# Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

Authors: Weiliang Xiao; Jiecheng Chen; Dashan Fan; Xuhuan Zhou
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spacesFB˙p,q1-2β+3/p′. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical case β=1/2. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.


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## Body

## 1. Introduction

We study the mild solutions to the fractional Navier-Stokes equations inR+×R3 as follows:
(1)ut+μ(-Δ)βu+(u·∇)u+∇π=0,(t,x)∈R+×R3;∇·u=0,(t,x)∈R+×R3;u(0,x)=u0(x),x∈R3.
Here u(t,x)=(u1(t,x),u2(t,x),u3(t,x)) denotes the velocity vector, μ>0 is the viscosity coefficient, and the scalar function π denotes the pressure. The initial data u0 is a divergence free vector field and the operator (-Δ)β is the Fourier multiplier with symbol |ξ|2β.The fractional Navier-Stokes equations, which are also called generalized Navier-Stokes equations, enjoy an invariance under the scaling(2)uλ(t,x)=λ2β-1u(λ2βt,λx),pλ(t,x)=λ4β-2p(λ2βt,λx),u0,λ=λ2β-1u0(λx).
We say that a function space is β-critical for (1) if its norm is invariant under the scaling u0(x)→λ2β-1u0(λx). There are many examples of critical spaces, for instance, BMO-(2β-1), B˙∞,∞-(2β-1), and the spaces we will discuss in this paper.The classical incompressible Navier-Stokes equations (i.e.,β=1) have been intensively studied. Leray first [1] introduced the concept of weak solutions and obtained the global existence of weak solutions. Fujita and Kato [2] gave a different approach to study the equations in their equivalent form of integral equations and proved the well-posedness in the space frame H˙1/2. A series study of mild solutions in different function spaces then arose, for instance, Kato [3] in Lebesgue space L3(R3), Cannone [4] in Besov space B˙p,∞-1+3/p, and the important well-posedness in BMO-1 by Koch and Tataru [5]. These works naturally lead one to study the well-posedness in the largest critical space B˙∞,∞-1. In fact, all the above spaces are critical spaces and satisfy the following continuous embeddings in the 3 dimensions:
(3)H˙1/2↪L3↪B˙p,∞(p<∞)-1+3/p↪BMO-1↪B˙∞,∞-1.
However, in the space B˙∞,∞-1, the Navier-Stokes equations are ill-posedness (see Bourgain and Pavlović [6] and Cheskidov and Shvydkoy [7]).As for the generalized case (1), Lions [8] proved the global existence of classical solutions in 3 dimensions when β≥5/4 (see also Wu [9] in n dimensions). For the important case β<5/4, Wu [10, 11] studied the well-posedness in B˙p,q1-2β+3/p. Inspired by Xiao [12] in the classical case (β=1), Li and Zhai [13, 14] studied (1) in some critical Q-type spaces for β∈(1/2,1), and Zhai [15] showed the well-posedness in BMO-(2β-1) when β∈(1/2,1). For the biggest critical space B˙∞,∞-(2β-1), Yu and Zhai [16] proved the well-posedness when β∈(1/2,1), Cheskidov and Shvydkoy [17] showed the ill-posedness when β∈[1,5/4). Very recently, Deng and Yao [18] studied (1) in Triebel-Lizorkin spaces F˙α,r-β and obtained the well-posedness in F˙3/(β-1),2-β and ill-posedness in F˙3/(β-1),r-β (r>2) in the case β∈(1,5/4).In this paper, we will study (1) in the Fourier-Besov spaces FB˙p,qs. We observe that although the Fourier-Besov spaces FB˙p,qs appear in the literature very recently, they have received a lot of attentions in studying Navier-Stokes equations, although sometime people gave these spaces several different names. An early paper by Cannone and Karch [19] worked in the space PMa, which is in fact the space FB˙∞,∞a (see Section 2 for details). Biswas and Swanson [20] studied the Gevrey regularity of Navier-Stokes equations in FB˙p,p2-3/p. Konieczny and Yoneda [21] used FB˙p,qs to study the Navier-Stokes equations with Coriolis (see also Fang et al. [22]). Lei and Lin [23] proved global existence of mild solutions in X-1, which is in fact equal to the space FB˙1,1-1. Cannone and Wu [24] extended the result in [23] to the Fourier-Herz spaces B˙qs. We may notice that B˙qs=FB˙1,q-1. Also, some properties of solutions in the space X-1 have been studied recently; see Zhang and Yin [25] for the blow-up criterion and Benameur [26] for the long time decay. All the above-mentioned works are involved in the classical Navier-Stokes equations. Those indicate that the Fourier-Besov spaces FB˙p,qs might be good work frames in the study of Navier-Stokes equations. Inspired by these observations, in this paper, we will study generalized Navier-Stokes equations in FB˙p,qs. We obtain a global well-posedness result which is more general than those in [23, 24]. Particularly, our well-posedness is also valid in the critical case β=1/2. Moreover, the long time decay of the solutions in Fourier-Besov spaces is also proved, which fully extends the result of [26].Throughout this paper, the notationA~B means that there exist positive constants C1≤C2 such that C1A≤B≤C2A. We use B˙p,qs to denote the classical homogenous Besov spaces and H˙s the homogenous Sobolev spaces. Also, C denotes a positive constant which may differ in lines if not being specified; p′ is the number satisfying 1/p+1/p′=1 for 1≤p≤∞. The inverse Fourier transform is denoted by F-1.We organize the paper as follows. In Section2 we give the definition of Fourier-Besov spaces and discuss some basic properties of these spaces. Our main results are also stated in this section. In Section 3 we prove the global well-posedness and in Section 4 we prove the long time decay property.


