# Algebra Properties in Fourier-Besov Spaces and Their Applications

Authors: Xuhuan Zhou; Weiliang Xiao
Journal: Journal of Function Spaces
 (2018)

---

## Abstract

We estimate the norm of the product of two scale functions in Fourier-Besov spaces. As applications of these algebra properties, we establish the global well-posedness for small initial data and local well-posedness for large initial data of the generalized Navier-Stokes equations. Particularly, we give a blow-up criterion of the solutions in Fourier-Besov spaces as well as a space analyticity of Gevrey regularity.


---

## Body

## 1. Introduction

In this paper, we study the mild solutions to the generalized Navier-Stokes equations (NSE) inR+×Rn(1)ut+μ-△βu=Qu,u,t,x∈R+×Rn;u0,x=u0x,x∈Rn.Here u(t,x)=(u1(t,x),⋯,un(t,x)), μ>0 is a constant, and the operator -△β is the Fourier multiplier with symbol ξ2β. The bilinear operator Q denotes the map of the form (2)Qju,v≔∑k,l,m=1nqk,lj,m∂mukvl,j=1,⋯,n,where qk,lj,mg≔∑a,b=1n‍αk,lj,m,a,bF-1(ξaξb/|ξ|2g^(ξ)) and αk,lj,m,a,b are real numbers.The incompressible NSE is a particular case of (1), by taking QNS=-1/2P(div(u⊗v)+div(v⊗u)) with the Leray projector P defined as (3)FPfjξ=∑k=1nδj,k-1ξjξkξ2f^kξ,where δj,k=1 if j=k and δj,k=0 if j≠k; that is, (4)ut+μ-△βu=-Pdivu⊗u,t,x∈R+×Rn;u0,x=u0x,x∈Rn.If we set the initial data u0 to be a divergence free vector field, the above system is exactly the (fractional) incompressible NSE.For the classical incompressible NSE (β=1,Q=QNS), the study of mild solution started by Fujita and Kato [1] in the space frame H˙1/2(R3) and was extended by many mathematicians in different spaces [2–5]. As for the generalized case β>0,Q=QNS, Lions [6] proved the global existence of classical solutions in 3-dimension when β≥5/4 (see also Wu [7] in n-dimension). For the important case β<5/4, Wu [8, 9] studied the well-posedness in B˙p,q1-2β+3/p(R3). Inspired by Xiao [10] in the classical case (β=1), Li and Zhai [11, 12] studied (1) in some critical Q-type spaces for β∈(1/2,1), and Zhai [13] showed the well-posedness in BMO-(2β-1) when β∈(1/2,1). For the biggest critical space B˙∞,∞-(2β-1), Yu and Zhai [14] proved the well-posedness when β∈(1/2,1), and Cheskidov and Shvydkoy [15] showed the ill-posedness when β∈[1,5/4). Deng and Yao [16] studied (1) in Triebel-Lizorkin spaces F˙α,r-β in 3-dimension 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).We focus on the results in Fourier-Besov spaces. Early results came from Cannone and Karch [17] in pseudomeasure type spaces PMα, in which they discuss the singular and regular solution of NSE. Iwabuchi [18] introduced a Besov type space Bqs (named Fourier-Herz space in [19]) to study the well-posedness and ill-posedness of Keller-Segel system. Scapellato and Ragusa in [20] introduced a Morrey-type sapce (named mixed Morrey space) to study qualitative properties of partial differential equations with discontinuous coefficients. Lei and Lin [21] study the global well-posedness for NSE in spaces X˙-1. All those spaces are special cases of Fourier-Besov spaces, which were first introduced by Konieczny and Yoneda [22], to study the dispersive effect of the Coriolis for NSE.Our first result on the estimates of the productuv in Fourier-Besov spaces has the similar form with that in Sobolev spaces [23], which can be seen as follows.Theorem 1.
Let1≤p,q≤∞ and s1,s2∈R such that(5)s1<np′,s2<np′,s1+s2>max⁡np′-np,0,where 1/p+1/p′=1. Then for u∈FB˙p,qs1,v∈FB˙p,qs2, one has (6)uvFB˙p,qs1+s2-n/p′≤CuFB˙p,qs1vFB˙p,qs2.Remark 2.
