# On the Blowup for the 3D Axisymmetric Incompressible Chemotaxis-Euler Equations

Authors: Shangyong An; Qian Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we investigate the 3D incompressible chemotaxis-Euler equations. Taking advantage of the structure of axisymmetric fluids, we establish the blowup criterion of the system using the Fourier localization method.


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## Body

## 1. Introduction

The effect of oxygen attraction on the emergence of bioconvective patterns is studied in [1, 2]. Some experiments, such as a colony of Bacillus subtilis suspending in a drop of water, are carried out to identify this phenomenon. From paper [3–6], we can also find the important role of chemotaxis between sperm and eggs. The following model, in [1], is introduced to analyze the above phenomenon:
(1)nt+u·∇n=Δn+χ∇·n∇Δ−1c+κn−μn2,nx,0,x∈d.Here,n and u represent the concentration of bacteria and the velocity field of the transported water, respectively. Besides, the vector field u is divergence free and independent of n. The equation describes the evolution of the bacteria transported by the velocity field of the fluid. Moreover, these cells are attracted by the oxygen concentration generated by chemotaxis. For the term χ∇·n∇Δ−1c, χ>0 is a parameter controlling the influence of the chemotactic effect. In addition, κ is the strength growth rate of the population and μ is a parameter regulating death by overcrowding.Apart from Equation (1), there are a lot of other models illustrating the procedure of oxygen attraction in biology. An increasing number of mathematicians studied the process in the past years, see [7–15]. Our aim in this paper is to further explore model (1), combined with an oxygen equation and a Navier-Stokes equation, see [16]. Then, we obtain the following model in d, d=2,3,
(2)nt+u·∇n=Δn−χ∇·n∇c+κn−μn2,ct+u·∇c=Δc−nc,ut+u·∇u−∇P=ηΔu−n∇Φ,∇·u=0,n0,x=n0x,c0,x=c0x,u0,x=u0x.The unknowns aren, c, u, and P, standing for the bacteria, the oxygen, the velocity field, and the pressure of the fluid separately. The third equation of the above system contained an extra force, buoyancy, which is produced by the density and a given gravitational potential Φ. η is the dissipation coefficient. If κ=μ=0 and η>0, the global existence of weak solutions in 2 was shown in [8, 12].In this paper, we chooseχ=κ=μ=1 and η=0, then (2) can be changed into the following one:
(3)nt+u·∇n=Δn−∇·n∇c+n−n2,ct+u·∇c=Δc−nc,ut+u·∇u−∇P=−n∇Φ,∇·u=0,n0,x=n0x,c0,x=c0x,u0,x=u0x.The Euler equation is shown as the following form:(4)ut+u·∇u+∇P=0,∇·u=0,u0,x=u0x.In three dimensional space, the vorticity equation has the form(5)ωt+u·∇ω=ω·∇u.But the chief difficulty is we are lacking information on the vortex-stretching termω·∇u. Although the global existence of classical solutions for the 3D Euler equation is an open problem, some known results are obtained under the circumstances of axisymmetric flows without swirl. That a vector field u is axisymmetric without swirl is defined as follows:
(6)ut,x=urt,r,zer+uzt,r,zez,x=x1,x2,z,r=x12+x221/2,where er,eθ,ez is the cylindrical basis of 3 and the components ur and uz do not depend on the angular variable. With this structure, vorticity takes the form
(7)ω=∂zur−∂ruzeθ≔ωθeθand satisfies
(8)∂tω+u·∇ω=urrω.Hence, the quantityΓ≔ωθ/r obeys to the equation
(9)∂tΓ+u·∇Γ=0.The goal of this paper is to build the blowup criterion of smooth solutions for (3) by the Fourier localization technique. Here, we follow ideas introduced in [17–21]. Our result reads as the following:Theorem 1.
Fors>3, suppose the triple n0,c0,u0∈Hs×Hs+1×Hs+1 and Φ∈Hs+2. Let u0 be an axisymmetric divergence-free vector field and its vorticity satisfies ω0/r∈L3,1. Assume that
(10)n∈C0,T;Hs∩L20,T;Hs+1,c∈C0,T;Hs+1∩L20,T;Hs+2,u∈C0,T;Hs+1are the smooth solutions to (3). If the condition
(11)∫0T∥nτ∥B∞,102dτ<∞holds true, then the solutions n,c,u can be extended beyond T>0.Remark 2.
In paper [20], a regularity criterion in terms of two items is established. But in Theorem 1, we give a different criterion using the only bacteria concentration in 3. The bacteria concentration plays a more important role in this model, and the nonlinear term −∇·n∇c is difficult to estimate. Hence, using bacteria concentration to show the regularity is natural and physical.Notation.
Throughout the paper,C means a harmless constant and may vary from line to line; CT denotes a constant C relating to T; ∥·∥p stands for the norm of the Lebesgue space Lp.


