 3. Lower Bouns In this section, we exten the pointwise ineuality of Córoba an Córoba to a more general form an prove the lower boun (1.5) for D(f ). The extene pointwise ineuality is given in Proposition 3.2 an the lower boun in Theorem 3.4. In [1] an [11], Córoba an Córoba prove the following pointwise ineuality. Proposition 3.1. Let α 1 an assume f C 2 (R ) ecays sufficiently fast at infinity. Then, for any x R, 2 f(x)( ) α f(x) ( ) α f 2 (x). As they pointe out, the conition that f C 2 (R ) can be weakene. This proposition actually hols for any f such that f, ( ) α f an ( ) α f 2 are efine, an are, respectively, the limits of f m, ( ) α f m an ( ) α f 2 m for f m C 2 (R ) that ecays sufficiently fast at infinity. For later applications, we exten the ineuality in Proposition 3.1 to a general form state in the following proposition. Proposition 3.2. Let α 1. Let p 1 = k 1 l1 an p 2 = k 2 l2 1 be rational numbers with l 1 an l 2 being o, an with k 1 l 2 + k 2 l 1 being even. Then, for any x R an any function f C 2 (R ) that ecays sufficiently fast at infinity, (p 1 + p 2 )f p 1 (x) ( ) α f p 2 (x) p 2 ( ) α f p 1+p 2 (x). (3.1) We make several remarks. First, this theorem also applies to functions that are not necessarily in C 2 (R ). The estimate still hols if f, ( ) α f p 2 an ( ) α f p 1+p 2 are efine, an are respective limits of f m, ( ) α f p 2 m an ( ) α f p 1+p 2 m for a seuence f m C 2 (R ) that ecays sufficiently fast at infinity. Seconly, the conition that k 1 l 2 + k 2 l 1 is even can not be remove.

8 81 J. Wu Proof. When α = orα = 1, (3.1) can be irectly verifie. When p 1 an p 2 are integers, (3.1) can be proven by slightly moifying the proof of Córoba an Córoba for Proposition 3.1. As shown in [11], ( ) α f can be represente as the integral ( ) α f(x)= C α P.V. f(x) f(y) R x y +2α y, x R, where C α is a constant epening on α only an P.V. means the principal value. Therefore, f p 1 (x) ( ) α f p 2 (x) f(x) p1+p2 f(x) p1 f(y) p2 = C α P.V. R x y +2α y p 1 f(x) p1+p2 (p 1 + p 2 )f (x) p1 f(y) p2 + p 2 f(y) p1+p2 = C α P.V. R (p 1 + p 2 ) x y +2α y + p 2 f(x) p1+p2 f(y) p1+p2 C α P.V. p 1 + p 2 R x y +2α y. (3.2) When k 1 l 2 + k 2 l 1 is even, p 1 + p 2 is even an we have Young s ineuality p 1 f(x) p 1+p 2 (p 1 + p 2 )f (x) p 1 f(y) p 2 + p 2 f(y) p 1+p 2. Conseuently, the first integral on the right of (3.2) is nonnegative. The secon integral is p simply the integral representation of 2 p 1 +p 2 ( ) α f p 1+p 2 (x). Therefore, (3.2) implies (3.1). Now, we consier the case when p 1 = k 1 l1 an p 2 = k 2 l2 1 are rational numbers. For notational convenience, we write F(x) = f(x) l 1 1 l 2. Since l 1 an l 2 are o, f C 2(R ) implies F C 2(R ). Because (3.1) has been shown for integers, we have f p 1 (x) ( ) α f p 2 (x) = F k 1l 2 (x) ( ) α F k 2l 1 (x) k 2 l 1 ( ) α F k 1l 2 +k 2 l 1 (x) k 1 l 2 + k 2 l 1 = p 2 ( ) α f p 1+p 2 (x). p 1 + p 2 That is, (3.1) hols in this case. This completes the proof of Proposition 3.2. If we are willing to assume the function f, then the assumption on the inices p 1 an p 2 can be reuce. The following proposition is ue to N. Ju [14]. Proposition 3.3. Let α 1 an let p. Then, for any f C 2 (R ) that ecays sufficiently fast at infinity, f, an any x R, (p + 1)f p (x) ( ) α f(x) ( ) α f p+1 (x). We now erive the lower boun (1.5) for D(f ). Theorem 3.4. Assume either α an p = 2 or α 1 an 2 <p<. If f C 2 (R ) ecays sufficiently fast at infinity an satisfies supp f {ξ R : K 1 2 ξ K 2 2 } (3.3)

