In this article, we study the blow-up slutions for the 2D Euler- Boussinesq equation

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

BLOW-UP CRITERION FOR THE 2D EULER-BOUSSINESQ SYSTEM IN TERMS OF TEMPERATURE

CHENYIN QIAN

Abstract. In this article, we study the blow-up slutions for the 2D Euler- Boussinesq equation. In particular, it is shown that if

Z T^∗

0

sup

r≥2

kΛ^1−αθ(t)k_Lr

√rlogr dt <∞ or Z T^∗

0

kΛ^1−αθkB˙⁰_∞,∞dt <∞,

then the local solution can be continued to the global one. This is an improve- ment of classical Lipschitz-type blow-up criterion (k∇θk_L1

tL^∞) in terms of the temperatureθ.

1. Introduction

The 2D incompressible generalized Boussinesq equations with the fractional Laplacian dissipation is of the type

∂_tv+ (v· ∇)v+νΛ^βv+∇π=θe₂, (x, t)∈R²×(0,∞),

∂tθ+ (v· ∇)θ+κΛ^αθ= 0,

∇ ·v= 0,

v(x,0) =v⁰, θ(x,0) =θ⁰,

(1.1)

where v = v(x, t) = (v1(x, t), v2(x, t)), π = π(x, t) and θ = θ(x, t) stand for, respectively, the velocity vector field, the pressure and the temperature. Here, the constants ν ≥ 0 and κ ≥ 0 denote viscosity coefficient and thermal diffusivity coefficient respectively, and e2= (0,1). Λ =√

−∆ is the Zygmund operator, and Λ^α is defined by the Fourier transform,

Λd^αf(ξ) =|ξ|^αfb(ξ), fb(ξ) = Z

R²

e^−ix·ξf(x) dx.

The study of the standard 2D incompressible Boussinesq system (withν >0, κ >0 and α= β = 2 in (1.1)) can be traced to 1980s (see [5]). Later, there are many works considering the global well-posedness problem for the standard 2D Boussinesq system without the viscosity (ν = 0, κ > 0) or without the thermal diffusivity (ν >0,κ= 0), see [1, 7, 14, 15, 19].

2010Mathematics Subject Classification. 35Q35, 35B35, 35B65, 76D05.

Key words and phrases. 2D Boussinesq equation; blow-up criterion; Besov space;

transport equation.

c

2016 Texas State University.

Submitted December 18, 20915. Published March 15, 2016.

1

Recently, the 2D incompressible Boussinesq system with fractional Laplacian generalizations have attracted considerable attention. For example, Hmidi and Zerguine obtained the global well-posedness of Euler-Boussinesq system (1.1) with 1< α≤2 in [17]. Hmidi, Keraani and Rousset obtained the global well-posedness for the Euler-Boussinesq system with critical dissipation (namely, α= 1) in [16].

There are many other related results to the Boussinesq equations system (1.1), we refer the reader to [2, 3, 6, 9, 11, 18, 12, 13, 21]. From above mentioned result, we see that it is difficult for us to get the global well-posedness for the system (1.1) with super-critical dissipation (α+β < 1). Therefore, people may turn to the blow-up criteria in terms of velocity (v) and temperature (θ). As for the blow-up criterion for Boussinesq equations in terms of θ, we refer readers to [8] (see also [7]), in which the authors Chae and Nam obtained the blow-up criterionk∇θk_L1

tL^∞

in the framework of L². The purposes of this article is to establish some blow-up criteria better thank∇θk_L1

tL^∞.

In view of [17], one can establish the local well-posedness results for the system (1.1) with 0< α≤2, κ >0 and

v⁰∈B^s_p,1(R²) with divv⁰= 0, θ⁰∈B_p,1^s−α(R²), (1.2) wheres≥1 +²_p withp∈]1,∞[. Here, we define the function space of solutions as follows

X_T^s,p:=C([0, T];B^s_p,1(R²))× C([0, T];B_p,1^s−α(R²))∩L¹([0, T];B_p,1^s (R²)

, (1.3) whereB^s−α_p,1 (R²) andB_p,1^s (R²) are Besov spaces (see Section 2). Now, we give the main results of this article.