## 2. Preliminaries

In this section, we give the definition of some function spaces and recall some useful lemmas.Firstly, we use the dynamic partition of the unity to give the definition of Besov spaces. One may check [22] for exact details. Let φ∈C0∞d be set in C=ξ∈d,3/4≤∣ξ∣≤8/3 satisfying
(12)∑q∈Zφ2−qξ=1,forξ≠0.Letχξ=1−∑q∈Nφ2−qξ. For f∈S′, Littlewood-Paley operators are defined as follows:
(13)Δ−1f=χDf;∀q∈Δqf=φ2−qDfand∀q∈Δ̇qf=φ2−qDf.The low-frequency cut-offs are denoted:(14)Sqf=∑−1≤q′≤q−1Δq′f,Ṡqf=∑q′≤q−1Δ̇q′f.Now, we introduce the definition of the Besov space. Fors∈ℝ,1≤p,r≤∞, the homogenous Besov space Ḃp,rs is defined as the set of tempered distributions of f∈S′/P satisfying
(15)fḂp,rs≔∑q∈Z2qsrΔ̇qfpr1/r<∞,where P is the polynomial space. The inhomogeneous space Bp,rs is the set of tempered distribution f with the norm
(16)fBp,rs≔∑q≥−12qsrΔqfpr1/r<∞.It is worthwhile to remark thatB2,2s and B∞,∞s coincide with the usual Sobolev spaces Hs and the usual Hölder space Cs for s∈ℝ\ℤ, respectively.In our study, we require the space-time Besov spaces as the following manner: forT>0 and ρ≥1, we denote by LTρBp,rs the set of all tempered distribution f such that
(17)fLTρBp,rs≜∑q∈ℤ2qsrΔ̇qfLpℝdr1/rLTρ<∞.Lemma 3 (see [22]).
Let1≤p≤q≤∞. Suppose that f∈Lp, then there exists a constant C independent of f, j such that
(18)suppf^⊂∣ξ∣≤C2j⇒∥∂αf∥q≤C2j∣α∣+dj1/p−1/q∥f∥p,suppf^⊂1C2j≤∣ξ∣≤C2j⇒∥f∥p≤C2−j∣α∣sup∣β∣=∣α∣∥∂βf∥p.Lemma 4 (see [22]).
There exists a constantC>0 such that for s>0, we have
(19)∥uv∥Hs≤C∥u∥∞∥v∥Hs+C∥u∥Hs∥v∥∞.Lemma 5 (see [23]).
Letu be a solution of the transport equation
(20)ut+v·∇u=0,ux,0=u0,and define Rq≔v·∇Δqu−Δqv·∇u, 1≤p≤p1≤∞, 1≤r≤∞, and s∈suchthats>−dmin1/p1,1/p′ors>−1−dmin1/p1,1/p′ifdivv=0. There exists a sequence cq∈ℓrℤ such that ∥cq∥ℓr=1 and a constant C depending only on d,r,s,p, and p1, which satisfy
(21)∀q∈ℤ,2qs∥Rq∥p≤CcqZ′t∥u∥Bp,rs,with
(22)Z′t≔∥∇v∥Bp1,∞d/p1∩L∞,ifs<1+dp1,∥∇v∥Bp1,rs−1,ifeithers>1+dp1ors=1+dp1forr=1.Lemma 6 (see [24]).
Letp,r∈1,∞2, v be a divergence-free vector-field belonging to the space Lloc1ℝ+;Lipℝd and let a be a smooth solution of the following transport equation:
(23)at+v·∇a=f,ax,0=a0.
If the initial dataa0∈Bp,r0, then we have for all t∈+ℝ(24)∥a∥L~t∞Bp,r0≤C∥a0∥Bp,r0+∥f∥L~t1Bp,r01+∫0t∥∇vτ∥L∞dτ.