9 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 811 for some K 1,K 2 > an some integer, then D(f ) f p 2 f ( ) α fx C2 2α f p R L p (R ), (3.4) where C is a constant epening on, p, K 1 an K 2 only. The restriction that α 1 comes from applying the pointwise ineualities. When p = 2, (3.4) is a irect conseuence of Plancherel s theorem an thus α is not reuire to satisfy α 1. We also remark that the reuirement f C 2 (R ) can be reuce. In fact, this theorem applie to any function f with the property that f an ( ) α f ( = 1, 2,..., p 2 ) are efine, an are, respectively, limits of f m an ( ) α fm with each f m C 2 (R ). This also explains why we o not assume the functions are in C 2 (R ) when we apply this theorem in the subseuent sections. Proof. When p = 2, the lower boun in (3.4) is a irect conseuence of Plancherel s theorem. When p>2, Proposition 3.3 implies D(f ) f p 2 f ( ) α fx C α (f p 2 2 ) x = C α (f p 2 ) 2 R R L 2. It then follows from 2) of Proposition 2.3 that D(f ) C 2 2α f p 2 2 L 2 C 2 2α f p L p. Other useful lower bouns for D(f ) can also be establishe using the pointwise ineuality in Proposition 3.3. These lower bouns which o not reuire the support of f satisfy the conition (3.3). Theorem 3.5. Assume either α an p = 2 or α 1 an 2 <p<. Then D(f ) can be boune as follows. (a) If p = 2 an 2α =, then D(f ) C f 2 L (R (3.5) ) for any [2, ) an some constant epening on only. (b) If p = 2 an 2α <, then D(f ) C f 2 L 2α 2 (R ) for some constant C epening on α an only. (c) If p>2an 2α =, then, for any f L 2 (R ), D(f ) C f (1+β)p f L p (R βp ) L 2 (R ), where β = 2 4 p 2 for any [p, ), an C is a constant epening on p an only. () If p>2an 2α <, then for any f L 2 (R ), D(f ) C f (1+γ)p p f L p (R γ ) L 2 (R ), where γ = (p 2) 4α an C is a constant epening on, α an p only. The proof of this theorem is left to the appenix.

10 812 J. Wu 4. A Priori Estimates in B r This section erives two a priori bouns in B r : one for solutions of the GNS euations t u + u u + P = ν( ) α u, x R, t >, (4.1) u =, x R, t >, (4.2) an one for solutions of the euation u(x, ) = u (x), x R, (4.3) t F + ν( ) α F = P(v w), x R, t >, (4.4) F(x,) = F (x), x R, (4.5) where P = I 1 is the matrix operator proecting onto ivergence free vector fiels an I is the ientity matrix. The bouns are given in Theorems 4.1 an 4.3. We work with the following form of the GNS euations: t u + ν( ) α u = P(u u), (4.6) which is euivalent to (4.1) an (4.2). This form of the GNS euations can be seen as a special case of (4.4). Before stating an proving the theorems, we first briefly introuce the spaces L ρ ((a, b); B s ) an L ρ ((a, b); B s ). For ρ, [1, ] an s,a,b R, L ρ ((a, b); B s ) is efine in the stanar fashion. That is, L ρ ((a, b); B s ) enotes the space of Lρ -integrable functions from (a, b) to B s. The space L ρ ((a, b); B s ) was introuce in [6] an later use in [5] an [13]. We say that f L ρ ((a, b); B s ) if where µ is efine by µ = 2 s ( b a {µ } l, ) f(,t) ρ ρ 1 L p t µ = 2 s sup f(,t) L p t (a,b) The norm in L ρ ((a, b); B s ) is given by f L ρ ((a,b); B s ) = µ l. for ρ<, for ρ =. For notational convenience, we sometimes write L ρ t ( B s ) for L ρ ((,t); B s ). We now state the first maor theorem of this section.

11 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 813 Theorem 4.1. Let r R an 1. Assume either <αan p = 2 or <α 1 an 2 <p<. Then any solution u of the GNS euations (4.1) an (4.2) or of (4.6) satisfies t u B r + Cν u B r+ 2α C u B 1 2α+ p u B r+ 2α for [1, ) (4.7) an, for =, u(,t) B p, r +Cν u L1 t ( B p, r+2α ) u B p, r +C sup u(,τ) τ t B 1 2α+ p u L1 t ( B p, r+2α ), p, (4.8) where C s are constants epening on, α, r an p only. The ineuality in (4.7) contains rich information about solutions of the GNS euations. For example, if we know u is comparable to ν in B 1 2α+ p, then u is boune in B r for any r R. In particular, when r = 1 2α + p, (4.7) becomes a close ineuality. We state the result in this special case as a corollary. Corollary 4.2. Let 1. Assume either <αan p = 2 or <α 1 an 2 <p<. Then any solution u of (4.1) an (4.2) or of (4.6) satisfies t u B +Cν u B 1 2α+ p 1 2α+ p + 2α an, for =, C u B 1 2α+ p u B 1 2α+ p + 2α for [1, ) u(,t) B 1 2α+ p p, + Cν u L1 t ( B 1+ p u p, ) B 1 2α+ p p, +C sup τ t u(,τ) B 1 2α+ p p, u L1 t ( B 1+ p, p, ) where C s are constants epening on, α an p only. A special inication of this ineuality is that any solution of the GNS euations will remain in the Besov space B 1 2α+ p for all time if the corresponing initial atum u is in this Besov space an is comparable to ν. Proof of Theorem 4.1. Let Z. Applying to (4.6) yiels t u + (u ) u + ν( ) α u = [P,u ]u, (4.9) where the brackets represent the commutator operator, namely [P,u ]u P (u u) u P u. We then ot both sies of (4.9) by p u p 2 u an integrate over R. Since R (u ) u u p 2 ux =,