An important case isp=q=2, since by Plancherel’s identity we know FB˙2,2s=H˙s. This theorem gives that if s1<n/2,s2<n/2,s1+s2>0, then uvH˙s1+s1-n/2≤CuH˙s1vH˙s2. This is exactly the case in homogeneous Sobolev spaces [24]. A similar result in Besov space can be seen in p. 61 of [25].As an application of this theorem, we study the Cauchy problem of (1) in Fourier-Besov spaces FB˙p,q1-2β+n/p′.Theorem 3.
Let1≤p,q≤∞,q′/1+q′<β<q′/1+q′min⁡{1+n/p′,1+n/2}. Then for any u0∈FB˙p,q1-2β+n/p′ with ∇·u0=0, the Cauchy problem (1) admits a unique mild solution u and (7)u∈ET≔C0,T;FB˙p,q1-2β+n/p′∩Lq0,T;FB˙p,q1-2β+n/p′+2β/q∩L2q0,T;FB˙p,q1-2β+n/p′+β/q.Moreover, let T∗ denote the maximal time of existence of such a solution, then
(i) there is a constantC0 such that if u0FB˙p,q1-2β+n/p′≤C0μ, then (8)T∗=∞;(ii) if T∗<∞, then (9)uL2q0,T∗;FB˙p,q1-2β+n/p′+β/q=∞.Particularly, our result also holds in the case p>1,q=1,β=1.This well-posedness result corresponds to the classical case for initial datau0∈H˙1/2(R3) [2] if we take n=3,p=q=2,β=1,μ=1 in Theorem 3. Unfortunately, our result is not suitable for the case p=q=1,β=1, which has been proved in [26]. To address this special case, we also prove the following theorem.Theorem 4.
Let1≤p≤q≤2,β∈(1/1+1/q′,1+n/2p′/1+1/q′]. Then for any u0∈FB˙p,q1-2β+n/p′ with ∇·u0=0, the Cauchy problem (1) admits a unique mild solution u∈ET. Moreover, let T∗ denote the maximal time of existence of such a solution, then
(i) there is a constantC0 such that if u0FB˙p,q1-2β+n/p′≤C0μ, then (10)T∗=∞;(ii) if T∗<∞, then (11)uL2q0,T∗;FB˙p,q1-2β+n/p′+β/q=∞.Particularly, our result also holds in the case p=q=1,β=1.Now we focus on the space analyticity. Our main method is Gevrey estimate, which was introduced by Foias and Temam [27], since that Gevrey class technique has become an effective approach in the study of space analyticity of solutions. Ferrari and Titi [28] established Gevrey regularity for a very large class of parabolic equation with analytic nonlinearity. Grujić and Kukavica [29] prove the Gevrey regularity for NSE in Lp. More results on the analyticity of solution for NSE can be seen in Lemarie-Rieusset [5] and references therein. Biswas [30] established Gevrey class regularity of solutions to a large class of dissipative equations in Besov type spaces defined via caloric extension. Bae [31] proved the Gevrey estimate of solution for NSE in the spaces X˙-1. Inspired by this, we establish the Gevrey class regularity for the generalized NSE in the Fourier-Besov spaces. We indicate that any order derivative of the solution u enjoys the same behavior with u in some sense. In fact, denote by et|D|β the Fourier multiplier with symbol et|ξ|β, then we have the following result.Theorem 5.
Let1≤p,q≤∞,q′/1+q′<β<q′/1+q′min{1+n/p′,1+n/2},β≤1. Then for any u0∈FB˙p,q1-2β+n/p′, the Cauchy problem (1) admits a unique mild solution u∈ET such that etDβu∈ET. Particularly, our result also holds in the case p>1,q=1,β=1.Theorem 6.
Let1≤p≤q≤2,q′/1+q′<β<q′/1+q′(1+n/2p′),β≤1. Then for any u0∈FB˙p,q1-2β+n/p′, the Cauchy problem (1) admits a unique mild solution u∈ET such that etDβu∈ET. Particularly, our result also holds in the case p=q=1,β=1.Remark 7.
In our later proof, we can also obtain that conclusions (i), (ii) in Theorems3 and 4 are also valid for et|D|βu in Theorems 5 and 6, respectively.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 homogeneous Besov spaces and H˙s the homogeneous Sobolev spaces. Also, C denotes a positive constant which may differ in lines if not being specified, and 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 some basic properties of Fourier-Besov spaces. Then we prove Theorem 1 as well as a corollary. In Section 3, we give the proof of Theorems 3 and 4. And in Section 4, we prove the space analyticity of Theorems 5 and 6.