## 3. Proof of Theorems

### 3.1. Local Well-Posedness

We construct the following smoothing system:(25)ntk+uk·∇nk=Δnk−∇·nk∇ck+nk−nk2,k∈N,ctk+uk·∇ck=Δck−cknk,utk+uk·∇uk−∇Pk=Δuk+nk∇Φ,∇·uk=0,nk,ck,ukt=0=Skn0,Skc0,Sku0.Step 1.
Uniform boundedness.Taking the operationΔq with q≥−1 on the first equation of (25), we obtain
(26)Δqntk+Δquk·∇nk=ΔΔqnk−∇·Δqnk∇ck+Δqnk−Δqnk2.Making theL2-inner product for (26) with Δqnk yields
(27)12ddt∥Δqnk∥22+∥∇Δqnk∥22=−∫ℝdΔquk·∇nkΔqnkdx−∫ℝd∇·Δqnk∇ckΔqnkdx+∫ℝdΔqnkΔqnkdx−∫ℝdΔqnk2Δqnkdx≤∥Δquk·∇nk∥2∥Δqnk∥2+∥Δqnk∇ck∥2∥∇Δqnk∥2+∥Δqnk∥22+∥Δqnk2∥2∥Δqnk∥2.Multiplying22qs on both sides of the above inequality, then taking the ℓ1 norm, using Hölder’s inequality and Young’s inequality together with Lemma 4, we have
(28)12ddt∥nk∥Hs2+∥nk∥Hs+12≤∥uk·∇nk∥Hs∥nk∥Hs+∥nk∇ck∥Hs∥nk∥Hs+1+∥nk∥Hs2+∥nk2∥Hs∥nk∥Hs≤C∥uk∥Hs∥nk∥Hs+1∥nk∥Hs+C∥nk∥Hs∥ck∥Hs+1∥nk∥Hs+1+∥nk∥Hs2+C∥nk∥Hs2∥nk∥Hs≤C∥uk∥Hs2∥nk∥Hs2+18∥nk∥Hs+12+C∥nk∥Hs2∥ck∥Hs+12+18∥nk∥Hs+12+∥nk∥Hs2+C∥nk∥Hs4+∥nk∥Hs2.Then, we conclude(29)ddt∥nk∥Hs2+∥nk∥Hs+12≤C∥uk∥Hs2∥nk∥Hs2+∥nk∥Hs2∥ck∥Hs+12+∥nk∥Hs2+∥nk∥Hs4.In a similar way to (29), we obtain
(30)12ddt∥ck∥Hs+12+∥ck∥Hs+22≤C∥uk∥Hs+12∥ck∥Hs+12+18∥ck∥Hs+22+C∥ck∥Hs+14+18∥nk∥Hs+12.Thus, we have(31)ddt∥ck∥Hs+12+∥ck∥Hs+22≤C∥uk∥Hs+12∥ck∥Hs+12+∥ck∥Hs+14+18∥nk∥Hs+12.OperatingΔq with q≥−1 to the third equation of (25) implies
(32)Δqutk+uk·∇Δquk−∇ΔqPk=uk·∇Δquk−Δquk·∇uk−Δqnk∇Φ.Taking theL2-inner product for the above equality with Δquk gives
(33)12ddt∥Δquk∥22=∫ℝduk·∇Δquk−Δquk·∇ukΔqukdx−∫ℝdΔqnk∇ΦΔqukdx≤∥uk·∇Δquk−Δquk·∇uk∥2∥Δquk∥2+∥Δqnk∇Φ∥2∥Δquk∥2.Multiplying22qs+1 on both sides of the above inequality and taking the ℓ1 norm, we conclude
(34)ddt∥uk∥Hs+12≤∥∇uk∥∞∥uk∥Hs+12+18∥nk∥Hs+12+∥uk∥Hs+12≤C∥uk∥Hs+14+∥uk∥Hs+12+18∥nk∥Hs+12.Collecting (29)–(34), we have
(35)ddt∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12+∥nk∥Hs+12+∥ck∥Hs+22≤C∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+121+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12≤1+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+122.We obtain from the Gronwall inequality that(36)1+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12≤1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+121−C1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12t.Let(37)T=12C1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12>0,then we obtain
(38)supt∈0,T∥nkt∥Hs2+∥ckt∥Hs+12+∥ukt∥Hs+12+∫0t∥nk∥Hs+12+∥ck∥Hs+22τdτ≤21+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12.Step 2.
Extracting sequences.According to (38), we get
(39)nk∈L∞0,T,Hs∩L20,T,Hs+1,ck∈L∞0,T,Hs+1∩L20,T,Hs+2,uk∈L∞0,T,Hs+1.In order to prove the convergence, we require uniform boundedness for∂tnk, ∂tck, and ∂tuk. By the first equation of (25), we infer
(40)∥∂tnk∥Lt∞H−1≤∥Δnk∥Lt∞H−1+∥uk·∇nk∥Lt∞H−1+∥∇·nk∇ck∥Lt∞H−1+∥nk∥Lt∞H−1+∥nk2∥Lt∞H−1≤∥nk∥Lt∞Hs+∥uk∥Lt∞Hs+1∥nk∥Lt∞Hs+∥nk∥Lt∞Hs∥ck∥Lt∞Hs+1+∥nk∥Lt∞Hs+∥nk∥Lt∞Hs2≤C.In a similar process, we have(41)∥∂tck∥Lt∞H−1≤∥Δck∥Lt∞H−1+∥uk·∇ck∥Lt∞H−1+∥cknk∥Lt∞H−1≤∥ck∥Lt∞Hs+1+∥uk∥Lt∞Hs+1∥ck∥Lt∞Hs+1+∥ck∥Lt∞Hs+1∥nk∥Lt∞Hs≤C.∥∂tuk∥Lt∞H−1≤∥uk·∇uk∥Lt∞H−1+∥nk∇Φ∥Lt∞H−1≤∥uk∥Lt∞Hs+12+∥nk∥Lt∞Hs≤C.SinceL2 is locally compactly embedded in H−1, we can apply the Aubin-Lions Lemma to deduce that, extracting a subsequence, the approximate solution sequence nk,ck,uk strongly converges in L∞0,T;H−1 to some function n,c,u such that