12 814 J. Wu we obtain t u p L p + pν u p 2 u ( ) α ux R = p u p 2 u [P,u ]ux. (4.1) R Accoring to Theorem 3.4, the issipative part amits the following lower boun: pν u p 2 u ( ) α ux Cν2 2α u p R L p. (4.11) We now estimate the nonlinear part I p u p 2 u [P,u ]ux. R By Höler s ineuality, I C u p 1 L p [P,u ]u L p. To estimate [P,u ]u L p, we use Bony s notion of paraprouct to write where [P,u ]u = I 1 + I 2 + I 3 + I 4 + I 5, (4.12) I 1 = k Z P ((S k 1 u ) k u) (S k 1 u )P k u, I 2 = k Z P (( k u )S k 1 u), I 3 = k Z( k u ) S k 1 u, I 4 = I 5 = k l 1 k l 1 P (( k u ) l u), ( k u ) l u. We now estimate the L p -norms of these terms. Accoring to (2.6), the summation in I 1 is only over those k satisfying k 2. Let be the convolution kernel associate with the operator P. Since each entry in P is the ifference between 1 an a prouct of two Riesz transforms, is smooth. Using, we write I 1 = (x y)(s k 1 u(y) S k 1 u(x)) k u(y) y. k 2 R We integrate by parts an use the fact that u = to obtain I 1 = (x y) (S k 1 u(y) S k 1 u(x)) k uy. R k 2

13 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 815 By Young s ineuality, I 1 L p C k 2 = C k 2 S k 1 u L k u L p R x (x) x S k 1 u L k u L p. (4.13) Similarly, the summations in I 2 an I 3 are also only over k satisfying k 2. Applying Young s ineuality an using the fact that the norm of the operator P in L p with 1 <p< is 1 yiel I 2 L p C S k 1 u L k u L p, (4.14) I 3 L p C C k 2 k 2 k 2 k u L p S k 1 u L k u L p S k 1 u L. (4.15) The summation in I 4 is over k with k 3. The L p norm of I 4 is boune by I 4 L p C k u L p l u L. (4.16) k 3, k l 1 The estimate of I 5 is similar, namely, I 5 L p C k 2, k l 1 k u L p l u L. (4.17) Collecting the estimates (4.13), (4.14), (4.15), (4.16) an (4.17), we fin that [,u ]u L p C S k 1 u L k u L p +C k 2 k 3, k l 1 k u L p l u L. (4.18) We emphasize that the summations in (4.18) are only over a finite number of k s. It suffices to consier the term with k = in the first summation an the term with k = l = in the secon summation. By Bernstein s ineuality, Therefore, S 1 u L u L C 2 (1+ p ) u L p, [,u ]u L p C u L p m u L C 2 (1+ p )m m u L p. 2 (1+ p )m m u L p + C 2 (1+ p ) u 2 L p.

14 816 J. Wu Conseuently, we obtain the following boun for the nonlinear term in (4.1): I C u p L p 2 (1+ p )m m u L p + C 2 (1+ p ) u p+1 L p. (4.19) Combining (4.1), (4.11) an (4.19), we fin Euivalently, t u p L p + Cν2 2α u p L p C u p L p 2 (1+ p )m m u L p + C 2 (1+ p ) u p+1 L p. (4.2) t u L p + Cν2 2α u L p C u L p 2 (1+ p )m m u L p + C 2 (1+ p ) u 2 Lp. (4.21) For 1 <, we multiply both sies by 2 r u 1 L p where J 1 an J 2 are given by J 1 C t u B r + Cν u B r+ 2α 2 r u L p an sum over Z to get J 1 + J 2, (4.22) 2 (1+ p )m m u L p, J 2 C 2 (1+ p ) u L p 2 r u L p. To estimate J 1, we write J 1 = C 2 (r+ 2α ) u L p Applying the elementary ineuality (2.7), we have 2 (1+ p 2α)m m u L p 2 2α(m ) 1 2 (1+ p 2α)m m u L p 2 (1+ p 2α)m m u L p 2 2α(m ). 2 2α(m ) = C u B 1 2α+ p, (4.23) 1