## 2. Algebra Properties in Fourier-Besov Spaces

Letφ∈Cc∞(Rn) be a radial real-valued smooth function such that 0≤φ(ξ)≤1 and (12)suppφ⊂ξ∈Rn:34≤ξ≤83,∑j∈Zφ2-jξ=1foranyξ≠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 8.
Fors∈R,1≤p,q≤∞, set (13)
f
F
B
˙
p
,
q
s
=
∑
j
∈
Z
2
j
s
q
φ
j
f
^
L
p
q
1
/
q
,
q
<
∞
;
s
u
p
j
∈
Z
2
j
s
φ
j
f
^
L
p
,
q
=
∞
.We define the homogeneous Fourier-Besov space FB˙p,qs as (14)FB˙p,qs=f∈S′P:fFB˙p,qs<∞.The Fourier-Besov spaces look similar to the classical Besov spaces, but without the inverse Fourier transform. In fact, there are close relationships between them [32]. These spaces are, also, similar to central Morrey spaces studied in [33]. In order to apply in PDE, we also need to derive the properties of Fourier-Besov spaces with space-time norm.Definition 9.
Lets∈R,1≤p,q≤∞ and I=[0,T),T∈(0,∞]. The space-time norm is defined on f(t,x) by (15)ft,xLrI;FB˙p,qs≔∑j∈Z2jsqφjf^LrI;Lpq1/q;ft,xLrI;B˙p,qs≔∑j∈Z2jsqF-1φjf^LrI;Lpq1/q.Here B˙p,qs denotes the classical homogeneous Besov space. Besides, we also define the norm f(t,x)Lr(I;X) for some Banach space X with norm ·X by (16)ft,xLrI;X≔∫Ift,·Xrdt1/r.Remark 10.
By the definition and Minkowski’s inequality, we easily deduce that(17)ft,xLrI;FB˙p,qs≤ft,xLrI;FB˙p,qs,forr≤q;ft,xLrI;FB˙p,qs≤ft,xLrI;FB˙p,qs,forr≥q.Proposition 11.
Lets∈R,1≤p,q,r≤∞,I=[0,T),T∈(0,∞]. The following inclusions hold.
( 1 ) If p=2, then Lr(I;FB˙2,qs)=Lr(I;B˙2,qs).
( 2 ) If p<2, then Lr(I;FB˙p,qs)⊂Lr(I;B˙p′,qs).
( 3 ) If p>2, then Lr(I;B˙p,qs)⊂Lr(I;FB˙p′,qs).
( 4 ) If q1≤q2, then Lr(I;FB˙p,q1s)⊂Lr(I;FB˙p,q2s).
( 5 ) If 1≤q≤∞,1≤p1≤p2≤∞,s1,s2∈R satisfy s1+n/p1=s2+n/p2, then (18)LrI;FB˙p2,qs2⊂LrI;FB˙p1,qs1.(6) If 1≤p1,p2,q1,q2,r1,r2≤∞ satisfy s=(1-θ)s1+θs2,1/p=1-θ/p1+θ/p2,1/q=1-θ/q1+θ/q2 and 1/r=1-θ/r1+θ/r2 for 0≤θ≤1, then (19)fLrI;FB˙p,qs≤fLr1I;FB˙p1,q1s11-θfLr2I;FB˙p2,q2s2θ.The special casep=q has an interesting equivalent norm, which can be seen by the following proposition (see [22] for the proof).Proposition 12.
Define the spacesX˙s,p as (20)X˙s,p=f∈S′P:∫Rnξspf^pdξ1/p<∞.Then we have X˙s,p=FB˙p,ps and the norms are equivalent (21)fFB˙p,ps~∫Rnξspf^pdξ1/p.Now we give the proof of Theorem1. The tools we use are the paraproduct and Bony’s decomposition, which can be found in [8, 17].Proof of Theorem1.