(42)nk∈L∞0,T;Hs∩L20,T,Hs+1,ck∈L∞0,T;Hs+1∩L20,T,Hs+2,uk∈L∞0,T;Hs+1.By the above estimates, we can easily have the limit in the approximate system (25) and n,c,u solve (3) in the sense of distribution. Using a classical method [12], we have n∈C0,T;Hs, c∈C0,T;Hs+1, and u∈C0,T;Hs+1.Step 3.
Uniqueness.Let us consider the two solutionsn1,c1,u1 and n2,c2,u2 associated with the same initial data and satisfy (3). We use the notation δn=n1−n2, δc=c1−c2, and δu=u1−u2. Then, we have
(43)∂tδn+δu·∇n1+u2·∇δn=Δδn−∇·δn∇c1−∇·n2∇δc+δn−n1δn−n2δn,∂tδc+δu·∇c1+u2·∇δc=Δδc−n1δc−c2δn,∂tδu+δu·∇u1+u2·∇δu−∇P1−P2=Δδu+δn∇Φ.Multiplying the first equation of (43) by δn and integrating in spaces, we obtain
(44)12ddt∥δnt∥22+∥∇δnt∥22=−∫ℝdδu·∇n1δndx−∫ℝd∇·δu∇c1δndx−∫ℝd∇·n2∇δcδndx+∫ℝdδnδndx−∫ℝdn1δnδndx−∫ℝdn2δnδndx≤C∥δu∥22+∥δn∥22∥n1∥Hs2+C∥δn∥22∥c1∥Hs+12+18∥∇δn∥22+C∥n2∥Hs2∥∇δc∥22+18∥∇δn∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22,from which we conclude
(45)ddt∥δnt∥22+∥∇δnt∥22≤C∥δu∥22+∥δn∥22∥n1∥Hs2+∥δn∥22∥c1∥Hs+12+∥n2∥Hs2∥∇δc∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22.Then, multiplying the second equation of (43) by δc and integrating in spaces, we know
(46)12ddt∥δct∥22+∥∇δct∥22=−∫ℝdδu·∇c1δcdx−∫ℝdδcn1δcdx−∫ℝdδnc2δcdx≤C∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12.Hence, we get(47)ddt∥δct∥22+∥∇δct∥22≤C∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12.Applying∂i on both sides of the second equation of Equation (43) gives
(48)∂t∂iδc+u2·∇∂i∇c−Δ∂iδc=−∂iδu·∇c1−∂iu2·∇δc−∂in1δc−∂ic2δn.Taking theL2-inner product for the above equation with ∂iδc, we obtain
(49)12ddt∥∇δct∥22+∥Δδct∥22=−∑i∫ℝd∂iδu·∇c1∂iδcdx−∑i∫ℝd∂iu2·∇δc∂iδcdx−∑i∫ℝd∂in1δc∂iδcdx−∑i∫ℝd∂ic2δn∂iδcdx=∫ℝdδu·∇c1Δδcdx−∫ℝd∇δc·∇u2·∇δcdx+∫ℝdn1δcΔδcdx+∫ℝdc2δnΔδcdx≤C∥δu∥22∥c1∥Hs+12+18∥Δδc∥22+C∥∇δc∥22∥u2∥Hs+1+C∥δc∥22∥n1∥Hs2+18∥Δδc∥22+C∥δn∥22∣c2∥Hs+12+18∥Δδc∥22.Hence, we have(50)ddt∥∇δct∥22+∥Δδct∥22≤C∥δu∥22∥c1∥Hs+12+∥∇δc∥22∥u2∥Hs+1+∥δc∥22∥n1∥Hs2+∥δn∥22∣c2∥Hs+12.Multiplying the third equation of system (43) by δu and integrating in spaces, we get
(51)12ddt∥δut∥22=−∫ℝdδu·∇u1·δudx+∫ℝdδn∇Φ·δudx≤C∥δu∥22∥u1∥Hs+1+C∥δn∥22+∥δu∥22.Thus,(52)ddt∥δut∥22≤C∥δu∥22∥u1∥Hs+1+∥δn∥22+∥δu∥22.From (45)–(52), we obtain
(53)ddt∥δnt∥22+∥δct∥22+∥∇δct∥22+∥δut∥22+∥∇δn∥22+∥∇δc∥22+∥Δδc∥22≤C∥δu∥22+∥δn∥22∥n1∥Hs2+∥δn∥22∥c1∥Hs+12+∥n2∥Hs2∥∇δc∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22+∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12+∥δu∥22∥c1∥Hs+12+∥∇δc∥22∥u2∥Hs+1+∥δc∥22∥n1∥Hs2+∥δn∥22∣c2∥Hs+12+∥δu∥22∥u1∥Hs+1+∥δn∥22+∥δu∥22.Then, we have(54)ddt∥δnt∥22+∥δct∥22+∥∇δct∥22+∥δut∥22≤CFt∥δn∥22+∥δc∥22+∥∇δc∥22+∥δu∥22,where
(55)Ft=1+∥n1∥Hs+∥n2∥Hs+∥u1∥Hs+1+∥u2∥Hs+1+∥n1∥Hs2+∥n2∥Hs2+∥c1∥Hs+12+∥c2∥Hs+12.From (3), we infer that Ft is integrable. Using the Gronwall inequality gives the uniqueness.