15 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 817 where is the conugate of, namely = 1. Therefore, J 1 is boune by J 1 C u B 1 2α+ p To boun J 2, we first write u B r+ 2α. J 2 = C 2 (1 2α+ p ) u L p 2 (r+ 2α ) u L p. It is clear that 2 (1 2α+ p ) u L p Therefore, J 2 is boune by 2 (1 2α+ p ) u L p 1 = u B 1 2α+ p. (4.24) J 2 C u B 1 2α+ p u B r+ 2α Finally, we insert the estimates for J 1 an J 2 in (4.22) to obtain (4.7). We now eal with the case when =. We multiply (4.21) by 2 r, integrate over (,t)an take the supremum over Z to get u(,t) B p, r + Cν u L1 t ( B p, r+2α ) u B p, r + J 1 + J 2, (4.25) where J 1 an J 2 are given by J 1 = C sup 2 r u L p To estimate J 1, we rearrange its terms as J 1 = C sup It is then clear that J 1 C sup sup. 2 (1+ p )m m u L p τ, J 2 = C sup 2 r 2 (1+ p ) u 2 L p τ. 2 (r+2α) u L p sup 2 (r+2α) u L p τ τ t 2 2α(m ) 2 (1 2α+ p )m m u L p τ. 2 2α(m ) 2 (1 2α+ p )m m u(,τ) L p C sup 2 (r+2α) u L p τ sup τ t = C u L1 t ( B r+2α p, ) sup τ t u(,τ) B 1 2α+ p p, sup 2 (1 2α+ p ) u(,τ) L p. (4.26)

16 818 J. Wu J 2 can be boune as follows. J 2 = C sup 2 (1 2α+ p ) u L p 2 (r+2α) u L p τ C sup C sup τ t sup τ t 2 (1 2α+ p ) u L p sup 2 (r+2α) u L p τ u(,τ) B 1 2α+ p p, u L1 t ( B r+2α p, ). (4.27) Combining the estimates in (4.25), (4.26) an (4.27) yiels (4.8). This completes the proof of Theorem 4.1. We now present the priori boun for solutions of the general euation (4.4). This type of estimates will be neee in Sect. 6 when we stuy the existence an uniueness of the solutions to the GNS euations. To establish the estimate, we nee to restrict to α> 1 2. Theorem 4.3. Let r R an [1, ]. Assume either 1 2 <αan p = 2 or 1 2 <α 1 an 2 <p<. Assume that v an w are in the class L ([,T); B r ) L ([,T); B r+ 2α ) for <T. Then any solution F of (4.4) satisfies t F B + Cν F B r r+ 2α ( ), (4.28) C v B 1 2α+ p w B r+ 2α + w B 1 2α+ p v B r+ 2α where C is a constant epening on an p only. For the sake of conciseness, we i not inclue in this theorem the ineuality for the case when =. It can be state an erive analogously as (4.28). Proof. We apply to (4.4) an then multiply by p F p 2 F. Bouning the issipative part by the lower boun as in (4.11) an estimating the right-han sie by Höler s ineuality, we obtain t F p L p + Cpν2 2α F p L p Cp (v )w L p F p 1 L p. That is, t F L p + Cν2 2α F L p C (v )w L p. (4.29) To estimate the term on the right, we write (v )w = K 1 + K 2 + K 3,

17 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 819 where K 1, K 2 an K 3 are given by K 1 = k ((S k 1 v ) k w), K 2 = (( k v )S k 1 w), k K 3 = (( k v ) l w). k l 1 To estimate these terms, we first notice that the summations in K 1, K 2 an K 3 are only over k satisfying k 2. Therefore, it suffices to estimate the representative term with k =. Applying Young s ineuality an Bernstein s ineuality, we obtain K 1 L p C S 1 v L w L p C m v L 2 w L p C 2 p m m v L p 2 w L p, K 2 L p C S 1 w L v L p C 2 (1+ p )m m w L p v L p, K 3 L p C v L p w L C 2 (1+ p ) v L p w L p. Inserting these estimates in (4.29), then multiplying by 2 r F 1 L p an summing over all, we obtain t F B + Cν F B r r+ 2α where L 1, L 2 an L 3 are given by L 1 = C L 2 = C L 3 = C 2 r F 1 L p 2 w L p 2 r F 1 L p v L p L 1 + L 2 + L 3, (4.3) 2 p m m v L p, 2 (1+ p )m m w L p, 2 r F 1 L p 2 (1+ p ) v L p w L p. To boun L 1, we write L 1 = C 2 (r+ 2α ) ( 1) F 1 L p 2 (r+ 2α ) w L p 2 (1+ p 2α)m m v L p 2 (2α 1)(m ).

