Dynamics of PDE, Vol.4, No.3, 227-245, 2007
Homothetic Variant of Fractional Sobolev Space with Application to Navier-Stokes System
Jie Xiao
Communicated by Y. Charles Li, received October 28, 2006.
n Abstract. It is proved that for α ∈ (0, 1), Qα(R ), not only as an interme- diate space of W 1,n(Rn) and BMO(Rn) but also as a homothetic variant of 2n ˙ 2 Rn n−2α Rn Sobolev space Lα( ) which is sharply imbedded in L ( ), is isomor- phic to a quadratic Morrey space under fractional differentiation. At the same n n time, the dot product ∇·(Qα(R )) is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system R1+n ∞ × Rn in + = (0, ) .
Contents 1. Introduction and Summary 227 2. Proof of Theorem 1.2 233 3. Proof of Theorem 1.4 239 References 244
1. Introduction and Summary We begin by the square form of John-Nirenberg’s BMO space (cf. [13]) which plays an important role in harmonic analysis and applications to partial differen- tial equations. For a locally integrable complex-valued function f defined on the Euclidean space Rn, n 2, with respect to the Lebesgue measure dx, we say that f is of BMO class, denoted≥ f BMO = BMO(Rn), provided ∈ 1 n 2 2 f = sup `(I) − f(x) f dx < . k kBMO − I ∞ I ZI 1991 Mathematics Subject Classification. Primary 35Q30, 42B35; Secondary 46E30. 1,n n n n Key words and phrases. Homothety, W (R ), Qα(R ), BMO(R ), quadratic Morrey space, sharp Sobolev imbedding, incompressible Navier-Stokes system. The author was supported in part by both NSERC (Canada) and Dean of Science Startup Fund (MUN, Canada).
c 2007 International Press 227 228 JIE XIAO
Here and elsewhere sup means that the supremum ranges over all cubes I Rn I ⊂ with edges parallel to the coordinate axes; `(I) is the sidelength of I; and fI = n − `(I) I f(x)dx stands for the mean value of f over I. On the basis of the semi-norm , a large scale of function spaces has R BMO been introduced in [11], as defined bkelo· kw.
Definition 1.1. For α ( , ), let Qα be the space of all measurable complex-valued functions f on∈ R−∞n obeying∞
1 2 2 2α n f(x) f(y) f = sup `(I) − | − | dxdy < . k kQα x y n+2α ∞ I ZI ZI | − | This Qα is a natural extension of BMO according to the following result (proved in [11] and [30]):
BMO, α ( , 0), 1,n ∈ −∞ Qα = New space between W and BMO, α [0, 1), ∈ C, α [1, ). ∈ ∞ Here W 1,n = W1,n(Rn) is the homothetic energy space of all C1 complex-valued functions f on Rn with
1 n n f W 1,n = f(x) dx < . k k Rn |∇ | ∞ Z More importantly, Qα, α (0, 1), can be regarded as the homothetically invariant ∈ ˙ 2 ˙ 2 Rn counterpart of the homogeneous Sobolev space Lα = Lα( ) which consists of all complex-valued functions f on Rn with the α-energy
1 f(x) f(y) 2 2 f L˙ 2 = | − n+2α| dxdy < . k k α Rn Rn x y ∞ Z Z | − |