### 3.2. Blowup Criterion

OperatingΔq with q≥−1 to the first equation of (3) gives
(56)Δqnt+u·∇Δqn=ΔΔqn+u·∇Δqn−Δqu·∇n−∇·Δqn∇c+Δqn−Δqn2.Taking theL2-inner product for the above equation with Δqn, we get
(57)12ddt∥Δqn∥22+∥∇Δqn∥22=∫du·∇Δqn−Δqu·∇nΔqndx+∫dΔqn∇c·∇Δqndx+∫dΔqnΔqndx−∫dΔqn2Δqndx≤∥u·∇Δqn−Δqu·∇n∥2∥Δqn∥2+∥Δqn∇c∥2∥∇Δqn∥2+∥Δqn∥22+∥Δqn2∥2∥Δqn∥2.Multiplying22qs on both sides of the above inequality and performing ℓ1 norm, we have
(58)ddt∥n∥Hs2+∥n∥Hs+12≤∥∇u∥∞∥n∥Hs2+∥n∇c∥Hs∥n∥Hs+1+∥n∥Hs2+∥n2∥Hs∥n∥Hs.Using Young’s inequality, we conclude(59)ddt∥n∥Hs2+∥n∥Hs+12≤C∥∇u∥∞∥n∥Hs2+C∥n∥∞2∥∇c∥Hs2+C∥n∥Hs2∥∇c∥∞2+18∥n∥Hs+12+∥n∥Hs2+C∥n∥∞2∥n∥Hs2+18∥n∥Hs+12.In terms of the second equation of (3), we know
(60)Δqct+u·∇Δqc=ΔΔqc+u·∇Δqc−Δqu·∇c−Δqnc.Multiplying the above equality byΔqc and integrating in spaces mean
(61)12ddt∥Δqc∥22+∥∇Δqc∥22=∫ℝdu·∇Δqc−Δqu·∇cΔqcdx+∫ℝdΔqncΔqcdx≤∥u·∇Δqc−Δqu·∇c∥2∥Δqc∥2+∥Δqnc∥2∥Δqc∥2.Multiplying22qs+1 on both sides of the above inequality and taking ℓ1 norm, we obtain
(62)ddt∥c∥Hs+12+∥c∥Hs+22≤∥∇u∥∞∥c∥Hs+12+∥nc∥Hs+1∥c∥Hs+1.Utilizing Young’s inequality, we have(63)ddt∥c∥Hs+12+∥c∥Hs+22≤C∥∇u∥∞∥c∥Hs+12+C∥n∥∞2∥c∥Hs+12+C∥c∥∞2∥c∥Hs+12+18∥n∥Hs+12.According to the third equation of (3), we get
(64)Δqut+u·∇Δqu−∇ΔqP=u·∇Δqu−Δqu·∇u−Δqn∇Φ.Taking theL2-inner product for the above equality with Δqu implies
(65)12ddt∥Δqu∥22=∫ℝdu·∇Δqu−Δqu·∇uΔqudx−∫ℝdΔqn∇ΦΔqudx≤∥u·∇Δqu−Δqu·∇u∥2∥Δqu∥2+∥Δqn∇Φ∥2∥Δqu∥2.Multiplying22qs+1 on both sides of the above inequality and taking the ℓ1 norm, we have
(66)ddt∥u∥Hs+12≤C∥∇u∥∞∥u∥Hs+12+18∥n∥Hs+12+C∥u∥Hs+12.Collecting (59)–(66), we deduce
(67)ddt∥n∥Hs2+∥c∥Hs+12+∥u∥Hs+12+∥n∥Hs+12+∥c∥Hs+22≤∥n∥Hs2+∥c∥Hs+12+∥u∥Hs+121+∥∇u∥∞+∥n∥∞2+∥∇c∥∞2.The Gronwall inequality implies(68)∥nt∥Hs2+∥ct∥Hs+12+∥ut∥Hs+12+∫0t∥nτ∥Hs+12+∥cτ∥Hs+22dτ≤Cexp∫0t1+∥∇uτ∥∞+∥nτ∥∞2+∥∇cτ∥∞2dτ.Next, we turn to prove condition (11). Applying ∇ on both sides of the second equation of (3) means
(69)∂t∇c+u·∇2c−Δ∇c=−∇nc−∇u·∇c.Multiplying the above equality with∇cp−2∇c, we obtain
(70)1pddt∥∇c∥pp+∫ℝdu·∇2c∇cp−2∇cdx+4p−1p2∥∇∇cp/2∥22=−∫ℝd∇nc∇cp−2∇cdx−∫ℝd∇u·∇c∇cp−2∇cdx≤p−1∫ℝdnc∇cp−2Δcdx+∥∇u∥∞∥∇c∥pp≤2p−1p∥n∥∞∥c∥p∥∇cp−2/2∥2p/p−2∥∇∇cp/2∥2+∥∇u∥∞∥∇c∥pp≤2p−1pp2p−1ε∥n∥∞2∥c∥p2∥∣∇c∣∥p−22+2p−1εp∥∇∇cp/2∥22+∥∇u∥∞∥∇c∥pp.Because of(71)∫ℝdu·∇2c∇cp−2∇cdx=1p∫ℝdu·∇∇cpdx=0,we get
(72)ddt∥∇c∥p2≤C∥n∥∞2+∥∇u∥∞∥∇c∥p2.Utilizing Gronwall’s inequality, we have(73)∥∇ct∥p2≤C∥∇c0∥p+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ≤C∥c0∥Hs+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ.Settingp→∞, we conclude
(74)∥∇ct∥∞2≤∥c0∥Hs+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ.Submitting (74) into (68) gives
(75)∥nt∥Hs2+∥ct∥Hs+12+∥ut∥Hs+12+∫0t∥nτ∥Hs+12+∥cτ∥Hs+22dτ≤Cexpexp∫0t1+∥∇uτ∥∞+∥nτ∥∞2dτ.On the other hand, using the inhomogeneous dynamic partition of the unity, we have(76)∥∇u∥L∞=∥∑j≥−1∇Δju∥L∞≤∥∇Δ−1u∥L∞+∑j=0∞∥∇Δj∥L∞≤C∥u0∥L2+∑j=0∞∥Δjω∥L∞≤C∥u0∥L2+∥ω∥B∞,10.Taking the curl to the third equation of (3) implies
(77)ωt+u·∇ω=urrω−curln∇ϕ.Using Lemma6, we obtain
(78)∥ω∥B∞,10≤C∥ω0∥B∞,10+∫0t∥curln∇ϕτ∥B∞,10+∥urrω∥B∞,10×1+∫0t∥∇uτ∥L∞dτ≤C∥ω0∥B∞,10+∫0t∥curln∇ϕτ∥B∞,10+∥urr∥L∞∥ω∥B∞,10×1+∫0t∥∇uτ∥L∞dτ.For the term∥curln∇ϕτ∥B∞,10, using Bony’s decomposition, we have
(79)∥curln∇ϕτ∥B∞,10=∑∣q−q′∣≤42q∥ΔqSq′−1nΔq′∇ϕ∥L∞+∑∣q−q′∣≤42q∥ΔqSq′−1∇ϕΔq′n∥L∞+∑∣q′′−q′∣≤1q′≥q−32q∥ΔqΔq′nΔq′′∇ϕ∥L∞≜I1+I2+I3.For the termI1, we have
(80)I1≤C∑∣q−q′∣≤4∑j≤q′−22q∥Δq′∇ϕ∥L∞∥Δjn∥L∞≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Similarly,(81)I2≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.As forI3,
(82)I3≤C∑∣q′′−q′∣≤1q′′≥q−32q−q′′2q′′∥Δq′′∇ϕ∥L∞∥Δq′n∥L∞≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Plugging (80)–(82) into (79) yields
(83)∥curln∇ϕτ∥B∞,10≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Putting (83) into (78), using the fact ∥urt/r∥L∞≤C∥ω0/r∥L3,1 [25] and Gronwall’s inequality, we have
(84)∥ω∥B∞,10≤C∥ω0∥B∞,10+∫0t∥nτ∥B∞,10dτe∫0t∥∇uτ∥L∞dτ.Substituting (84) into (76) gives
(85)∥∇u∥L∞≤C∥u0∥Hs+∫0t∥nτ∥B∞,10dτe∫0t∥∇uτ∥L∞dτ.Applying the Gronwall inequality, we get(86)∥∇u∥L∞≤CeexpC∫0t∥nτ∥B∞,10.Substituting (86) into (75) and using the fact B∞,10°L∞, we obtain the desired result.This completes the proof of Theorem1.