Theorem 1.1. Let 1/2 < α≤1 and (v, θ)∈χ^s,p_T∗ be the local unique solution of (1.1)with initial data satisfying (1.2), whereT^∗ is the maximal existence time. If

Z T^∗

0

sup

r≥2

kΛ^1−αθ(t)kL^r

√rlogr dt <∞, (1.4)

or

Z T^∗

0

kΛ^1−αθkB˙_∞,∞⁰ dt <∞, (1.5) then the solution (v, θ)can be continued beyond T^∗.

2. Notation and preliminaries

We begin this section with dyadic decomposition. Let S be the Schwartz class of rapidly decreasing functions. Let functionsχ, ϕ∈ S(R^d) supported inB={ξ∈ R^d:|ξ| ≤4/3} andC={ξ∈R^d : 3/4≤ |ξ| ≤8/3} respectively, such that

X

j∈Z

ϕ(2^−jξ) = 1,∀ξ∈R^d\{0},

χ(ξ) +X

j≥0

ϕ(2^−jξ) = 1,∀ξ∈R^d.

Foru∈ S⁰, we set

∆₋₁u=χ(D)u ∀q∈N, ∆_qu=ϕ(2^−qD)u ∀q∈Z,

∆˙_qu=ϕ(2^−qD)u.

The following low-frequency cut-off will be also used:

Squ= X

−1≤j≤q−1

∆ju and S˙qu= X

j≤q−1

∆˙ju

We now recall the definitions of Besov spaces through the dyadic decomposition.

Let s ∈ R, p, q ∈ [1,∞], the inhomogeneous Besov space B_p,q^s (R^d) is the set of tempered distributionusuch that

kukB^s_p,q := 2^jsk∆jukL^p

`^q <∞.

To define the homogeneous Besov spaces, we first denote by S⁰/P the space of tempered distributions modulo polynomials. Thus we define the space ˙B_p,q^s (R^d) as the set of distributionu∈ S⁰/P such that

kukB˙^s_p,q := 2^jsk∆˙_jukL^p

`^q <∞.

We point out that ifs >0 then we haveB^s_p,q(R^d) = ˙B^s_p,q(R^d)∩L^p(R^d) and kukB^s_p,q ≈ kukB˙^s_p,q+kukL^p.

In our next study we require two kinds of coupled space-time Besov spaces. The first is defined in the following manner: forT >0 andq≥1, we denote byL^r_TB˙^s_p,q the set of all tempered distribution u satisfying

kuk_Lr

TB˙^s_p,q:=k

2^qsk∆˙qukL^p

`^q

kL^r_T <∞.

The second mixed space is Le^r_TB˙_p,q^s which is the set of tempered distribution u satisfying

kukLe^r_TB˙_p,q^s :=

2^qsk∆˙_qukL^r_TL^p

`^q

<∞.

3. Proof of main results In this section we use Φ_k to denote function of the form

Φk(t) =C0exp(. . .exp

| {z }

ktimes

(C0t). . .),

whereC₀depends on the involved norms of the initial data and its value may vary from line to line up to some absolute constants. We will make an intensive use (without mentioning it) of the following trivial facts

Z t

0

Φk(τ)dτ ≤Φk(t) and expZ t 0

Φk(τ)dτ

≤Φk+1(t).

Firstly, we introduce a pseudo-differential operatorRαdefined byRα:= Λ^−α∂₁= Λ^1−αR,0 < α < 1, where R := ^∂_Λ¹ is the usual Riesz transform. For (1.1), the vorticity equation is

∂_tω+v· ∇ω=∂₁θ, (3.1)

and the acting ofRα on the temperature equation, we have

∂tRαθ+v· ∇Rαθ+κΛ^αRαθ=−[Rα, v· ∇]θ. (3.2) Denotez=ω+Rαθ. Thus we obtain

∂_tz+v· ∇z=−[R_α, v· ∇]θ. (3.3) Firstly, we give the following crucial Lemmas which are useful for us to proof the main results.

Lemma 3.1([4]). Assume thatvis divergence-free and thatfsatisfies the transport equation onR^d,

∂tf +v· ∇f =g,

f|t=0=f0. (3.4)

There exists a constantC, depending only ond, such that for all 1≤p, r≤ ∞and t∈R+, we have

kfk

Le^∞_t (B_p,r⁰ )≤C kf0k_B0 p,r+kgk

Le¹_t(B_p,r⁰ )

1 +

Z t

0

k∇v(τ)kL^∞dτ

. (3.5)

Lemma 3.2 ([17]). Let v be a solution of the incompressible Euler system onR²

∂tv+v· ∇v+∇π=f, v(x,0) =v⁰,

divv= 0.