## 2. Preliminaries and Main Results

We first introduce the definition of Fourier-Besov spaces inn dimensions. Let φ∈Cc∞(Rn) be a radial real-valued smooth function such that 0≤φ(ξ)≤1 and
(4)supp⁡φ⊂{ξ∈Rn:34≤|ξ|≤83},∑j∈Z‍φ(2-jξ)=1,foranyξ≠0.
We denote φj(ξ)=φ(2-jξ) and P the set of all polynomials. The space of tempered distributions is denoted by S′.Definition 1.
Fors∈R,1≤p,q≤∞, set
(5)∥f∥FB˙p,qs={(∑j∈Z‍2jsq∥φjf^∥Lpq)1/q,q<∞;sup⁡j∈Z2js∥φjf^∥Lp,q=∞.
One defines the homogeneous Fourier-Besov space FB˙p,qs as
(6)FB˙p,qs={f∈S′P:∥f∥FB˙p,qs<∞}.We see that the Fourier-Besov spaces are defined in a similar way with the classical homogeneous Besov spaces, but there are lack of the inverse Fourier transform. This allows us to derive estimates by Hölder's inequality directly, instead of using Bernstein's inequality. Now we explain that Fourier-Besov spaces contain some known spaces applied in studying Navier-Stokes equations.Cannone and Karch [19] introduced the spaces PMa as follows:
(7)PMa={v∈S′:v^∈Lloc⁡1,0∥v∥PMa=esssup⁡ξ∈Rn|ξ|a|v^(ξ)|<∞}.
We easily see that PMa=FB˙∞,∞a.The norm of Fourier-Herz spacesB˙qs in [24] is defined as
(8)∥f∥B˙qs={(∑j∈Z2jsq∥φjf^∥L1q)1/q,q<∞;sup⁡j∈Z2js∥φjf^∥L1,q=∞.
Obviously, we have B˙qs=FB˙1,qs.The spaceX-1 introduced by Lei and Lin [23] is
(9)X-1={f∈S′(Rn):∫Rn‍|ξ|-1|f^|dξ<∞}.
We claim that X-1=FB˙1,1-1. This fact can be seen by the following proposition (proof in [21]).Proposition 2.
Define the spacesXs as
(10)Xs={f∈S′P:(∫Rn‍|ξ|sp|f^|pdξ)1/p<∞}.
Then one has Xs=FB˙p,ps and the norms are equivalent:
(11)∥f∥FB˙p,ps~(∫Rn|ξ|sp|f^|pdξ‍)1/p.We discuss some inclusion relationships inFB˙p,qs.Proposition 3.
Lets∈R, 1≤p, q≤∞. One has the follwoing.(1)
Ifp=2, then FB˙2,qs=B˙2,qs.(2)
Ifp<2, then FB˙p,qs⊂B˙p′,qs.(3)
Ifq1≤q2, then FB˙p,q1s⊂FB˙p,q2s.(4)
If1≤q≤∞, 1≤p1≤p2≤∞, and s1,s2∈R satisfy s1+n/p1=s2+n/p2, then
(12)FB˙p2,qs2⊂FB˙p1,qs1.(5)
Ifs=(1-θ)s1+θs2, 1/p=(1-θ)/p1+(1-θ)/p2, and 1/q=(1-θ)/q1+(1-θ)/q2 for 0≤θ≤1, then
(13)∥f∥FB˙p,qs≤∥f∥FB˙p1,q1s11-θ∥f∥FB˙p2,q2s2θ.Proof.
(1) is a consequence of Plancherel’s identity, and Hausdorff-Young’s inequality gives (2). Equation (3) is just the inclusionlq1⊂lq2 for 1≤q1≤q2≤∞. To conclude (4), we use Hölder’s inequality to get
(14)∥φjf^∥Lp1≤C2jn(1/p1-1/p2)∥φjf^∥Lp2.
Since s1 and s2 satisfy s1+n/p1=s2+n/p2, we immediately get
(15)2js1∥φjf^∥Lp1≤C2js2∥φjf^∥Lp2.
Taking the lq-norm on the above inequality we have
(16)∥f∥FB˙p1,qs1≤C∥f∥FB˙p2,qs2.