The reason for saying this is at least that Qα and ˙ 2 enjoy the following k · k k · kLα property: n α 2 f φ Qα = f Qα and f φ ˙ 2 = λ − f ˙ 2 k ◦ k k k k ◦ kLα k kLα n for any homothetic map x φ(x) = λx + x0; λ > 0, x0 R . In the six year period 7→since the paper [11] appeared,∈ it has been found that Qα is a useful and interesting concept; see also [1], [2], [24], [8], [9], [16], [23], [7] and [10]. This means that the study of this new space has not yet ended up – in fact, there are many unexplored problems related to Qα. In this paper, although not attacking one of those open problems in Section 8 in [11], we go well beyond the previous results by studying the relation between this space and the quadratic Morrey space, but also giving an application of the induced facts to the incompressible Navier-Stokes system. To deal with the former, it is necessary to consider the following variant of [8, Theorem 3.3] that expands Fefferman-Stein’s basic result for BMO in [12]: Given n a C∞ real-valued function ψ on R with
1 (n+1) n x ψ L , ψ(x) . (1 + x )− , ψ(x)dx = 0 and ψt(x) = t− ψ( ). ∈ | | | | Rn t Z HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM229
Then for a measurable complex-valued function f on Rn, r 2α n 2 (1+2α) (1.1) f Qα sup r − f ψt(y) t− dydt < . ∈ ⇐⇒ x Rn,r (0, ) 0 y x On the other hand, if 2 n xj x ψ (x) = (4π)− 2 exp | | for j = 1, ..., n, j − 2 − 4 then t2∆ (ψj )t(x) = t∂j e (x, 0) and so, for α (0, 1) and a measurable complex-valued function f on Rn, ∈ r 2α n t2∆ 2 1 2α (1.3) f Qα sup r − e f(y) t − dydt < . ∈ ⇐⇒ x Rn,r (0, ) 0 y x α (i) Qα = ( ∆)− 2 2,n 2α , BMO is proper with − L − → n+2α f n 2 sup k kBMO , f Q >0 f Qα ≤ s 2 k k α k k n where a measurable complex-valued function f on R belongs to 2,n 2α if and only if L − 1 2α n 2 − 2 f 2,n−2α = sup `(I) f(x) fI dx < . k kL | − | ∞ I ZI α 2n 2 2 − (ii) L˙ = ( ∆)− 2 L , L n 2α is sharp with α − → 1 α 1 2 n 2n n 2α 2πy (1,0,...,0) 2 2 f − Γ − k kL n 2α −2 Γ(n) e− · 1 sup = n+2α n | n+2α − | dy . f 2 Rn y f L˙ 2 >0 L˙ Γ 2 ! Γ 2 ! k k α k k α Z | | 1 n Rn 1 (iii) Qα−; = (Qα) , where a tempered distribution f on belongs to Qα−; if and only∞if ∇ · ∞ 1 r2 2 2α n t∆ 2 α f −1 = sup r − e f(y) t− dydt < . Qα;∞ k k x Rn,r (0, ) 0 y x Note that 2,n 2α is the so-called Morrey space of square form (cf. [3] and L − [22]) and 2,n = BMO. So Theorem 1.2 (i) keeps true for α = 0 in the sense of ( ∆)0BLMO = BMO. Quite surprisingly, this part corresponds nicely to − α Strichartz’s ( ∆)− 2 BMO-equivalence [26, Theorem 3.3]: − 1 2 2 α n f(x) f(y) f ( ∆)− 2 BMO sup `(I) − | − | dxdy < . ∈ − ⇐⇒ x y n+2α ∞ I ZI ZI | − | The imbedding without best constant in Theorem 1.2 (ii) is well-known (see for example [21, Theorem 1] and the related references therein) and very useful in the study of the semi-linear wave equations (cf. [20]). A close look at both (i) and (ii) reveals that Qα behaves like a homothetic Sobolev space. In addition, Theorem 1.2 