(3.6) Then for s >−1,(p, r)∈(1,∞)×[1,∞] we have

kvkLe^∞_t (B^s_p,r)≤Ce^CV(t)

kv⁰kB^s_p,r+ Z t

0

e^−CV(τ)kf(τ)kB^s_p,rdτ

, (3.7) withV(t) :=Rt

0k∇v(τ)kL^∞dτ.

Lemma 3.3. Let α∈(0,1),v be a smooth divergence-free vector field.

(i) for every(s, p, r)∈(−1, α)×[2,∞]×[1,∞], there exists a constant C >0 such that

k[Rα, v· ∇]θkB^s_p,r ≤Ck∇vkL^p

kθk_B^s+1−α

∞,r +kθkL^p

, (3.8)

(ii) for every (r, %)∈[1,∞]×(1,∞) and >0, there exists a constantC > 0 such that

k[Rα, v· ∇]θkB⁰_∞,r ≤C(kωkL^∞+kωkL^%)

kθk_B^+1−α

∞,r +kθkL^%

. (3.9) Proof. Part (i) can be found in [21, Proposition 3.3]. For the second part, we can imitate the proof of [16, Theorem 3.3(2)] to get (ii). We omit the detail here.

Lemma 3.4. Let α∈[0,2],κ >0 andv be a smooth divergence-free vector-field of R². Letθ be a smooth solution of

∂tθ+v· ∇θ+κΛ^αθ=f,

θ(x,0) =θ⁰. (3.10)

Then (i) for every ρ∈[1,∞],p∈[1,∞],s >−1 one has (see [17]) kθk

Le^ρ_t(B^s+

α ρ

p,1 )≤Ce^CV(t)

kθ⁰kB_p,1^s (1 +t^1ρ) + Z t

0

Ts(τ)dτ+kfk_L1 t(B^s_p,1)

(3.11) with

V(t) :=

Z t

0

k∇v(τ)kL^∞dτ, Ts(t) :=k∇θ(t)kL^∞kv(t)kB^s_p,11_[1,∞)(s).

(ii) for everyp∈[1,∞], we have theL^p estimates (see[10]) kθkL^p≤ kθ⁰kL^p+

Z t

0

kf(τ)kL^pdτ. (3.12)

(iii) for every p∈[1,∞](see [16]) kθkLe^∞_t B⁰_p,1 ≤C kθ⁰k_B⁰

p,1+kfk_L¹

tB⁰_p,1

1 +

Z t

0

k∇v(τ)kL^∞dτ

. (3.13)

(iv) for p∈(1,∞), ρ∈ [1,∞] and f = 0, there exists a constant C such that (see [17])

sup

q∈N

2^qαρk∆qθk_L^ρ

tL^p≤Ckθ⁰kL^p+Ckθ⁰kL^∞kωk_L¹

tL^p. (3.14) Proof of Theorem 1.1. By using the above Lemmas and the equation (3.3), we first estimatek∇v(t)kL^∞. We do this in several steps.

Step 1: Estimation ofkω(t)kL^a, for some max{2, p}< a <∞. From (3.3), for any t < T^∗, by Lemma 3.4 (ii) we have

kz(t)k_La ≤ kz(0)k_La+ Z t

0

k[R_α, v· ∇]θ(τ)k_Ladτ. (3.15) Note that (see [10, p 516])Rt

0Λ^αRαθ|Rαθ|^a−2Rαθdτ ≥0, by (3.2) we have kRαθ(t)kL^a ≤ kRαθ⁰kL^a+

Z t

0

k[Rα, v· ∇]θ(τ)kL^adτ. (3.16) Since v⁰ ∈ B_p,1^s , s ≥ 1 + 2/p, we see that ω⁰ ∈ L^p ∩L^∞, and for θ⁰ ∈ B^s−α_p,1 with 0 < α ≤ 1, we have θ⁰ ∈ L^p∩L^∞ note that p < a < ∞, the embedding B_p,1^s−α ,→B^s−α−