We will use the technique of the paraproduct. Set(22)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(23)Δ˙juv=∑k-j≤4Δ˙jS˙k-1uΔ˙kv+∑k-j≤4Δ˙jS˙k-1vΔ˙ku+∑k≥j-3Δ˙jΔ˙kuΔ˙~kv≔Ij+IIj+IIIj.Thus we can divide the norm by(24)uvFB˙p,qs1+s2-n/p′≤∑j2s1+s2-n/p′jqIj^Lpq1/q+∑j2s1+s2-n/p′jqIIj^Lpq1/q+∑j2s1+s2-n/p′jqIIIj^Lpq1/q.The terms Ij and IIj are symmetrical. Using Young’s inequality and Hölder’s inequality(25)Ij^Lp≤∑k-j≤4S˙k-1uΔ˙kv^Lp≤∑k-j≤4vk^Lp∑l≤k-2ul^L1≤∑k-j≤4vk^Lp∑l≤k-22s1-n/p′lqul^L1q1/q∑l≤k-22n/p′-s1lq′1/q′≤C∑k-j≤42n/p′-s1kvk^LpuFB˙1,qs1-n/p′.Using the inclusion FB˙p,qs1⊂FB˙1,qs2-n/p′, we have(26)∑j2s1+s2-n/p′jqIj^Lpq1/q≤CuFB˙p,qs1vFB˙p,qs2.In a similar way, we can prove (27)∑j2s1+s2-n/p′jqIIj^Lpq1/q≤CuFB˙p,qs1vFB˙p,qs2.For the remaining term, we first consider the case p≤2, in which s1+s2>0. 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(28)2s1+s2-n/p′jIIIj^Lp≤C∑k≥j-32s1+s2-n/p′j2n/p′juk^∗∑l-k≤1vl^Lpp′/p′-p≤C∑k≥j-32s1+s2juk^Lp∑l-k≤1vl^Lp≤C∑k≥j-32s1+s2j-k2s1kuk^Lp∑l-k≤12s2lvl^Lp.When q>2, we take lq-norm of both sides of (28) and use Young’s inequality with 1+1/q=1/q′+2/q to get (29)∑j2s1+s2-n/p′jqIIIj^Lpq1/q≤C2s1kuk^Lp∑l-k≤12s2lvl^Lplq/2k≤CuFB˙p,qs1vFB˙p,qs2.When q≤2, then FB˙p,qs2⊂FB˙p,q′s2; by definition, taking lq-norm of both sides of (28) and using Young’s inequality with 1+1/q=1+1/q, we get (30)∑j2s1+s2-n/p′jqIIIj^Lpq1/q≤C2s1kuk^Lp∑l-k≤12s2lvl^Lpl1k≤CuFB˙p,qs1vFB˙p,q′s2≤CuFB˙p,qs1vFB˙p,qs2.For the case p>2, we have s1+s2>n/p′-n/p. By Hölder’s inequality, there holds(31)2s1+s2-n/p′jIIIj^Lp≤C∑k≥j-32s1+s2-n/p′j2n/pjuk^∗∑l-k≤1vl^L∞≤C∑k≥j-32s1+s2-n/p′+n/pjuk^Lp′∑l-k≤1vl^Lp≤C∑k≥j-32s1+s2-n/p′+n/pj-k2s1kuk^Lp∑l-k≤12s2lvl^Lp.Following the same steps as in the case p≤2, we obtain the same estimate. Collecting the above estimates we finish our proof.By a slight modification of the proof, we can also obtain the following.Corollary 13.
Let1≤p≤∞ and s1,s2∈R such that(32)s1≤np′,s2≤np′,s1+s2>max⁡np′-np,0,where 1/p+1/p′=1. Then for u∈FB˙p,1s1,v∈FB˙p,1s2, one has(33)uvFB˙p,1s1+s1-n/p′≤CuFB˙p,1s1vFB˙p,1s2.