1 n 1 (iii) extends [15, Theorem 1]: BMO− = (BMO) that just says: f BMO− if and only if there are f BMO such that∇ · f = n ∂ f . ∈ j ∈ j=1 j j As with the latter, we recall that the Cauchy problem for the incompressible Navier-Stokes system on the half-space R1+n = (0P, ) Rn: + ∞ × ∂ u ∆u + (u )u p = 0, in R1+n; t − · ∇ − ∇ + R1+n (1.4) u = 0, in + ; ∇ · n u t=0 = a, in R | is to establish theexistence of a solution (velocity) u = u(t, x) = u1(t, x), ..., un(t, x) with a pressure p = p(t, x) of the fluid at time t [0, ) and position x Rn ∈ ∞ ∈ that assumes the given data (initial velocity) a = a(x) = (a1(x), ..., an(x)). If the solution exists, is unique, and depends continuously on the initial data (with respect to a given topology), then we say that the Cauchy problem is well-posed in that topology. HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM231 Of particularly significant is the invariance of (1.4) under the scaling changes: u(t, x) u (t, x) = λu(λ2t, λx); 7→ λ p(t, x) p (t, x) = λ2p(λ2t, λx); 7→ λ a(x) aλ(x) = λa(λx). 7→ So if (u(t, x), p(t, x), a(x)) satisfies (1.4) then (uλ(t, x), pλ(t, x), aλ(x)) is a solution of (1.4) for any λ > 0. This leads to a consideration of the well-posedness for (1.4) with a Cauchy data being of the scaling invariance. Through the scale invariance n n n a n n = (a ) n = (a ) n = a n = a n n , k λk(L ) k λ j kL k j λkL k jkL k k(L ) j=1 j=1 j=1 X X X Kato proved in [14] that (1.4) has mild solutions locally in time if a (Ln)n and ∈ globally if a (Ln)n is small enough (for some generalizations of Kato’s result, see e.g. [28] andk k[31]). Furthermore, in [15], Koch-Tataru found, among other results, 1 n that (1.4) still has mild solutions locally in time if a (V MO− ) and globally once ∈ 1 n n r2 2 n t∆ 2 aj BMO−1 = sup r− e aj(y) dydt k k Rn | | j=1 j=1 x ,r (0, ) 0 y x 1 r2 2 − 2α n t∆ 2 α f Q 1 = sup r − e f(y) t− dydt < ; k k α;T x Rn,r (0,T ) 0 y x ∞ g Xα;T = sup √t g(t, ) L k k t (0,T ) k · k ∈ 1 r2 2 2α n 2 α + sup r − g(t, y) t− dydt < . x Rn,r2 (0,T ) 0 y x In particular, we write 1 1 1 1 Q0;−T = BMOT− , V Q0− = V MO− and X0;T = XT . Two immediate comments are given below: If f (x) = λf(λx) and g (t, x) = λg(λ2t, λx) for λ, t > 0 and x Rn, λ λ ∈ then − − fλ 1 = f 1 and gλ X ∞ = g X ∞ ; k kQα;∞ k kQα;∞ k k α; k k α; − that is, 1 and X ∞ are scaling invariant. Second, we have k · kQα;∞ k · k α; n 1 1 L Q− BMO− and X X . ⊆ α;1 ⊆ 1 α;1 ⊆ 1 To see this, note that f BMO−1 = f −1 f −1 . Additionally, recall that k k k kQ0;∞ ≤ k kQα;∞ n n−p 1+ p t∆ f B˙ p,− , p > n if and only if e f Lp . t 2p for t (0, 1]; see also [4]. Using ∞ n ∈ k k 1+ p ∈ H¨older’s inequality, we obtain that if f B˙ p,− , p > n and r (0, 1), then ∈ ∞ ∈ r2 r2 t∆ 2 α n(p 2) t∆ 2 α n 2α e f(y) t− dydt . r − e f Lp t− dt . r − 0 y x 2. Proof of Theorem 1.2 To verify Theorem 1.2 (i), we