18 82 J. Wu When α> 2 1, (2α 1)(m ) < an we obtain as in (4.23) 2 (1+ p 2α)m m v L p 2 (2α 1)(m ) C v B 1 2α+ p. After applying the elementary ineuality (2.7), we fin L 1 C v B 1 2α+ p w B r+ 2α F 1 B r+ 2α. L 2 is boune similarly, but we o not nee α> 1 2, L 2 C w B 1 2α+ p v B r+ 2α F 1 B r+ 2α. To boun L 3, we write L 3 = C 2 (r+ 2α ) ( 1) F 1 L p 2 (r+ 2α ) w L p 2 (1+ p 2α) v L p. Estimating 2 (1+ p 2α) v L p as in (4.24), we obtain L 3 C v B 1 2α+ p w B r+ 2α F 1 B r+ 2α Combining these estimates for L 1, L 2 an L 3 with (4.3), we obtain t F B + Cν F B r r+ 2α ( ) C v B 1 2α+ p w B r+ 2α + w B 1 2α+ p v B r+ 2α F 1 Applying Young s ineuality then yiels (4.28).. B r+ 2α. 5. A Priori Estimates in L ((,T); B s ) an in L ((,T); B s ) In this section, we establish a priori estimates for solutions of the GNS euations an for those of (4.4) in two ifferent type of spaces: L ((,T); B s ) an L ((,T); B s ). The maor results are state in Theorems 5.1, 5.2, 5.3 an 5.4. We will nee these estimates when we stuy solutions of the GNS euations in the next section. The first theorem provies an a priori estimate for solutions of the GNS euations in L ([,T); B r+ 2α ). The erivation of this boun reuires >2. Theorem 5.1. Let 2 <. Assume either <αan p = 2 or <α 1an 2 <p<. Let r = 1 + p 2α + 2α. Let u B r an let u be a solution of the GNS euations with the initial atum u. Set A(t) = u. L ((,t); B r+ 2α )

19 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 821 Then, for any T>, A(T ) Cν 1 (1 E (T )) 2 r u L p + Cν ( 2) A 2 (t) t, (5.1) where E (t) exp( Cν2 2α t) an C s are constants epening on,p an. In particular, A(T ) Cν 1 u B + Cν ( 2) A 2 (t) t. r Proof. As erive in the proof of Theorem 4.1, u satisfies (4.21), namely, t u L p + Cν2 2α u L p C u L p 2 (1+ p )m m u L p + C 2 (1+ p ) u 2 L p. m 1 We first convert it into the following integral form: u(,t) L p CE (t) u L p + C +C E (t s) u(,s) L p E (t s)2 (1+ p ) u(,s) 2 L ps 2 (1+ p )m m u(,s) L ps, where E (t) exp( Cν2 2α t). Multiplying both sies by 2 (r+ 2α ), raising them to the th power, summing over all an integrating over (,T), we obtain u L ( (,T ); B r+ 2α where M 1, M 2 an M 3 are given by with M 1 C M 2 C M 3 C ) M 1 + M 2 + M 3, (5.2) E (t) 2α 2(r+ ) u L p t, 2 (1+ p +r+ 2α ) M 21 (t)t, 2 (r+ 2α ) M 31 (t)t ( ) M 21 = E (t s) u(,s) 2 L ps (5.3)

20 822 J. Wu an M 31 = E (t s) u(,s) L p 2 (1+ p )m m u(,s) L ps. (5.4) We now estimate M 1, M 2 an M 3. Inserting in M 1,wehave E (t) t = Cν 1 2 2α (1 E (T )) M 1 = Cν 1 (1 E (T )) 2 r u L p. (5.5) In particular, M 1 Cν 1 u B r. To boun M 2, we start with an estimate for M 21. For >2, Höler s ineuality implies ( ) ( E (t s) u(,s) 2 ) 2 ( ) 2 L ps 2 E (t s)s u L p s ( )) 2 ( 2 Cν ( 2) 2 (1 2α ( 2) E 2 t u L s) p. (5.6) Therefore, M 2 is boune by ( 2 M 2 Cν ( 2) 2 (1+ p 2α+ 4α +r+ 2α ) u L s) p t. For r = 1 + p 2α + 2α,wehave ( 2 M 2 Cν ( 2) 2 (r+ 2α ) u L s) p t Cν ( 2) = Cν ( 2) 2 (r+ 2α ) u L p s u 2 L ( (,t);b r+ 2α 2 t ) t. (5.7) We now boun M 3. First, we estimate M 31. As in the estimate of (5.6), we obtain M 31 Cν ( 2) 2 2α ( 2) u L p s 2 (1+ p )m m u L p s = Cν ( 2) u L p s 2 (1+ p 2α+ 4α )m m u L p2 2α(1 2 )(m ) s.