## 3. The Well-Posedness

To prove the well-posedness, we invoke the fix point principle. We consider the mild solution which means the equivalent integral equation(34)ut,x=e-μt-△βu0+∫0te-μt-τ-△βQu,uτ,xdτ≔e-μt-△βu0+Bu,u.Lemma 14 (linear estimate).
LetI=[0,T),0<T≤∞,s∈R,1≤p,q≤∞. Set Lu0=e-μt(-△)βu0; we have(35)Lu0L∞I;FB˙p,qs+μ1/qLu0LqI;FB˙p,qs+2β/q≤Cu0FB˙p,qs.Proof.
By the Fourier transform, we have(36)Lu0∧ξ=e-μtξ2βu0^.Multiplying φj and taking the Lp-norm imply (37)φjLu0∧Lp≤e-μt22jβ3/42βu0j^Lp,where we denote uj^=φju^. Multiplying 2js and taking lq-norm (38)Lu0L∞I;FB˙p,qs≤u0FB˙p,qs.Similarly, multiplying 2j(s+2β/q) and taking Lq-norm with respect to time on I, (39)2js+2β/qφjLu0∧LqI;Lp≤∫Ie-μqt22jβ3/42β22jβdt1/q2jsu0j^Lp.Since ∫0∞‍e-μqt22jβ(3/4)2β22jβdt=μq(3/4)2β, taking lq-norm we get (40)Lu0LqI;FB˙p,qs+2β/q≤Cμ-1/qu0FB˙p,qs.Lemma 15 (nonlinear estimate).
LetI=[0,T),0<T≤∞,s∈R,1≤p,q≤∞. Set Gf=∫0t‍e-μ(t-τ)(-△)βf(τ,x)dτ; we have (41)μ1/q′GfL∞I;FB˙p,qs+μGfLqI;FB˙p,qs+2β/q≤CfLqI;FB˙p,qs-2β/q′.Proof.
Fourier transform gives(42)Gf∧ξ=∫0te-μt-τξ2βfτ^dτ.Multiplying φj and taking the Lp-norm(43)φjGf∧Lp≤∫0te-μt-τ22jβ3/42βfj^Lpdτ,where we denote uj^=φju^. Multiplying 2js and using Hölder’s inequality, we get(44)2jsφjGf∧Lp≤∫0te-μt-τ22jβ3/42β2j2β/q′2js-2β/q′fj^Lpdτ≤∫Ie-μq′t22jβ3/42β22jβdt1/q′2js-2β/q′fj^LqI;LpUsing ∫0∞‍e-μq′t22jβ(3/4)2β22jβdt=μq′(3/4)2β and taking lq-norm, we conclude(45)GfL∞I;FB˙p,qs≤Cμ-1/q′fLqI;FB˙p,qs-2β/q′.Similarly, multiplying 2j(s+2β/q)(46)2js+2β/qφjGf∧Lp≤∫0te-μt-τ22jβ3/42β2j2β/q+2β/q′2js-2β/q′fj^Lpdτ.Taking Lq-norm with respect to time on I and using Young’s inequality, (47)2js+2β/qφjGf∧LqI;Lp≤∫Ie-μt22jβ3/42β22jβdt2js-2β/q′fj^LqI;Lp.Now taking lq-norm, we obtain our desired inequality (48)GfLqI;FB˙p,qs+2β/q≤Cμ-1fLqI;FB˙p,qs-2β/q′.Lemma 16 (bilinear estimate).
Let1≤p,q≤∞,q′/1+q′<β<q′/1+q′min⁡{1+n/p′,1+n/2}. We have(49)Qu,vLqI;FB˙p,q1-2β+n/p′-2β/q′≤CuL2qI;FB˙p,q1-2β+n/p′+β/qvL2qI;FB˙p,q1-2β+n/p′+β/q.Particularly, the result also holds in the case p>1,q=1,β=1.Proof.
By Remark7 and Hölder’s inequality, it is sufficient to prove(50)Qu,vFB˙p,q1-2β+n/p′-2β/q′≤CuFB˙p,q1-2β+n/p′+2β/qvFB˙p,q1-2β+n/p′+2β/q.Note that all indices in the definition of Q(u,v) are finite; we have (51)Qu,vFB˙p,q1-2β+n/p′-2β/q′≤C∑F-1ξaξbξ2F∂mukvlFB˙p,q1-2β+n/p′-2β/q′.On the other hand, by Definition 8, we know(52)F-1ξaξbξ2F∂mukvlFB˙p,q1-2β+n/p′-2β/q′≤CukvlFB˙p,q2-2β+n/p′-2β/q′.Thus, we just need to consider the estimate of the product uvFB˙p,q2-2β+n/p′-2β/q′. However, set s1=s2=1-2β+n/p′+β/q, then by Theorem 1 and Corollary 13, we obtain our desired result.Next we introduce an abstract lemma on the existence of fixed point solutions [14, 19].Lemma 17.