must understand the quadratic Morrey space in spirit of (1.2). Lemma 2.1. Given α (0, 1). Let f be a measurable complex-valued function on Rn. Then ∈ (2.1) r 2 2α n t√ ∆ f 2,n 2α sup r − ∂te− − f(y) tdydt < . ∈ L − ⇐⇒ x Rn,r (0, ) 0 y x Proof. Assume f 2,n 2α. Recall ∈ L − n+1 2 n+1 2 1 + x (n + 1)Γ( 2 )π− t√ ∆ ψ0(x) = | | − n+3 and (ψ0)t(x) = t∂te− − (x, 0). (1 + x 2) 2 | | For any ball B = y Rn : y x < r in Rn, let 2B be the double ball of B and 1 { ∈ | − | } f2B = 2B − 2B f be the mean value of f on 2B. Note that E stands for the Lebesgue| measure| of a set E. Let also | | R f1 = (f f2B)12B , f2 = (f f2B)1Rn 2B and f3 = f2B, − − \ where 1E stands for the characteristic function of a set E. Since Rn ψ0(x)dx = 0, we conclude R t√ ∆ t∂ e− − f(y) = (ψ ) f(y) = (ψ ) f (y) + (ψ ) f (y). t 0 t ∗ 0 t ∗ 1 0 t ∗ 2 Concerning the first term (ψ ) f (y), we have the following estimates 0 t ∗ 1 r 2 1 ∞ 2 1 (ψ ) f (y) t− dydt (ψ ) f (y) t− dtdy | 0 t ∗ 1 | ≤ | 0 t ∗ 1 | Z0 ZB ZB Z0 ∞ 2 1 (ψ0)t f1(y) t− dt dy ≤ Rn | ∗ | Z Z0 2 . f(y) f2B dy 2B | − | Zn 2α 2 . r − f 2,n−2α , k kL 2 where we have used the L -boundedness: (f1) L2 . f1 L2 for the Littlewood- Paley -function of f kG k k k G 1 1 2 2 ∞ t√ ∆ 1 (f1)(y) = t∂te− − f1(y) t− dt . G 0 Z At the same time, if (t, y) (0, r) B and Bk is the ball with center x and radius 2kr, then we take the Cauc∈ hy-Sc×hwarz inequality into account, and obtain the following inequalities for (ψ ) f (y): 0 t ∗ 2 234 JIE XIAO t f(z) f2B (ψ0)t f2(y) . | − n+1| dz | ∗ | Rn 2B (t + x z ) Z \ | − | t f(z) f2B . | − n+1| dy Rn 2B (r + x z ) Z \ | − | ∞ t f(z) f2B . t | − n+1| dz Bk (r + x z ) Xk=1 Z | − | ∞ k (n+1) . t (2 r)− f(z) f dz | − 2B| k=1 ZBk Xα n 1 . tr − − f 2,n−2α . k kL Consequently, r 2 1 n 2α 2 (ψ0)t f2(y) t− dydt . r − f 2,n−2α . | ∗ | k kL Z0 ZB The above-established estimates yield that the supremum in (2.1) is finite. To handle the converse part, denote by S(I) = (t, x) R1+n : t (0, `(I)] and x I . ∈ + ∈ ∈ n o the Carleson box based on a given cube I Rn. Suppose the supremum condition in (2.1)⊂ is satisfied. Then 2α n t√ ∆ 2 (2.2) sup `(I) − ∂ e− − f(y) tdydt < . t ∞ I ZS(I) In order to verify f 2,n 2α, we consider the projection operator ∈ L − 1 Πψ0 F (x) = F (t, y)(ψ0)t(x y)t− dydt, R1+n − Z + and prove that if 1 2 2α n − 2 1 F α = sup `(I) F (t, y) t− dydt < k kC | | ∞ I ZS(I) ! then for any cube J Rn, ⊂ n 2α 2 − 2 (2.3) Πψ0 F (x) Πψ0 F dx . `(J) F α . | − J | k kC ZJ Given a cube J Rn and λ > 0, define λJ as the cube concentric with J and with sidelength `(λJ)⊂= λ`(J). Let F = F and F = F F . Then by a result of 1 |S(2J) 2 − 1 Coifman-Mayer-Stein [6, p. 328-329], HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM235 2 2 Πψ0 F1(x) dx Πψ0 F1(x) dx | | ≤ Rn | | ZJ Z 2 1 . F1(t, y) t− dydt R1+n | | Z + 2 1 . F (t, y) t− dydt | | ZS(2J) n 2α 2 − . F α `(J) . k kC On the other hand, from the definition of Πψ0 , the boundedness of ψ0 and the Cauchy-Schwarz inequality it follows that 2 Πψ0 F2(x) dx J | | Z 2 1 = (ψ0)t(x y)F2(t, y)t− dydt dx J R1+n − Z Z + 2 1 (ψ0)t(x y) F2(t, y ) t− dydt dx ≤ J R1+n S(2J) | − || | Z Z + \ ! 