## 3.1. Local Well-Posedness

We construct the following smoothing system:(25)ntk+uk·∇nk=Δnk−∇·nk∇ck+nk−nk2,k∈N,ctk+uk·∇ck=Δck−cknk,utk+uk·∇uk−∇Pk=Δuk+nk∇Φ,∇·uk=0,nk,ck,ukt=0=Skn0,Skc0,Sku0.Step 1.
Uniform boundedness.Taking the operationΔq with q≥−1 on the first equation of (25), we obtain
(26)Δqntk+Δquk·∇nk=ΔΔqnk−∇·Δqnk∇ck+Δqnk−Δqnk2.Making theL2-inner product for (26) with Δqnk yields
(27)12ddt∥Δqnk∥22+∥∇Δqnk∥22=−∫ℝdΔquk·∇nkΔqnkdx−∫ℝd∇·Δqnk∇ckΔqnkdx+∫ℝdΔqnkΔqnkdx−∫ℝdΔqnk2Δqnkdx≤∥Δquk·∇nk∥2∥Δqnk∥2+∥Δqnk∇ck∥2∥∇Δqnk∥2+∥Δqnk∥22+∥Δqnk2∥2∥Δqnk∥2.Multiplying22qs on both sides of the above inequality, then taking the ℓ1 norm, using Hölder’s inequality and Young’s inequality together with Lemma 4, we have
(28)12ddt∥nk∥Hs2+∥nk∥Hs+12≤∥uk·∇nk∥Hs∥nk∥Hs+∥nk∇ck∥Hs∥nk∥Hs+1+∥nk∥Hs2+∥nk2∥Hs∥nk∥Hs≤C∥uk∥Hs∥nk∥Hs+1∥nk∥Hs+C∥nk∥Hs∥ck∥Hs+1∥nk∥Hs+1+∥nk∥Hs2+C∥nk∥Hs2∥nk∥Hs≤C∥uk∥Hs2∥nk∥Hs2+18∥nk∥Hs+12+C∥nk∥Hs2∥ck∥Hs+12+18∥nk∥Hs+12+∥nk∥Hs2+C∥nk∥Hs4+∥nk∥Hs2.Then, we conclude(29)ddt∥nk∥Hs2+∥nk∥Hs+12≤C∥uk∥Hs2∥nk∥Hs2+∥nk∥Hs2∥ck∥Hs+12+∥nk∥Hs2+∥nk∥Hs4.In a similar way to (29), we obtain
(30)12ddt∥ck∥Hs+12+∥ck∥Hs+22≤C∥uk∥Hs+12∥ck∥Hs+12+18∥ck∥Hs+22+C∥ck∥Hs+14+18∥nk∥Hs+12.Thus, we have(31)ddt∥ck∥Hs+12+∥ck∥Hs+22≤C∥uk∥Hs+12∥ck∥Hs+12+∥ck∥Hs+14+18∥nk∥Hs+12.OperatingΔq with q≥−1 to the third equation of (25) implies
(32)Δqutk+uk·∇Δquk−∇ΔqPk=uk·∇Δquk−Δquk·∇uk−Δqnk∇Φ.Taking theL2-inner product for the above equality with Δquk gives
(33)12ddt∥Δquk∥22=∫ℝduk·∇Δquk−Δquk·∇ukΔqukdx−∫ℝdΔqnk∇ΦΔqukdx≤∥uk·∇Δquk−Δquk·∇uk∥2∥Δquk∥2+∥Δqnk∇Φ∥2∥Δquk∥2.Multiplying22qs+1 on both sides of the above inequality and taking the ℓ1 norm, we conclude
(34)ddt∥uk∥Hs+12≤∥∇uk∥∞∥uk∥Hs+12+18∥nk∥Hs+12+∥uk∥Hs+12≤C∥uk∥Hs+14+∥uk∥Hs+12+18∥nk∥Hs+12.Collecting (29)–(34), we have
(35)ddt∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12+∥nk∥Hs+12+∥ck∥Hs+22≤C∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+121+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12≤1+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+122.We obtain from the Gronwall inequality that(36)1+∥nk∥Hs2+∥ck∥Hs+12+∥uk∥Hs+12≤1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+121−C1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12t.Let(37)T=12C1+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12>0,then we obtain
(38)supt∈0,T∥nkt∥Hs2+∥ckt∥Hs+12+∥ukt∥Hs+12+∫0t∥nk∥Hs+12+∥ck∥Hs+22τdτ≤21+∥n0k∥Hs2+∥c0k∥Hs+12+∥u0k∥Hs+12.Step 2.
Extracting sequences.According to (38), we get
(39)nk∈L∞0,T,Hs∩L20,T,Hs+1,ck∈L∞0,T,Hs+1∩L20,T,Hs+2,uk∈L∞0,T,Hs+1.In order to prove the convergence, we require uniform boundedness for∂tnk, ∂tck, and ∂tuk. By the first equation of (25), we infer
(40)∥∂tnk∥Lt∞H−1≤∥Δnk∥Lt∞H−1+∥uk·∇nk∥Lt∞H−1+∥∇·nk∇ck∥Lt∞H−1+∥nk∥Lt∞H−1+∥nk2∥Lt∞H−1≤∥nk∥Lt∞Hs+∥uk∥Lt∞Hs+1∥nk∥Lt∞Hs+∥nk∥Lt∞Hs∥ck∥Lt∞Hs+1+∥nk∥Lt∞Hs+∥nk∥Lt∞Hs2≤C.In a similar process, we have(41)∥∂tck∥Lt∞H−1≤∥Δck∥Lt∞H−1+∥uk·∇ck∥Lt∞H−1+∥cknk∥Lt∞H−1≤∥ck∥Lt∞Hs+1+∥uk∥Lt∞Hs+1∥ck∥Lt∞Hs+1+∥ck∥Lt∞Hs+1∥nk∥Lt∞Hs≤C.∥∂tuk∥Lt∞H−1≤∥uk·∇uk∥Lt∞H−1+∥nk∇Φ∥Lt∞H−1≤∥uk∥Lt∞Hs+12+∥nk∥Lt∞Hs≤C.SinceL2 is locally compactly embedded in H−1, we can apply the Aubin-Lions Lemma to deduce that, extracting a subsequence, the approximate solution sequence nk,ck,uk strongly converges in L∞0,T;H−1 to some function n,c,u such that