To prove (5), we have
(17)∥f∥FB˙p,qs=(∑j2jsq∥|φjf^|1-θ|φjf^|θ∥Lpq)1/q≤(∑j2j(1-θ)s1q∥φjf^∥Lp1(1-θ)q2jθs2q∥φjf^∥Lp2θq)1/q≤∥f∥FB˙p1,q1s11-θ∥f∥FB˙p2,q1s2θ.Now we are ready to state our main results. From now on in this paper we take the dimensionn=3.Definition 4.
Lets∈R, 1≤p, q≤∞, and I=[0,T), T∈(0,∞]. The space-time norm is defined on f(t,x) by
(18)∥f(t,x)∥Lr(I;FB˙p,qs)∶=(∑j∈Z2jsq∥φjf^∥Lr(I;Lp)q)1/q.Our first result is on the well-posedness of (1).Theorem 5.
Let1≤p, q≤∞, and 1/2<β<min⁡{1+3/p′,5/2}. Then there exists a constant C0=C0(β,p,q) such that, for any u0∈FB˙p,q1-2β+3/p′ with ∇·u0=0 satisfying
(19)∥u0∥FB˙p,q1-2β+3/p′<C0μ,
the Cauchy problem (1) admits a unique global mild solution u and
(20)u∈C([0,∞);FB˙p,q1-2β+3/p′)∩L1([0,∞);FB˙p,q1+3/p′),
and it satisfies
(21)∥u∥L∞([0,∞);FB˙p,q1-2β+3/p′)+μ∥u∥L1([0,∞);FB˙p,q1+3/p′)≤2(1+(169)β)∥u0∥FB˙p,q1-2β+3/p′.
Particularly, our result also holds in the critical case q=1 and β=1/2.Remark 6.
We emphasize that the caseβ=1/2 is important, since it is also the critical case for the fractional Navier-Stokes equations. Note that when β=1/2, the function spaces we work on are FB˙p,13/p′. All these spaces are embedded into FB˙1,10, which is the space that consists of all functions whose Fourier transforms are in L1 (see Proposition 2).Remark 7.
Note thatFB˙p,q1-2β+3/p′⊂B˙∞,∞-2β-1 by Proposition 3 and the space FB˙p,q1-2β+3/p′ are also critical spaces. In fact, for u0,λ=λ2β-1u0(λx), we have u0,λ^=λ2β-4u0^(λ-1ξ). Set
(22)gj(ξ)∶=φ(2-j+[log⁡2λ]-log⁡2λξ)u0,λ^(ξ).
Then we have
(23)2j(1-2β+3/p′)∥gj∥Lp=2j(1-2β+3/p′)∥φ(2-j+[log⁡2λ]-log⁡2λξ)λ2β-4u0^(λ-1ξ)∥Lp=2([log⁡2λ]-log⁡2λ)(1-2β+3/p′)×2(j-[log⁡2λ])(1-2β+3/p′)∥φ(2-j+[log⁡2λ]η)u0^(η)∥Lp~2(j-[log⁡2λ])(1-2β+3/p′)∥φ(2-j+[log⁡2λ]η)u0^(η)∥Lp.
This implies that
(24)(∑j∈Z2jq(1-2β+3/p′)∥gj∥Lpq)1/q~∥u0∥FB˙p,q1-2β+3/p′.
On the other hand, by
(25)φj(ξ)u0,λ^(ξ)=∑|k-j|≤2‍φj(ξ)gk(ξ),
we can easily deduce that
(26)∥u0,λ∥FB˙p,q1-2β+3/p′~‍(∑j∈Z2jq(1-2β+3/p′)∥gj∥Lpq)1/q.Unfortunately, Theorem5 is not suitable for the case β=1,p=1, in which similar existence has been proved by Cannone and Wu [24]. To address this case, we also get the following theorem.Theorem 8.
Let1≤p≤q≤2 and β∈(1/2,1+3/2p′]. Then there exists a constant C0=C0(β,p,q) such that, for any u0∈FB˙p,q1-2β+3/p′ with ∇·u0=0 satisfying
(27)∥u0∥FB˙p,q1-2β+3/p′<C0μ,
the Cauchy problem (1) admits a unique global mild solution u and
(28)u∈C([0,∞);FB˙p,q1-2β+3/p′)∩L1([0,∞);FB˙p,q1+3/p′),
and it satisfies
(29)∥u∥L∞([0,∞);FB˙p,q1-2β+3/p′)+μ∥u∥L1([0,∞);FB˙p,q1+3/p′)≤2(1+(169)β)∥u0∥FB˙p,q1-2β+3/p′.
Particularly, our result also holds in the critical case p=q=1 and β=1/2.Remark 9.
In comparison with Theorem5, although we have a limitation 1≤p≤q≤2, the regularity index β in Theorem 10 lies in a larger interval when p=1.Our third result is on the decay property of the global solutionsTheorem 10.
Let1≤p≤q≤2 and β∈(5/6,1]. Assume that u∈C([0,∞);FB˙p,q1-2β+3/p′) is a global solution of (1). One has
(30)limsup⁡t→∞∥u(t)∥FB˙p,q1-2β+3/p′=0.Remark 11.