LetX be a Banach space with norm ·X and B:X×X↦X be a bounded bilinear operator satisfying(53)Bu,vX≤ηuXvX,for all u,v∈X and a constant η>0. Then for any fixed y∈X satisfying yX<ϵ<1/4η, the equation x≔y+B(x,x) has a solution x¯ in X such that x¯X≤2yX. Also, the solution is unique in B¯(0,2ϵ). Moreover, the solution depends continuously on y in the sense: if y′X<ϵ,x′=y′+Bx′,x′,x′X<2ϵ, then (54)x¯-x′X≤11-4ϵηy-y′X.This lemma allows us to solve the Cauchy problem (1) with bounded bilinear form and small data. Now we begin our proof.Proof of Theorem3.
We first seek the solution in the spacesL2q(I;FB˙p,q1-2β+n/p′+β/q). By (6) of Proposition 11(55)fL2qI;FB˙p,q1-2β+n/p′+β/q≤fL∞I;FB˙p,q1-2β+n/p′1/2fLqI;FB˙p,q1-2β+n/p′+2β/q1/2.By Lemma 15 with s=1-2β+n/p′ and Lemma 16(56)Bu,vL2qI;FB˙p,q1-2β+n/p′+β/q≤Cμ-1/2-1/2q′uL2qI;FB˙p,q1-2β+n/p′+β/qvL2qI;FB˙p,q1-2β+n/p′+β/q.By Lemma 17, we know that if e-μt(-△)βu0L2q(I;FB˙p,q1-2β+n/p′+β/q)<R with R=μ1/2+1/2q′/4C, then (34) has a unique solution in B(0,2R), where (57)B0,2R≔x∈L2qI;FB˙p,q1-2β+n/p′+β/q:xL2qI;FB˙p,q1-2β+n/p′+β/q≤2R.
Now we need to derivee-μt(-△)βu0L2q(I;FB˙p,q1-2β+n/p′+β/q)<R. First, we consider small initial data. Lemma 14 and (55) imply that(58)e-μt-△βu0L2qI;FB˙p,q1-2β+n/p′+2β/q≤Cμ-1/2qu0FB˙p,q1-2β+n/p′.Thus we can take u0 such that u0FB˙p,q1-2β+n/p′<C0μ with C0=(4C2)-1. Next, for the large initial data u0, we divide u0 by u0=F-1χ{ξ≤γ}u0^+F-1χξ>γu0^≔u10+u20, where γ=γ(u0)>0 is a large real number determined later. Since F-1χξ>γu0^ converges to 0 in FB˙p,q1-2β+n/p′ as γ→+∞, by (58) there exists some γ large enough such that (59)eμt-△βu20L2qI;FB˙p,q1-2β+n/p′+β/q≤Cμ-1/2qu20FB˙p,q1-2β+n/p′≤μ1/2+1/2q′8C.Now for u01, there holds (60)eμt-△βu10L2qI;FB˙p,q1-2β+n/p′+β/q=∑j∈Z2j1-2β+n/p′+β/qqφje-μtξ2βχξ≤γu0^L2qI;Lpq1/q≤∑j∈Z2j1-2β+n/p′+β/qqsupξ≤γe-μtξ2βξβ/qL2q0,Tqφjξ-β/qu0^Lpq1/q≤Cγβ/qT1/2qu0FB˙p,q1-2β+n/p′.Thus we can choose T small enough such that (61)T≤μ1/2+1/2q′8C2γβ/qu0FB˙p,q1-2β+n/p′2q.We now conclude that (34) has a unique solution u∈L2q(I;FB˙p,q1-2β+n/p′+β/q). By (34), Lemmas 14 and 15, and Theorem 1, we conclude (62)u∈L∞I;FB˙p,q1-2β+n/p′∩LqI;FB˙p,q1-2β+n/p′+2β/q.The continuity with respect to time is standard and thus we prove Theorem 3 up to the blow-up criterion. Next we prove the blow-up criterion. Suppose T∗ is the maximal time of existence of mild solution associated with u0. If we have a solution of (1) on [0,T∗) such that(63)uL2q0,T∗;FB˙p,q1-2β+n/p′+β/q<∞,then the integral equation (34), Lemmas 14 and 15, and Theorem 1 imply that for all t∈[0,T∗)(64)utFB˙p,q1-2β+n/p′<∞.Using the integral equation (34), and by similar method with the proof of Lemmas 14, 15, and 16(65)ut′-utFB˙p,q1-2β+n/p′≤I-e-μt-t′-△βu0FB˙p,q1-2β+n/p′+Qu,uLqt,t′;FB˙p,q1-2β+n/p′-2β/q′+I-e-μt-t′-△βQut′,ut′Lt′q0,t′;FB˙p,q1-2β+n/p′-2β/q′.Since (I-e-μ(t-t′)-△β)u0FB˙p,q1-2β+n/p′ converges to 0 as t′→t, and from (63) we know that (66)Qu,uLqI;FB˙p,q1-2β+n/p′-2β/q′≤uL2qI;FB˙p,q1-β+n/p′-β/q2.Thus we can conclude that the other two terms also converge to 0 as t′→t, so u(t) satisfies the Cauchy criterion at T∗. Thus there exists an element u∗ in FB˙p,q1-2β+n/p′ such that u(t)→u∗ as t→T∗. Now set u(T∗)=u∗, and consider the fractional Navier-Stokes equations starting by u∗. Using Theorem 1 we obtain that a solution exists on a larger time interval than [0,T∗), which is a contradiction.Proof of Theorem4.