(42)nk∈L∞0,T;Hs∩L20,T,Hs+1,ck∈L∞0,T;Hs+1∩L20,T,Hs+2,uk∈L∞0,T;Hs+1.By the above estimates, we can easily have the limit in the approximate system (25) and n,c,u solve (3) in the sense of distribution. Using a classical method [12], we have n∈C0,T;Hs, c∈C0,T;Hs+1, and u∈C0,T;Hs+1.Step 3.
Uniqueness.Let us consider the two solutionsn1,c1,u1 and n2,c2,u2 associated with the same initial data and satisfy (3). We use the notation δn=n1−n2, δc=c1−c2, and δu=u1−u2. Then, we have
(43)∂tδn+δu·∇n1+u2·∇δn=Δδn−∇·δn∇c1−∇·n2∇δc+δn−n1δn−n2δn,∂tδc+δu·∇c1+u2·∇δc=Δδc−n1δc−c2δn,∂tδu+δu·∇u1+u2·∇δu−∇P1−P2=Δδu+δn∇Φ.Multiplying the first equation of (43) by δn and integrating in spaces, we obtain
(44)12ddt∥δnt∥22+∥∇δnt∥22=−∫ℝdδu·∇n1δndx−∫ℝd∇·δu∇c1δndx−∫ℝd∇·n2∇δcδndx+∫ℝdδnδndx−∫ℝdn1δnδndx−∫ℝdn2δnδndx≤C∥δu∥22+∥δn∥22∥n1∥Hs2+C∥δn∥22∥c1∥Hs+12+18∥∇δn∥22+C∥n2∥Hs2∥∇δc∥22+18∥∇δn∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22,from which we conclude
(45)ddt∥δnt∥22+∥∇δnt∥22≤C∥δu∥22+∥δn∥22∥n1∥Hs2+∥δn∥22∥c1∥Hs+12+∥n2∥Hs2∥∇δc∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22.Then, multiplying the second equation of (43) by δc and integrating in spaces, we know
(46)12ddt∥δct∥22+∥∇δct∥22=−∫ℝdδu·∇c1δcdx−∫ℝdδcn1δcdx−∫ℝdδnc2δcdx≤C∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12.Hence, we get(47)ddt∥δct∥22+∥∇δct∥22≤C∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12.Applying∂i on both sides of the second equation of Equation (43) gives
(48)∂t∂iδc+u2·∇∂i∇c−Δ∂iδc=−∂iδu·∇c1−∂iu2·∇δc−∂in1δc−∂ic2δn.Taking theL2-inner product for the above equation with ∂iδc, we obtain
(49)12ddt∥∇δct∥22+∥Δδct∥22=−∑i∫ℝd∂iδu·∇c1∂iδcdx−∑i∫ℝd∂iu2·∇δc∂iδcdx−∑i∫ℝd∂in1δc∂iδcdx−∑i∫ℝd∂ic2δn∂iδcdx=∫ℝdδu·∇c1Δδcdx−∫ℝd∇δc·∇u2·∇δcdx+∫ℝdn1δcΔδcdx+∫ℝdc2δnΔδcdx≤C∥δu∥22∥c1∥Hs+12+18∥Δδc∥22+C∥∇δc∥22∥u2∥Hs+1+C∥δc∥22∥n1∥Hs2+18∥Δδc∥22+C∥δn∥22∣c2∥Hs+12+18∥Δδc∥22.Hence, we have(50)ddt∥∇δct∥22+∥Δδct∥22≤C∥δu∥22∥c1∥Hs+12+∥∇δc∥22∥u2∥Hs+1+∥δc∥22∥n1∥Hs2+∥δn∥22∣c2∥Hs+12.Multiplying the third equation of system (43) by δu and integrating in spaces, we get
(51)12ddt∥δut∥22=−∫ℝdδu·∇u1·δudx+∫ℝdδn∇Φ·δudx≤C∥δu∥22∥u1∥Hs+1+C∥δn∥22+∥δu∥22.Thus,(52)ddt∥δut∥22≤C∥δu∥22∥u1∥Hs+1+∥δn∥22+∥δu∥22.From (45)–(52), we obtain
(53)ddt∥δnt∥22+∥δct∥22+∥∇δct∥22+∥δut∥22+∥∇δn∥22+∥∇δc∥22+∥Δδc∥22≤C∥δu∥22+∥δn∥22∥n1∥Hs2+∥δn∥22∥c1∥Hs+12+∥n2∥Hs2∥∇δc∥22+∥δn∥22+∥n1∥Hs∥δn∥22+∥n2∥Hs∥δn∥22+∥δu∥22+∥δc∥22∥c1∥Hs+12+∥δc∥22∥n1∥Hs+∥δn∥22+∥δc∥22∥c2∥Hs+12+∥δu∥22∥c1∥Hs+12+∥∇δc∥22∥u2∥Hs+1+∥δc∥22∥n1∥Hs2+∥δn∥22∣c2∥Hs+12+∥δu∥22∥u1∥Hs+1+∥δn∥22+∥δu∥22.Then, we have(54)ddt∥δnt∥22+∥δct∥22+∥∇δct∥22+∥δut∥22≤CFt∥δn∥22+∥δc∥22+∥∇δc∥22+∥δu∥22,where
(55)Ft=1+∥n1∥Hs+∥n2∥Hs+∥u1∥Hs+1+∥u2∥Hs+1+∥n1∥Hs2+∥n2∥Hs2+∥c1∥Hs+12+∥c2∥Hs+12.From (3), we infer that Ft is integrable. Using the Gronwall inequality gives the uniqueness.