Recently, Benameur [26] obtained the same property in the space X-1=FB˙1,1-1 for the classical Navier-Stokes equations (β=1). Our result improves and extends his result.


## 3. The Well-Posedness

First, we study the linear estimates of (1). For this purpose we consider the dissipative equation:
(31)ut+μ(-Δ)βu=F(t,x),(t,x)∈R+×R3;u(x,0)=u0(x),x∈R3.
It is easy to see that the equivalent integral equation of (31) is
(32)u(t,x)=e-μt(-Δ)βu0-∫0t‍e-μ(t-τ)(-Δ)βF(τ,x)dτ.
By taking F(t,x)=0 or u0(x)=0, we obtain the linear term or the nonlinear term of the equation, respectively. This indicates that the following lemma is very useful in our later proof.Lemma 12 (linear estimate).
LetI=[0,T), 0<T≤∞, s∈R, 1≤p, and q≤∞. Assume that u0∈FB˙p,qs and F∈L1(I;FB˙p,qs). Then the solution u(t,x) to the Cauchy problem (31) satisfies
(33)∥u∥L∞(I;FB˙p,qs)+μ∥u∥L1(I;FB˙p,qs+2β)≤(1+(169)β)(∥u0∥FB˙p,qs+∥F∥L1(I;FB˙p,qs)).Proof.
By taking the Fourier transform we have(34)∂tu^+μ|ξ|2βu^=F^.
Multiplying φj and taking the Lp-norm on both sides,
(35)ddt∥uj^∥Lp+(916)βμ22βj∥uj^∥Lp≤∥Fj^∥Lp,
where we denote uj^=φju^. Integrating with respect to time on [0,t), we get
(36)∥uj^∥Lp+(916)βμ22βj∥uj^∥L1(0,t;Lp)≤∥u0j^∥Lp+∥Fj^∥L1(I;Lp).
By the definition of FB˙p,qs and the triangle inequality for lq, it is easy to obtain our desired inequality.Next we consider the bilinear estimate, which is the key estimate in solving the Navier-Stokes equations.Lemma 13 (bilinear estimate).
Let1≤p, q≤∞, and 1/2<β<min⁡{1+3/p′,5/2} and set
(37)X=L∞(I;FB˙p,q1-2β+3/p′)∩L1(I;FB˙p,q1+3/p′),
with the norm
(38)∥u∥X=∥u∥L∞(I;FB˙p,q1-2β+3/p′)+μ∥u∥L1(I;FB˙p,q1+3/p′).
Then there exists some constant C=C(β,p,q)>0 depending on β, p, and q such that
(39)∥∇·(u⊗v)∥L1(I;FB˙p,q1-2β+3/p′)≤Cμ-1∥u∥X∥v∥X.
Particularly, it is true for the cases q=1 and β=1/2.Proof.
We will use the technique of the paraproduct. Set(40)Δ˙ju=(F-1φj)*u,S˙j=∑k≤j-1‍Δ˙ku,Δ˙~ju=∑|k-j|≤1‍Δ˙ku,for∀j∈Z.
By Bony’s decomposition, we have for fixed j(41)Δ˙j(uv)=∑|k-j|≤4‍Δ˙j(S˙k-1uΔ˙kv)+∑|k-j|≤4‍Δ˙j(S˙k-1vΔ˙ku)+∑k≥j-3‍Δ˙j(Δ˙kuΔ˙~kv)∶=Ij+IIj+IIIj.
For simplicity,we can view ∇·(u⊗v), as the first derivative of two scale functions u,v. Consider
(42)∥∂(uv)∥L1(I:FB˙p,q1-2β+3/p′)≤83(∑j2(1-2β+3/p′)jq2jq∥Δ˙j(uv)^∥L1(I;Lp)q)1/q≤83[(∑j‍2(2-2β+3/p′)jq∥Ij^∥L1(I;Lp)q)1/q0000000+(∑j‍2(2-2β+3/p′)jq∥IIj^∥L1(I;Lp)q)1/q0000000+(∑j‍2(2-2β+3/p′)jq∥IIIj^∥L1(I;Lp)q)1/q].
The terms Ij and IIj are symmetrical. Using Young’s inequality and Hölder’s inequality we have
(43)∥Ij^∥L1(I;Lp)≤∑|k-j|≤4∥S˙k-1uΔ˙kv^∥L1(I;Lp)‍≤∑|k-j|≤4∥vk^∥L1(I;Lp)‍∑l≤k-2∥ul^∥L∞(I;L1)‍≤∑|k-j|≤4∥vk^∥L1(I;Lp)×(∑l≤k-22(1-2β)lq∥ul^∥L∞(I;L1)q‍)1/q×(∑l≤k-22(2β-1)lq′‍)1/q′≤C∑|k-j|≤42(2β-1)k∥vk^∥L1(I;Lp)∥u∥L∞(I;FB˙1,q1-2β).
Using the conclusion FB˙p,q1-2β+3/p′⊂FB˙1,q1-2β, we have
(44)(∑j‍2(2-2β+3/p′)jq∥Ij^∥L1(I;Lp)q)1/q≤C∥u∥L∞(I;FB˙p,q1-2β+3/p′)∥v∥L1(I;FB˙p,q1+3/p′).