Using the same method in proving Theorem3, to prove Theorem 4, it is sufficient to re-estimate B(u,v), and by Lemma 15 we only need to prove the following lemma.Lemma 18.
Let1≤p≤q≤2,β∈(1/1+1/q′,1+n/2p′/1+1/q′]. Then there exists some constant C=C(β,p,q)>0 depending on β,p,q such that(67)Qu,vLqI;FB˙p,q1-2β+n/p′-2β/q′≤CuL2qI;FB˙p,q1-2β+n/p′+β/qvL2qI;FB˙p,q1-2β+n/p′+β/q.Particularly, it is true for the case: p=q=1,β=1.Proof.
Similarly to Lemma16, we only need to estimate uvLq(I:FB˙p,q2-2β+3/p′-2β/q′). Using Bony’s decomposition(68)uvLqI:FB˙p,q2-2β+3/p′-2β/q′≤83∑j22-2β+n/p′-2β/q′jqIj^LqI;Lpq1/q+∑j22-2β+n/p′-2β/q′jqIIj^LqI;Lpq1/q+∑j22-2β+n/p′-2β/q′jqIIIj^LqI;Lpq1/q.The terms Ij and IIj are symmetrical. Using Young’s inequality and Hölder’s inequality we have (69)Ij^LqI;Lp≤∑k-j≤4S˙k-1uΔ˙kv^LqI;Lp≤∑k-j≤4vk^L2qI;Lp∑l≤k-2ul^L2qI;L1≤∑k-j≤4vk^L2qI;Lp∑l≤k-221-2β+β/qlqul^L2qI;L1q1/q∑l≤k-222β-1-β/qlq′1/q′≤C∑k-j≤422β-1-β/qkvk^L2qI;LpuL2qI;FB˙1,q1-2β+β/q.Recalling L2q(I;FB˙p,q1-2β+n/p′+β/q)⊂L2q(I;FB˙1,q1-2β+β/q) in Proposition 11, there holds(70)∑j22-2β+n/p′-2β/q′jqIj^LqI;Lpq1/q≤CuL2qI;FB˙p,q1-2β+n/p′+β/qvL2qI;FB˙p,q1-2β+n/p′+β/q.In a similar way, we can prove(71)∑j22-2β+n/p′-2β/q′jqIIj^LqI;Lpq1/q≤CuL2qI;FB˙p,q1-2β+n/p′+β/qvL2qI;FB˙p,q1-2β+n/p′+β/q.For the remaining term, we invoke Minkowski’s inequality. In fact when q≥p(72)
∑
j
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
j
q
I
I
I
j
^
L
q
I
;
L
p
q
1
/
q
≤
∑
k
>
j
-
3
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
j
φ
j
ξ
u
k
^
∗
∑
l
-
k
≤
1
v
l
^
L
q
I
;
L
p
l
q
j
≤
∑
k
∑
j
<
k
+
3
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
j
φ
j
ξ
u
k
^
∗
∑
l
-
k
≤
1
v
l
^
L
q
I
;
L
p
q
1
/
q
≤
∑
k
∑
j
<
k
+
3
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
j
φ
j
ξ
u
k
^
∗
∑
l
-
k
≤
1
v
l
^
L
ξ
p
q
1
/
q
L
t
q
≤
∑
k
∫
R
3
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
p
j
φ
j
ξ
p
u
k
^
∗
∑
l
-
k
≤
1
v
l
^
p
d
ξ
l
q
/
p
j
<
k
+
3
1
/
p
L
t
q
≤
s
u
p
ξ
∑
j
φ
j
ξ
q
1
/
q
∑
k
2
2
-
2
β
+
n
/
p
′
-
2
β
/
q
′
k
+
3
u
k
^
∗
∑
l
-
k
≤
1
v
l
^
L
q
I
;
L
p
≤
C
∑
k
2
1
-
2
β
+
β
/
q
k
u
k
^
L
2
q
I
:
L
1
∑
l
-
k
≤
1
2
1
-
2
β
+
n
/
p
′
+
β
/
q
k
-
l
2
1
-
2
β
+
n
/
p
′
+
β
/
q
l
v
l
^
L
2
q
I
;
L
p
≤
C
u
L
2
q
I
;
F
B
˙
p
,
q
1
-
2
β
+
n
/
p
′
+
β
/
q
v
L
2
q
I
;
F
B
˙
p
,
q
1
-
2
β
+
n
/
p
′
+
β
/
q
.In the last inequality we use L2q(I;FB˙p,q1-2β+n/p′+β/q)⊂L2q(I;FB˙p,q′1-2β+n/p′+β/q) for q∈[1,2] by (4) of Proposition 11. Collecting the above estimates we conclude our desired result.