21 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 823 For >2an r = 1 + p 2α + 2α, we obtain by applying (2.7), 2 (1+ p 2α+ 4α )m m u L p2 2α(1 2 )(m ) (5.8) 2 (r+ 2α )m m u L p where is the conugate of. Therefore, M 31 Cν ( 2) 1 = Cν ( 2) u L ( u B r+ 2α s (,t);b r+ 2α 2 2α(1 2 )(m ) ) u L p s 1 = C u B r+ 2α, u L p s. (5.9) Inserting this boun in M 3, we obtain M 3 Cν ( 2) 2 (r+ 2α ) = Cν ( 2) u 2 L ((,t);b r+ 2α ) u L p s u t L ((,t);b r+ 2α ) t. (5.1) Combining (5.5), (5.7), (5.1) with (5.2) yiels (5.1). This completes the proof of Theorem 5.1. We now provie the estimate for solutions of the GNS euations in L ([,t); B r+ 2α ). Theorem 5.2. Let 1 <. Assume either <αan p = 2 or <α 1an 2 <p<. Let r = 1 + p 2α. Let u B r an let u be a solution of the GNS euations with the initial atum u. Set B(t) = u L ( (,t); B r+ 2α ). Then, for any T>, B(T ) Cν 1 (1 E (T )) 2 r u L p + Cν ( 1) B 2 (T ). (5.11) In particular, B(T ) Cν 1 u B r + Cν ( 1) B 2 (T ).

22 824 J. Wu Proof. Following a similar proceure as in the proof of Theorem 5.1, we obtain u L ( where N 1, N 2 an N 3 are given by (,T ); B r+ 2α ) N 1 + N 2 + N 3, (5.12) N 1 C N 2 C N 3 C 2 (r+ 2α ) E (t) u L p t, 2 (1+ p +r+ 2α ) M 21 (t)t, 2 (r+ 2α ) M 31 (t)t, with M 21 efine in (5.3) an M 31 in (5.4). The boun for N 1 is the same as that for M 1, which is given in (5.5). To estimate N 2, we apply Young s ineuality to obtain ( M 21 (t) t E 1 If r = 1 + p 2α, then N 2 is boune by N 2 Cν ( 1) = Cν ( 1) Cν ( 1) = Cν ( 1) u 2 L ( ) 1 ( ) 2 (t) t u L p t ( 2 Cν ( 1) 2 2α ( 1) u L t) p. 2 (1+ p 2α+ 2α +r+ 2α ) ( (2 (r+ 2α ) ) 2 u L p t 2 (r+ 2α ) u L p t (,T ); B r+ 2α ) 2 u L p t 2 ). (5.13) To boun N 3, we again apply Young s ineuality to get M 31 t C E u L 1 L (,T ) 2 (1+ p )m m u L p (,T ) L (,T ) = Cν ( 1) 2 2α ( 1) u L p t 2 (1+ p )m m u L p t.

23 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 825 Since r = 1 + p 2α, the right-han sie can be written as Cν ( 1) u L p t 2 (r+ 2α )m m u L p2 2α(1 1 )(m ) t. For >1, we have, accoring to (2.7), 2 (r+ 2α )m m u L p 2 2α(1 1 )(m ) C 2 (r+ 2α )m m u L p Therefore, M 31 t Cν ( 1) u L p t 2 (r+ 2α )m m u L p t m = Cν ( 1) u L p t u L ( ). (,T ); B r+ 2α 1. Therefore, N 3 Cν ( 1) 2 (r+ 2α ) u L p t u L ( (,T ); B r+ 2α ) = Cν ( 1) u 2 L ( (,T ); B r+ 2α ). (5.14) Inserting the estimates (5.5), (5.13) an (5.14) in (5.12), we obtain (5.11). The a priori estimates of Theorems ( 5.1 an 5.2 can ) be extene to solutions of the euations in (4.4). The boun in L (,T); B r+ 2α is state in Theorem 5.3, while ( ) the boun in L (,T); B r+ 2α is provie in Theorem 5.4. These theorems can be shown by combining the arguments in the proofs of Theorems 4.3, 5.1 an 5.2, so we omit the etails. Theorem 5.3. Let 2 <. Assume either 2 1 <αan p = 2 or 2 1 <α 1 an 2 <p<. Let r = 1 + p 2α + 2α. Assume that F B r an ( ) v, w L (,T); B r+ 2α for some T>. Then any solution of (4.4) an (4.5) satisfies F L ( (,T ); B r+ 2α ) Cν 1 (1 E (T )) 2 r F L p +Cν ( 2) v ( L (,t); B r+ 2α ) w L ( (,t); B r+ 2α ) t. (5.15)