In a similar way we can prove that
(45)(∑j‍2(2-2β+3/p′)jq∥IIj^∥L1(I;Lp)q)1/q≤C∥v∥L∞(I;FB˙p,q1-2β+3/p′)∥u∥L1(I;FB˙p,q1+3/p′).
For the remaining term, we first consider the case p≤2 in which β<1+3/p′. By Hölder's inequality with 1/p=1/p′+1/p-1/p′ and by Young's inequality with 1+1/p-1/p′=1/p+1/p, we have
(46)2(2-2β+3/p′)j∥IIIj^∥L1(I;Lp)≤C∑k≥j-3‍2(2-2β+3/p′)j2(3/p′)j∥uk^*∑|l-k|≤1‍vl^∥L1(I;Lpp′/(p′-p))≤C∑k≥j-32(2-2β+6/p′)j∥uk^∥L1(I;Lp)∑|l-k|≤1‍∥vl^∥L∞(I;Lp)≤C∑k≥j-3‍2(2-2β+6/p′)(j-k)2(1-2β+3/p′)k∥uk^∥L∞(I;Lp)×∑|l-k|≤1‍2(1+3/p′)l∥vl^∥L1(I;Lp).
When q>2, we take lq-norm of both sides of (46) and use Young’s inequality with 1+1/q=1/q′+2/q to get
(47)(∑j2(2-2β+3/p′)jq∥IIIj^∥L1(I;Lp)q)1/q≤C∥2(1-2β+3/p′)k∥uk^∥L∞(I;Lp)×∑|l-k|≤1‍2(1+3/p′)l∥vl^∥L1(I;Lp)∥lq/2(k)≤C∥u∥L∞(I;FB˙p,q1-2β+3/p′)∥v∥L1(I;FB˙p,q1+3/p′).
When q≤2, since FB˙p,q1+3/p′⊂FB˙p,q′1+3/p′, we take lq-norm of both sides of (46) and use Young’s inequality with 1+1/q=1+1/q to get
(48)(∑j‍2(2-2β+3/p′)jq∥IIIj^∥L1(I:Lp)q)1/q≤C∥∑|l-k|≤1‍2(1-2β+3/p′)k∥uk^∥L∞(I;Lp)0000000×∑|l-k|≤1‍2(1+3/p′)l∥vl^∥L1(I;Lp)∥l1(k)≤C∥u∥L∞(I;FB˙p,q1-2β+3/p′)∥v∥L1(I;FB˙p,q′1+3/p′)≤C∥u∥L∞(I;FB˙p,q1-2β+3/p′)∥v∥L1(I;FB˙p,q1+3/p′).
Next we consider the case p>2 and hence β≤5/2. By Hölder's inequality we have
(49)2(2-2β+3/p′)j∥IIIj^∥L1(I;Lp)≤C∑k≥j-3‍2(2-2β+3/p′)j2(3/p)j∥uk^*∑|l-k|≤1‍vl^∥L1(I;L∞)≤C∑k≥j-3‍2(5-2β)j∥uk^∥L∞(I;Lp′)∑|l-k|≤1‍∥vl^∥L1(I;Lp)≤C∑k≥j-3‍2(5-2β)(j-k)2(1-2β+3/p′)k∥uk^∥L∞(I;Lp)×∑|l-k|≤1‍2(1+3/p′)l∥vl^∥L1(I:Lp).
Following the same steps as in the case p≤2, we obtain the same estimate for p>2. Collecting the above estimates we finish our proof.Next we introduce an abstract lemma on the existence of fixed point solutions [16, 24].Lemma 14.
LetX be a Banach space with norm ∥·∥X and let B:X×X↦X be a bounded bilinear operator satisfying
(50)∥B(u,v)∥X≤η∥u∥X∥v∥X,
for all u,v∈X and a constant η>0. Then for any fixed y∈X satisfying ∥y∥X<ϵ<1/4η, the equation x:=y+B(x,x) has a solution x- in X such that ∥x-∥X≤2∥y∥X. Also, the solution is unique in B-(0,2ϵ). Moreover, the solution depends continuously on y in the sense that if ∥y′∥X<ϵ, x′=y′+B(x′,x′), and ∥x′∥X<2ϵ, then
(51)∥x--x′∥X≤11-4ϵη∥y-y′∥X.This lemma allows us to solve the Cauchy problem (1) with bounded bilinear form and small data. The mild solution of (1) is the solution to the equivalent integral form:
(52)u(t,x)=e-μt(-Δ)βu0-∫0te-μ(t-τ)(-Δ)βP∇·(u⊗u)(τ,x)dτ=e-μt(-Δ)βu0+B(u,u),
where P=I+∇(-Δ)-1div⁡ is the Leray-Hopf projector. To make B(u,v) become a bilinear form, we simply take (1/2)u⊗v+(1/2)v⊗u instead of u⊗v in the integral.Proof of Theorem5.