## 3.2. Blowup Criterion

OperatingΔq with q≥−1 to the first equation of (3) gives
(56)Δqnt+u·∇Δqn=ΔΔqn+u·∇Δqn−Δqu·∇n−∇·Δqn∇c+Δqn−Δqn2.Taking theL2-inner product for the above equation with Δqn, we get
(57)12ddt∥Δqn∥22+∥∇Δqn∥22=∫du·∇Δqn−Δqu·∇nΔqndx+∫dΔqn∇c·∇Δqndx+∫dΔqnΔqndx−∫dΔqn2Δqndx≤∥u·∇Δqn−Δqu·∇n∥2∥Δqn∥2+∥Δqn∇c∥2∥∇Δqn∥2+∥Δqn∥22+∥Δqn2∥2∥Δqn∥2.Multiplying22qs on both sides of the above inequality and performing ℓ1 norm, we have
(58)ddt∥n∥Hs2+∥n∥Hs+12≤∥∇u∥∞∥n∥Hs2+∥n∇c∥Hs∥n∥Hs+1+∥n∥Hs2+∥n2∥Hs∥n∥Hs.Using Young’s inequality, we conclude(59)ddt∥n∥Hs2+∥n∥Hs+12≤C∥∇u∥∞∥n∥Hs2+C∥n∥∞2∥∇c∥Hs2+C∥n∥Hs2∥∇c∥∞2+18∥n∥Hs+12+∥n∥Hs2+C∥n∥∞2∥n∥Hs2+18∥n∥Hs+12.In terms of the second equation of (3), we know
(60)Δqct+u·∇Δqc=ΔΔqc+u·∇Δqc−Δqu·∇c−Δqnc.Multiplying the above equality byΔqc and integrating in spaces mean
(61)12ddt∥Δqc∥22+∥∇Δqc∥22=∫ℝdu·∇Δqc−Δqu·∇cΔqcdx+∫ℝdΔqncΔqcdx≤∥u·∇Δqc−Δqu·∇c∥2∥Δqc∥2+∥Δqnc∥2∥Δqc∥2.Multiplying22qs+1 on both sides of the above inequality and taking ℓ1 norm, we obtain
(62)ddt∥c∥Hs+12+∥c∥Hs+22≤∥∇u∥∞∥c∥Hs+12+∥nc∥Hs+1∥c∥Hs+1.Utilizing Young’s inequality, we have(63)ddt∥c∥Hs+12+∥c∥Hs+22≤C∥∇u∥∞∥c∥Hs+12+C∥n∥∞2∥c∥Hs+12+C∥c∥∞2∥c∥Hs+12+18∥n∥Hs+12.According to the third equation of (3), we get
(64)Δqut+u·∇Δqu−∇ΔqP=u·∇Δqu−Δqu·∇u−Δqn∇Φ.Taking theL2-inner product for the above equality with Δqu implies
(65)12ddt∥Δqu∥22=∫ℝdu·∇Δqu−Δqu·∇uΔqudx−∫ℝdΔqn∇ΦΔqudx≤∥u·∇Δqu−Δqu·∇u∥2∥Δqu∥2+∥Δqn∇Φ∥2∥Δqu∥2.Multiplying22qs+1 on both sides of the above inequality and taking the ℓ1 norm, we have
(66)ddt∥u∥Hs+12≤C∥∇u∥∞∥u∥Hs+12+18∥n∥Hs+12+C∥u∥Hs+12.Collecting (59)–(66), we deduce
(67)ddt∥n∥Hs2+∥c∥Hs+12+∥u∥Hs+12+∥n∥Hs+12+∥c∥Hs+22≤∥n∥Hs2+∥c∥Hs+12+∥u∥Hs+121+∥∇u∥∞+∥n∥∞2+∥∇c∥∞2.The Gronwall inequality implies(68)∥nt∥Hs2+∥ct∥Hs+12+∥ut∥Hs+12+∫0t∥nτ∥Hs+12+∥cτ∥Hs+22dτ≤Cexp∫0t1+∥∇uτ∥∞+∥nτ∥∞2+∥∇cτ∥∞2dτ.Next, we turn to prove condition (11). Applying ∇ on both sides of the second equation of (3) means
(69)∂t∇c+u·∇2c−Δ∇c=−∇nc−∇u·∇c.Multiplying the above equality with∇cp−2∇c, we obtain
(70)1pddt∥∇c∥pp+∫ℝdu·∇2c∇cp−2∇cdx+4p−1p2∥∇∇cp/2∥22=−∫ℝd∇nc∇cp−2∇cdx−∫ℝd∇u·∇c∇cp−2∇cdx≤p−1∫ℝdnc∇cp−2Δcdx+∥∇u∥∞∥∇c∥pp≤2p−1p∥n∥∞∥c∥p∥∇cp−2/2∥2p/p−2∥∇∇cp/2∥2+∥∇u∥∞∥∇c∥pp≤2p−1pp2p−1ε∥n∥∞2∥c∥p2∥∣∇c∣∥p−22+2p−1εp∥∇∇cp/2∥22+∥∇u∥∞∥∇c∥pp.Because of(71)∫ℝdu·∇2c∇cp−2∇cdx=1p∫ℝdu·∇∇cpdx=0,we get
(72)ddt∥∇c∥p2≤C∥n∥∞2+∥∇u∥∞∥∇c∥p2.Utilizing Gronwall’s inequality, we have(73)∥∇ct∥p2≤C∥∇c0∥p+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ≤C∥c0∥Hs+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ.Settingp→∞, we conclude
(74)∥∇ct∥∞2≤∥c0∥Hs+∫0t∥nτ∥∞2dτexp∫0t∥∇uτ∥∞dτ.Submitting (74) into (68) gives
(75)∥nt∥Hs2+∥ct∥Hs+12+∥ut∥Hs+12+∫0t∥nτ∥Hs+12+∥cτ∥Hs+22dτ≤Cexpexp∫0t1+∥∇uτ∥∞+∥nτ∥∞2dτ.On the other hand, using the inhomogeneous dynamic partition of the unity, we have(76)∥∇u∥L∞=∥∑j≥−1∇Δju∥L∞≤∥∇Δ−1u∥L∞+∑j=0∞∥∇Δj∥L∞≤C∥u0∥L2+∑j=0∞∥Δjω∥L∞≤C∥u0∥L2+∥ω∥B∞,10.Taking the curl to the third equation of (3) implies
(77)ωt+u·∇ω=urrω−curln∇ϕ.Using Lemma6, we obtain
(78)∥ω∥B∞,10≤C∥ω0∥B∞,10+∫0t∥curln∇ϕτ∥B∞,10+∥urrω∥B∞,10×1+∫0t∥∇uτ∥L∞dτ≤C∥ω0∥B∞,10+∫0t∥curln∇ϕτ∥B∞,10+∥urr∥L∞∥ω∥B∞,10×1+∫0t∥∇uτ∥L∞dτ.For the term∥curln∇ϕτ∥B∞,10, using Bony’s decomposition, we have
(79)∥curln∇ϕτ∥B∞,10=∑∣q−q′∣≤42q∥ΔqSq′−1nΔq′∇ϕ∥L∞+∑∣q−q′∣≤42q∥ΔqSq′−1∇ϕΔq′n∥L∞+∑∣q′′−q′∣≤1q′≥q−32q∥ΔqΔq′nΔq′′∇ϕ∥L∞≜I1+I2+I3.For the termI1, we have
(80)I1≤C∑∣q−q′∣≤4∑j≤q′−22q∥Δq′∇ϕ∥L∞∥Δjn∥L∞≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Similarly,(81)I2≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.As forI3,
(82)I3≤C∑∣q′′−q′∣≤1q′′≥q−32q−q′′2q′′∥Δq′′∇ϕ∥L∞∥Δq′n∥L∞≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Plugging (80)–(82) into (79) yields
(83)∥curln∇ϕτ∥B∞,10≤C∥∇ϕ∥B∞,∞1∥n∥B∞,10.Putting (83) into (78), using the fact ∥urt/r∥L∞≤C∥ω0/r∥L3,1 [25] and Gronwall’s inequality, we have
(84)∥ω∥B∞,10≤C∥ω0∥B∞,10+∫0t∥nτ∥B∞,10dτe∫0t∥∇uτ∥L∞dτ.Substituting (84) into (76) gives
(85)∥∇u∥L∞≤C∥u0∥Hs+∫0t∥nτ∥B∞,10dτe∫0t∥∇uτ∥L∞dτ.Applying the Gronwall inequality, we get(86)∥∇u∥L∞≤CeexpC∫0t∥nτ∥B∞,10.Substituting (86) into (75) and using the fact B∞,10°L∞, we obtain the desired result.This completes the proof of Theorem1.



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*Source: 5541713-2021-04-07.xml*