24 826 J. Wu In particular, (5.15) hols when the first term on the right of (5.15) is replace by Cν 1 F B r. Theorem 5.4. Let 1 <. Assume either 2 1 <αan p = 2 or <p<. Let r = 1 + p 2α. Assume that F B r an ( ) v, w L (,T); B r+ 2α <α 1 an for some T>. Then any solution of (4.4) an (4.5) satisfies F L ( (,T ); B r+ 2α ) Cν 1 (1 E (T )) 2 r F L p +Cν ( 1) v L ( (,T ); B r+ 2α ) w L ( (,T ); B r+ 2α ). (5.16) In particular, (5.16) hols when the first term on the right of (5.16) is replace by Cν 1 F B r. 6. Existence an Uniueness This section presents three existence an uniueness results for the GNS euations. The first one asserts the global existence an uniueness of solutions in B r (R ) with r = 1 2α + p for initial ata that are comparable to ν in B r (R ). The secon one establishes the local existence an uniueness of solutions in B r (R ) for arbitrarily large ata in B r (R ). The thir one concerns solutions in L ((,T); B s+ 2α (R )) with s = 1 2α + p + 2α. The precise statements are given in Theorems 6.1, 6.2 an 6.3. Theorem 6.1. Let 1. Assume either 2 1 <αan p = 2 or <p<. Let r = 1 2α + p. Assume that u B r (R ) satisfies u B r C ν <α 1 an for some suitable constant C epening on an p only. Then the GNS euations (4.1),(4.2) an (4.3) have a uniue global solution u satisfying u C([, ); B r ) L ((, ); B r+ 2α ) for 1 <, u C([, ); B p, r ) L 1 ((, ); B 1+ p p, ) for =. In aition, for any t>, u(,t) B r + Cν u L ((,t); B r+ 2α C 1 ν for 1 <, ) u(,t) B p, r + Cν u L1 ((,t); B 1+ p C 1 ν for =, p, ) for some constants C an C 1 epening on an p only.

25 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 827 Although this theorem oes not explicitly mention the case when 1 p<2, we can easily exten it to cover any initial atum u B r with 1 p<2. In fact, by the Besov embeing (see Part 2) of Proposition 2.1), B r 1 p 1, (R ) B r 2 p 2, (R ) for any 1,1 p 1 p 2 an r 1 = r 2 + ( p 1 1 p 1 2 ). Thus, if u B r with 1 p<2an r = 1 2α + p, then u B r 2 p 2, for any p 2 > 2 an r 2 = 1 2α + p 2. Theorem 6.1 then implies that the GNS euations have a uniue global solution associate with any u B r with 1 p<2an r = 1 2α + p. Proof of Theorem 6.1. To apply the contraction mapping principle, we write the GNS euations in the integral form u(t) = Gu(t) exp( ν( ) α t)u exp( ν( ) α (t s))p(u u)(s)s, where exp( ν( ) α t) for each t is a convolution operator with exp( ν( ) α t)(ξ) = exp( ν(2π ξ ) 2α t). We shall only provie the proof for the case when 1 < since the proof for = is analogous. To set up, we write ( ) X = C([, ); B r ), Y = L (, ); B r+ 2α, Z = X Y. For the norm in Z, we choose u Z = max{ u X + Cν u Y }, where C is a suitable constant epening on, p an only. In aition, we use D to enote the subset D ={u Z : u Z C 1 ν}. We aim to show that G is a contractive map from D to D. For notational convenience, we write Gu as where u an F(v,w)are given by F(v,w) = Gu = u F (u, u), u = exp( ν( ) α t)u, Obviously, u satisfies the euation exp( ν( ) α (t s))p(v w)(s)s. t u + ν( ) α u =, u (x, ) = u (x).