We begin with the bilinear operatorB(u,v). Observing that B(u,v) can be viewed as the solution to the dissipative equation (31) with u0=0,F=-P∇·(u⊗v). Thus we can use Lemma 12 with s=1-2β+3/p′ and Lemma 13 to obtain
(53)∥B(u,v)∥X≤(1+(169)β)∥-P∇·(u⊗v)∥L1(I;FB˙p,q1-2β+3/p′)≤(1+(169)β)Cμ-1∥u∥X∥v∥X.
By Lemma 14 we know that if ∥e-μt(-Δ)βu0∥X<R with R=μ/4(1+(16/9)β)C, then (52) has a unique solution in B(0,2R), where
(54)B(0,2R)∶={x∈X:∥x∥X≤2R}.
Now we need to derive∥e-μt(-Δ)βu0∥X<R. Similarly, e-μt(-Δ)βu0 is the solution to the dissipative equation (31) with u0=u0 and F=0. By Lemma 12 we obtain
(55)∥e-μt(-Δ)βu0∥X≤(1+(169)β)∥u0∥FB˙p,q1-2β+3/p′.
Thus we conclude that if ∥u0∥FB˙p,q1-2β+3/p′<C0μ with C0=(4(1+(16/9)β)2C)-1, then (52) has a unique global solution u∈X satisfying
(56)∥u∥X≤2∥e-μt(-Δ)βu0∥X≤2(1+(169)β)∥u0∥FB˙p,q1-2β+3/p′.
The continuity with respect to time is standard and thus we finish our proof.Proof of Theorem8.
The method is the same with the proof of Theorem5. But we substitute Lemma 13 by the following lemma.Lemma 15.
Let1≤p≤q≤2 and β∈(1/2,1+3/2p′] and X is the same as in Lemma 13. Then there exists some constant C=C(β,p,q)>0 depending on β, p, and q such that
(57)∥∇·(u⊗v)∥L1(I;FB˙p,q1-2β+3/p′)≤Cμ-1∥u∥X∥v∥X.Particularly, it is true for the casesp=q=1andβ=1/2.
Proof. The proof is also same with Lemma 13. In fact by Bony's decomposition, we divide Δ˙j(uv) into three parts Ij, IIj, and IIIj. The parts Ij and IIJ satisfy the same estimate. Hence it is sufficient to deal with the part IIIJ. In fact when q≥p, we have
(58)(∑j2(2-2β+3/p′)jq∥IIIj^∥L1(I;Lp)q‍)1/q≤∥∑k≥j-3∫I∥[uk^*∑|l-k|≤1‍vl^]2(2-2β+3/p′)jφj(ξ)0000000000000×[uk^*∑|l-k|≤1‍vl^]∥Lξpdt∥lq(j)≤∑k∫I∥[uk^*∑|l-k|≤1vl^]p∫R32(2-2β+3/p′)pj000000000000×|φj(ξ)|p‍[uk^*∑|l-k|≤1vl^]pdξ∥lq/p(j≤k+3)1/pdt≤sup⁡ξ(∑jφj(ξ)q‍)1/q∑k2(2-2β+3/p′)(k+3)×∫I∥uk^*∑|l-k|≤1‍vl^∥Lξpdt≤23(2-2β+3/p′)∑k2(1-2β)k∥uk^∥L∞(I:L1)×∑|l-k|≤1‍2(1+3/p′)(k-l)2(1+3/p′)l∥vl^∥L1(I;Lp)≤C∥u∥L∞(I;FB˙1,q1-2β)∥v∥L1(I;FB˙p,q1+3/p′).
In the last inequality we use a similar conclusion with (3) in Proposition 3; that is, L1(I;FB˙p,q1+3/p′)⊂L1(I;FB˙p,q′1+3/p′), when q∈[1,2].


## 4. The Decay Property

We introduce some lemmas which have interest in themselves.Lemma 16.
Letβ<5/4, s>5/2-2β, and 1≤p, q≤2. Then we have
(59)∥f∥FB˙p,q1-2β+3/p′≤C∥f∥L21-(5/2-2β)/s∥f∥H˙s(5/2-2β)/s.Proof.
By definition and Hölder’s inequality we have(60)∥f∥FB˙p,q1-2β+3/p′=(∑j∈Z2(1-2β+3/p′)jq∥φjf^∥Lpq)1/q≤(∑j∈Z2(1-2β+3/p′)jq∥φjf^∥L2q2(3/p-3/2)jq)1/q≤(∑j≤M2(5/2-2β)jq∥φjf^∥L2q‍)1/q+(∑j≥M2(5/2-2β-s)jq2sjq∥φjf^∥L2q)1/q≤C12(5/2-2β-s)M(∑j∈Z∥φjf^∥L22‍)1/2+C22(5/2-2β-s)M(∑j∈Z2sjq∥φjf^∥L22‍)1/2.
Since B˙2,2s=H˙s and B˙2,20=L2 and by Proposition 3, we know that FB˙2,2s=B˙2,2s; we finish our proof by taking M such that 2M=(∥f∥H˙s/∥f∥L2)1/s.Lemma 17.