## 4. Space Analyticity

The proof is similar with the proof of Theorem3. Let(73)Ut,x≔eμtDβut,x.By the integral equation (34), we have (74)Ut,x=eμt-1/2tDβe-1/2μtDβu0+∫0teμt-τ-1/2t-τDβe-1/2t-τDβeτDβQu,udτ.Since eμ(t-1/2t)|ξ|β and eμ(t-τ-1/2(t-τ))|ξ|β are uniformly bounded on t∈(0,∞) and τ∈[0,t], together with Lemmas 14 and 15 we only need to consider the estimate of eτ|D|βQ(u,v)Lq(I;FB˙p,q1-2β+n/p′-2β/q′). To apply Lemma 17, we need a bilinear estimate. To this end, we write Q(u,v) as Q(e-τ|D|βU,e-τ|D|βV) with V≔eτ|D|βv and we prove the following lemma.Lemma 19.
Let1≤p,q≤∞,q′/1+q′<β<q′/1+q′min⁡1+n/p′,1+n/2,β≤1. Then(75)eτDβQu,vLqI;FB˙p,q1-2β+n/p′-2β/q′≤CUL2qI;FB˙p,q1-2β+n/p′+β/qVL2qI;FB˙p,q1-2β+n/p′+β/q.Particularly, our result also holds in the case p>1,q=1,β=1.Proof.
Using the same discussion in the proof of Lemma16, it is sufficient to consider the estimate of eτ|D|β(uv)Lq(I;FB˙p,q2-2β+n/p′-2β/q′). By Bony’s decomposition, we have for fixed j(76)Δ˙jeτDβuv=∑k-j≤4Δ˙jeτDβS˙k-1uΔ˙kv+∑k-j≤4Δ˙jeτDβS˙k-1vΔ˙ku+∑k≥j-3Δ˙jeτDβΔ˙kuΔ˙~kv≔Rj1+Rj2+Rj3.The idea is that the Lp-norm of FΔ˙jeτ|D|β(uv) can be bounded by that of FΔ˙j(UV). In fact, (77)Rj1^Lp=∑k-j≤4φjeτξβ∑l≤k-2e-τξβUl^∗e-τξβVk^Lp=∑k-j≤4φj∫Rneτξβ-ξ-ηβ-ηβ∑l≤k-2Ul^ξ-ηVk^ηdηLp.Since eτ(|ξ|β-|ξ-η|β-|η|β) is uniformly bounded on τ if β∈[0,1], we can eliminate this term. Thus, we obtain (78)Rj1^Lp≤∑k-j≤4φj∑l≤k-2Ul^∗Vk^Lp.Similarly, we can also obtain that(79)Rj2^Lp≤∑k-j≤4φj∑l≤k-2Vl^∗Uk^Lp;Rj3^Lp≤∑k-j≤4φjUk^∗∑l-k≤1Vl^Lp.Following the same step in the proof of Theorem 1 and Lemma 16, it is easy to conclude our desired result.By this lemma and the above observations, we can obtain Theorem5 by following the proof of Theorem 3 line by line. Similarly, to prove Theorem 6, it is sufficient to obtain the following lemma.Lemma 20.
Let1≤p≤q≤2,q′/1+q′<β<q′/1+q′(1+n/2p′),β≤1. Then(80)eτDβQu,vLqI;FB˙p,q1-2β+n/p′-2β/q′≤CUL2qI;FB˙p,q1-2β+n/p′+β/qVL2qI;FB˙p,q1-2β+n/p′+β/q.Particularly, our result also holds in the case p=q=1,β=1.Proof.
Using the discussion in Lemma19 and following the same steps in the proof of Lemma 18, it is easy to conclude our desired result.



---

*Source: 3629179-2018-12-04.xml*