26 828 J. Wu As shown in the proof of Theorem 4.1, we have u (,t) B + Cν u(,s) B r r+ 2α s u B. r Conseuently, u Z C ν. To boun F in D, we first notice that F satisfies t F + ν( ) α F = P(u u), F (u, u)(x, ) =. Applying the result of Theorem 4.3, we have t F B r + Cν F B r+ 2α Therefore, for u D an suitable C 1, C u B r u B r+ 2α. F (u, u) Z C 1 ν. Thus, G maps D to D. To see that G is contractive, we write the ifference Gu Gv into two parts, namely, Since F (u, u v) satisfies Gu Gv = (F (u, u v) + F(u v, v)). t F + ν( ) α F = P(u (u v)), F (u, u)(x, ) =. Again the result of Theorem 4.3 shows F (u, u v) Z C u Z u v Z. F(u v, v) amits a similar boun. Therefore, Gu Gv Z C( u Z + v Z ) u v Z. For suitable C 1, C( u Z + v Z )<1 an G is contractive. The result of Theorem 6.1 then follows from the contraction mapping principle. We now state an prove the local existence an uniueness result. For the sake of conciseness, the statement for the case when = is omitte from the theorem. Theorem 6.2. Let 1 <. Assume either 2 1 <αan p = 2 or 2 1 <α 1 an 2 <p<. Let r = 1 2α + p. Assume u B r (R ). Then there exists a T > such that the GNS euations (4.1),(4.2) an (4.3) have a uniue solution u on [,T)in the class C([,T); B r ) L ((,T); B r+ 2α ).

27 Fractional Laplacians an Generalize Navier-Stokes Euations in Besov Spaces 829 Proof. The proof combines the contraction mapping principle with the a priori bouns in Theorems 5.2 an 5.4. The operator G is the same as in the proof of the previous theorem, but the functional setting is ifferent. For T> an R> to be selecte, we efine Z 1 = L ((,T); B r+ 2α ), D 1 = { u Z 1 : u Z1 R }. The goal is to show that G is a contractive map from D 1 to D 1. We again write Gu = u F (u, u). Since u satisfies t u + ν( ) α u =, u (x, ) = u (x), we can show as in the proof of Theorem 5.2, u Z 1 Cν 1 (1 E (T )) 2 r u L p, where E (t) exp( Cν2 2α t). Since u B r, this ineuality especially implies that u Z1 is finite. In aition, by the Dominate Convergence Theorem, (1 E (T )) 2 r u L p ast. Therefore, u Z1 is small for small T>. Since F (u, u) satisfies t F + ν( ) α F = P(u u), F (u, u)(x, ) =, we have, accoring to Theorem 5.4, F (u, u) Z1 C u 2 Z 1 CR 2 for any u D 1. Thus for small T> an suitable R>, Gu D 1. As in the proof of Theorem 6.1, we can show by applying Theorem 5.4 again that Gu Gv Z1 C( u Z1 + v Z1 ) u v Z1 2CR u v Z1, which implies G is a contraction for suitable selecte R. By applying the contraction mapping principle, we fin that the GNS euations have a solution in Z 1, namely, in L ((,T); B r+ 2α ). To show u B r, we can erive similarly as in the proof of Theorem 5.2 the boun u(,t) B r u B r + C u 2 Z 1 for any solution u of the GNS euations. Thus, for t [,T), u(,t) is in B r. This completes the proof of Theorem 6.2. The space L ((,T); B r+ 2α ) was use in the proof of the previous theorem. If we use L ((,T); B s+ 2α ) instea, we have the following local existence an uniueness theorem. This result can be proven by the contraction mapping principle combine with the a priori estimates in Theorems 5.1 an 5.3. We omit etails of the proof.

28 83 J. Wu Theorem 6.3. Let 2 <. Assume either 2 1 <αan p = 2 or 2 1 <α 1 an 2 <p<. Let s = 1 2α + p + 2α. Assume u B s (R ). Then there exists a T>such that the GNS euations have a uniue solution u on [,T)satisfying C([,T); B s ) L ((,T); B s+ 2α ). Appenix We provie the proof of Proposition 3.5 in Sect. 3. Proof of Proposition 3.5. (a) When p = 2, we have D(f ) = f( ) α fx= R α f 2 x. R (A.1) Since α = 1 an = 2, we obtain (3.5) by applying the Sobolev embeing W r,r (R ) L (R ) (A.2) vali for any r (1, ) an [r, ). The ineuality in (b) is a conseuence of (A.1) an the Sobolev embeing W s,r (R ) L r rs (R ) (A.3) for any sr <. To prove (c), we first apply the pointwise ineuality in Proposition 3.2 to obtain ( D(f ) C f p/2 ( ) α f p/2) ( x = C α f p/2) 2 x. R R By (A.2), we have for any [p, ), D(f ) C f p L p. 2 Since 2 <p< p 2, we have the following interpolation ineuality: 2 4 p 4 f L p C f L 2 p 2 p 4 f. L p 2 For any f L 2 (R ) with f L 2, this ineuality is euivalent to f p L p 2 C f (1+β)p L p f βp L 2 with β = 2 4 p 2. The proof of () is similar to that of (c) except that we use the Sobolev embeing (A.3) instea of (A.2). This completes the proof.

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