Letβ∈(1/2,1] and 1≤p, q≤2. Consider
(61)∥uv∥H˙1-β≤C∥u∥L2∥v∥FB˙p,q1-β+3/p′+C∥u∥H˙β∥v∥FB˙p,q1-2β+3/p′.Proof.
We use the equivalenceH˙s=B˙2,2s=FB˙2,2s. To conclude the result we only need to show that
(62)∥uv∥H˙1-β≤C∥u∥L2∥v∥FB˙p,21-β+3/p′+C∥u∥H˙β∥v∥FB˙1,21-2β,
since we have the conclusions FB˙p,21-2β+3/p′⊂FB˙1,21-2β and FB˙p,q1-2β+3/p′⊂FB˙p,21-2β+3/p′ by Proposition 3. The method is similar with the proof of Lemma 13. Consider
(63)∥uv∥FB˙2,21-β≤(∑j2(1-β)j2∥Ij^∥L22)1/2+(∑j2(1-β)j2∥IIj^∥L22‍)1/2+(∑j2(1-β)j2∥IIIj^∥L22‍)1/2,
where Ij, IIj, and IIIj are the same with the proof of Lemma 13. Consider
(64)∥Ij^∥L2≤∑|k-j|≤4∥S˙k-1uΔ˙kv^∥L2≤∑|k-j|≤4∥vk^∥L2∑l≤k-2∥ul^∥L1≤C∑|k-j|≤4∥vk^∥L2×(∑l≤k-22(1-2β)l2∥ul^∥L12)1/2×(∑l≤k-22(2β-1)l2)1/2≤C∑|k-j|≤42(2β-1)k∥vk^∥L2∥u∥FB˙1,21-2β.
Thus we get
(65)‍(∑j2(1-β)j2∥Ij^∥L22)1/2≤C∥v∥H˙β∥u∥FB˙1,21-2β.
To estimate the term IIj, we make a minor modification to get
(66)∥IIj^∥L2≤∑|k-j|≤4∥S˙k-1vΔ˙ku^∥L2≤∑|k-j|≤4∥uk^∥Lp∑l≤k-2∥vl^∥L2p/(3p-2)≤C∑|k-j|≤4∥uk^∥Lp×(∑l≤k-22-2l(3/p′)∥vl^∥L2p/(3p-2)2)1/2×(∑l≤k-222l(3/p′))1/2≤C∑|k-j|≤42(3/p′)k∥uk^∥Lp∥v∥FB˙2p/(3p-2),2-3/p′.
By (4) in Proposition 3, we know that L2=FB˙2,20⊂FB˙2p/(3p-2),2-3/p′. Thus
(67)(∑j2(1-β)j2∥IIj^∥L22)1/2≤C∥u∥FB˙p,21-β+3/p′∥v∥L2.
Finally we derive the estimate of the last part as
(68)(∑j2(1-β)j2∥IIIj^∥L22)1/2≤∥00000000000×[uk^*∑|l-k|≤1vl^]∥Lξ2∑k≥j-3∥[uk^*∑|l-k|≤1vl^]2(1-β)jφj(ξ)‍00000000000×[uk^*∑|l-k|≤1vl^]∥Lξ2∥l2(j)≤∑k(∫R3∑j≤k+32(1-β)2j‍‍|φj(ξ)|2‍0000000000×[uk^*∑|l-k|≤1vl^]2dξ)1/2≤sup⁡ξ(∑jφj(ξ)2)1/2×∑k2(1-β)(k+3)∥uk^*∑|l-k|≤1‍vl^∥Lξ2≤C∑k2βk∥uk^∥L2∑|l-k|≤12(1-2β)(k-l)2(1-2β)l∥vl^∥L1≤C∥u∥H˙β∥v∥FB˙1,21-2β.Now we can begin our proof of Theorem10. The method is based on the work from Gallagher-Iftimie-Planchon [27].Proof of Theorem10.
Letϵ>0 be any constant small enough such that ϵ≤C0μ, where C0 is the constant in Theorem 5 and μ is the viscosity coefficient in (1). For k∈N+, define
(69)Ak∶={ξ∈R3:|ξ|≤kand|u0^(ξ)|≤k}.
Obviously F-1(χAku0^) converges to u0 in FB˙p,q1-2β+3/p′. So there exists some k∈N+ such that
(70)∥u0-F-1(χAku0^)∥FB˙p,q1-2β+3/p′<ϵ2.
Set
(71)u0,k=F-1(χAku0^),w0,k=u0-F-1(χAku0^).
Thus u0,k∈FB˙p,q1-2β+3/p′∩L2 and we have shown that ∥w0,k∥FB˙p,q1-2β+3/p′<ϵ/2. Now we consider the following equations:
(72)wt+μ(-Δ)βw+(w·∇)w+∇π=0,(t,x)∈R+×R3;∇·w=0,(t,x)∈R+×R3;w(0,x)=w0,k(x),x∈R3.