2 ∞ 1 (ψ0)t(x y) F2(t, y) t− dydt dx ≤ J S(2k+1J) S(2kJ) | − || | Z k=1 Z \ ! X 2 ∞ k 1 . 2 `(J) − (ψ0)t(x y) F2(t, y) dydt dx J S(2k+1J) S(2k J) | − || | Z k=1 Z \ ! X 2 ∞ k (n+1) . 2 `(J) − F2(t, y) dydt dx J S(2k+1J) S(2k J) | | ! Z Xk=1 Z \ 1 2 2 ∞ n k 2 2 1 . 2 `(J) − F2(t, y) t− dydt dx J S(2k+1J) S(2k J) | | Z k=1 Z \ ! X n 2α 2 − . F α `(J) . k kC Thus 2 2 Π F (x) Π F dx . Π F (x) dx ψ0 − ψ0 J ψ0 ZJ ZJ 2 2 . Πψ0 F1(x) dx + Πψ0 F2(x) dx J J Z n 2α Z 2 − . F α `(J) , k kC namely, (2.3) holds. Applying (2.3) to Π (ψ ) f which equals f under (2.2), ψ0 0 t ∗ we achieve f 2,n 2α. ∈ L − PROOF OF THEOREM 1.2 (i). Since the imbedding with that constant can be α derived from a routine computation, it suffices to show Qα = ( ∆)− 2 2,n 2α. − L − 236 JIE XIAO 1+α t√ ∆ For f 2,n 2α, let F (t, y) = t ∂te− − f(y). Then by Lemma 2.1 we get ∈ L − r 2α n 2 1 2α 2 sup r − F (t, y) t− − dydt . f 2,n−2α , x Rn,r (0, ) 0 y x PROOF OF THEOREM 1.2 (ii). According to [27, p. 175], we use Fubini’s theo- 1 rem, Plancherel’s formula, the change of variables y = x − z and an orthonormal transform to obtain | | 2 2 (n+2α) f L˙ 2 = f(x + y) f(x) dx y − dy k k α Rn Rn | − | | | Z Z 2πy x 2 (n+2α) 2 = e− · 1 y − dy fˆ(x) dx Rn Rn | − | | | | | Z Z 2πz (1,0,...,0) 2 (n+2α) 2 2α = e− · 1 z − dz fˆ(x) x dx. Rn | − | | | Rn | | | | Z Z α ˙ 2 2 2 ˆ Accordingly, L = ( ∆)− L . Note that f(x) = n f(y) exp( 2πix y)dy and α − R − · t√ ∆ R e− − f(x) = fˆ(y) exp 2π(iy x + y t) dy. Rn − · | | Z So, by differentiation and integration (cf. [25 , p. 83]), we find t√ ∆ 2 2 2 ˆ 2 e− − f L2 = 8π x f(x) exp( 4π x t)dx. k∇ k Rn | | | | − | | Z Consequently, 2 ∞ t√ ∆ 2 1 2α 8π Γ 2(1 α) 2α 2 2 − ˆ (2.4) e− − f L t − dt = 2(1 α) x f(x) dx. k∇ k (4π) Rn | | | | Z0 − Z Moreover, the preceding consideration actually tells us that proving the desired sharp imbedding amounts to verifying the following best inequality 2 ∞ t√ ∆ 2 1 2α (2.5) f 2n τn,α e− − f L2 t − dt, k kL n−2α ≤ k∇ k Z0 where 2α 1 4α n 2α n 2 − Γ −2 Γ(n) τn,α = . n+2α n παΓ 2(1 α) Γ Γ 2 ! − 2 HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM237 To this end, we use f, g as f(x)g(x)dx, and then get h i Rn R 2 2 2α 2 2α 2 f, g = fˆ, gˆ x fˆ(x) dx x − gˆ(x) dx . |h i| |h i| ≤ Rn | | | | Rn | | | | Z Z Because (cf. [19, Corollary 5.10]) n 2α 2 n 2α 2α 2 π − Γ −2 g(x)g(y) x − gˆ(x) dx = n 2α dxdy, Rn | | | | Γ(α) Rn Rn x y Z Z Z | − | − it follows from Lieb’s sharp version [18] of the Hardy-Littlewood-Sobolev inequality that 2α παΓ n 2α n 2 −2 Γ(n) 2 2α ˆ 2 f, g n+2α n g 2n x f(x) dx. |h i| ≤ Γ Γ k kL n+2α Rn | | | | 2 2 ! Z