Since ϵ/2≤C0μ/2≤C0μ, by Theorem 5, there exists a unique global solution wk of (72) such that
(73)wk∈C([0,∞);FB˙p,q1-2β+3/p′)∩L1([0,∞);FB˙p,q1+3/p′).
Moreover,
(74)∥wk∥L∞([0,∞);FB˙p,q1-2β+3/p′)+μ∥wk∥L1([0,∞);FB˙p,q1+3/p′)≤C∥w0,k∥FB˙p,q1-2β+3/p′.
An easy computation gives wk∈C([0,∞);FB˙p,q1-2β+3/p′) and for all t∈[0,∞), we have
(75)∥wk(t)∥FB˙p,q1-2β+3/p′+μ∥wk∥L1([0,t);FB˙p,q1+3/p′)≤C∥w0,k∥FB˙p,q1-2β+3/p′.
Now put uk=u-wk. Then uk∈C([0,∞);FB˙p,q1-2β+3/p′) and it satisfies
(76)∂tuk+μ(-Δ)βuk+(uk·∇)uk+(uk·∇)wk+(wk·∇)uk+∇π-∇πk=0,∇·uk=0,uk(0,x)=u0,k(x),
where π and πk are the correspond pressures to the solutions u and wk, respectively. Taking the L2 inner product with uk, we have
(77)12ddt∥uk∥L22+μ∥(-Δ)β/2uk∥L22≤|〈(uk·∇)wk+(wk·∇)uk,uk〉|.
To estimate |〈(uk·∇)wk,uk〉|, we have
(78)|〈(uk·∇)wk,uk〉|=|〈∇·(uk⊗wk),uk〉|≤∥(-Δ)1/2-β/2(uk⊗wk)∥L2×∥(-Δ)β/2uk∥L2.
By Lemma 17, we have
(79)|〈(uk·∇)wk,uk〉|≤C∥wk∥FB˙p,q1-β+3/p′∥uk∥L2∥uk∥H˙β+C∥wk∥FB˙p,q1-2β+3/p′∥uk∥H˙β2.
By (75), we have ∥wk∥FB˙p,q1-2β+3/p′≤Cϵ/2. We further assume ϵ small enough such that C2ϵ≤μ/4. Using ab≤a2/2+b2/2, we have
(80)|〈(uk·∇)wk,uk〉|≤2C2μ∥wk∥FB˙p,q1-β+3/p′2∥uk∥L22+μ4∥uk∥H˙β2.
Thus we conclude that
(81)ddt∥uk∥L22+μ∥uk∥H˙β2≤8C2μ∥wk∥FB˙p,q1-β+3/p′2∥uk∥L22.
Gronwall lemma gives
(82)∥uk∥L22+μ∫0t‍∥uk∥H˙β2≤∥u0,k∥L22×exp⁡{8C2μ∫0t‍∥wk∥FB˙p,q1-β+3/p′2}.
Since q≤2, by Minkowski's inequality, we get
(83)∫0t‍∥wk∥FB˙p,q1-β+3/p′2≤(∑j∈Z‍2j(1-β+3/p′)q000000×(∫0t∥φjwk^∥Lp2)q/2)2/q≤(∑j∈Z‍2j(1-2β+3/p′)q/2∥φjwk^∥L∞([0,t);Lp)q/2000000×2j(1+3/p′)q/2∥φjwk^∥L1([0,t);Lp)q/2∑j∈Z‍)2/q≤∥wk∥L∞([0,t);FB˙p,q1-2β+3/p′)∥wk∥L1([0,t);FB˙p,q1+3/p′).
Thus, combining with (74) we can obtain
(84)∥uk∥L22+μ∫0t‍∥uk∥H˙β2≤∥u0,k∥L22exp⁡{8C4μ2∥w0,k∥FB˙p,q1-2β+3/p′2}.
Using Lemma 16 with s=β and (75), we know that uk∈L4β/(5-4β)([0,+∞);FB˙p,q1-2β+3/p′) and
(85)∫0∞‍∥uk∥FB˙p,q1-2β+3/p′4β/(5-4β)≤C4β/(5-4β)μ-1∥u0,k∥L24β/(5-4β)×exp⁡{16C4βμ2(5-4β)∥w0,k∥FB˙p,q1-2β+3/p′2}.
So by continuity, there is a time t0 such that ∥uk(t0)∥FB˙p,q1-2β+3/p′≤ϵ/2. Then we have
(86)∥u(t0)∥FB˙p,q1-2β+3/p′≤∥uk(t0)∥FB˙p,q1-2β+3/p′+∥wk(t0)∥FB˙p,q1-2β+3/p′≤ϵ2+ϵ2≤ϵ.
Finally, we consider the fractional Navier-Stokes equations starting at t=t0; by Theorem 5 and using a method as estimating (75) we conclude that
(87)∥u(t)∥FB˙p,q1-2β+3/p′+μ∥u∥L1([t0,t);FB˙p,q1+3/p′)≤C∥u(t0)∥FB˙p,q1-2β+3/p′≤Cϵ.
for all t>t0. Thus we finish our proof.



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*Source: 463639-2014-07-22.xml*