2 p+_a²

a,1 ,→B^1−α+_a,1 ^a2 ,→B^1−α_a,1 , we have Rαθ⁰ ∈L^a. From (3.15) and (3.16) we obtain

kωkL^a≤ kz(t)kL^a+kRαθ(t)kL^a

≤C kω⁰k_Lp∩L^∞+kθ⁰k_Bs−α p,1

+C Z t

0

k[R_α, v· ∇]θ(τ)k_Ladτ. (3.17) Using the classical embeddingB_a,1⁰ ,→L^a, and Lemma 3.3 (i), we have

k[Rα, v· ∇]θ(t)kL^a ≤Ck[Rα, v· ∇]θ(t)k_B0 a,1

≤Ck∇v(t)kL^a kθ(t)k_B^1−α

∞,1 +kθ(t)kL^a

. (3.18)

Fromkθ(t)kL^a≤ kθ⁰kL^a,∀t≥0, it follows that kω(t)kL^a ≤C

kω⁰kL^p∩L^∞+kθ⁰k_B^s−α

p,1

+C Z t

0

kω(τ)kL^a

kθ(τ)k_B^1−α

∞,1 +kθ⁰kL^a

dτ.

(3.19)

According to Gronwall’s inequality we obtain

kωkL^a ≤Ce^Cte^{CkθkL1tB1−α∞,1}. (3.20) Next we estimate kθk_L1

tB_∞,1^1−α, letN ∈ N, by the Littlewood-Paley decomposition, by condition (1.4) we see that

kθk_L1 tB^1−α_∞,1

≤ kSNθk_L1

tB^1−α_∞,1 +X

q≥N

2^q(1−α)k∆qθk_L1 tL^∞

≤Ctkθ⁰kL^∞+C Z t

0

X

0≤q<N

k∆qΛ^1−αθ(τ)kL^∞dτ +X

q≥N

2^q(1−α)k∆qθk_L1 tL^∞

≤Ctkθ⁰k_L^∞+C Z t

0

X

0≤q<N

2^q2bk∆_qΛ^1−αθ(τ)k_Lbdτ+X

q≥N

2^q(1−α)k∆_qθk_L1 tL^∞

≤C X

0≤q<N

2^q2bp blogb

Z t

0

sup

r≥2

kΛ^1−αθ(τ)kL^r

√rlogr dτ +Ctkθ⁰kL^∞+CX

q≥N

2^q(1−α+a2)k∆qθk_L1 tL^a

≤Ctkθ⁰kL^∞+C2^N2bp

blogb+CX

q≥N

2^q(1−α+a2)k∆qθk_L1 tL^a.

Since α >1/2, we choose a large enough such that 1−2α+ 4/a <0, and using Lemma 3.4 (iv), we have

X

q≥N

2^q(1−α+2a)k∆qθk_L¹

tL^a

≤ X

q≥N

2^{q(1−2α+a2)}

kθ⁰kL^a+kθ⁰kL^∞

Z t

0

kω(τ)kL^adτ

≤Ckθ⁰k_La+ 2^{N(1−2α+a2)}kθ⁰k_L^∞ Z t

0

kω(τ)k_Ladτ.

(3.21)

Therefore,

kθk_L1

tB_∞,1^1−α ≤C(t+ 1)kθ⁰kL^∞∩L^a+C2^N2bb¹²⁺ + 2^{N(1−2α+a2)}kθ⁰k_L^∞

Z t

0

kω(τ)k_Ladτ,

for any 0< <1/2. Settingb=N and selectingN as follows N =hlog e+Rt

0kω(τ)kL^adτ (2α−1−2/a) log 2

i + 2.

Then kθk_L1

tB^1−α_∞,1 ≤C(t+ 1)kθ⁰kL^∞∩L^a+Ch log

e+ Z t

0

kω(τ)kL^adτi¹₂+

. (3.22) Combining (3.20) with (3.22), we have

kθk_L1

tB_∞,1^1−α≤C(t+ 1)kθ⁰k_L^∞_∩La+Ch log

e+ Z t

0

kω(τ)k_Ladτi¹₂+

≤C(t+ 1)kθ⁰k_L^∞_∩La+Ch log

e+tCe^Cte^{CkθkL1tB1−α∞,1}i¹₂+

≤C(t²+t+ 1) kθ⁰kL^∞∩L^a+ 1

+Ckθk¹²⁺

L¹_tB^1−α_∞,1,

(3.23)

it follows that

kθk_L1

tB_∞,1^1−α≤C(1 +t+t²).

By (3.20), we obtain

kω(t)kL^a≤Φ₁(t), (3.24)

where Φk(t), k= 1,2,3, . . . is same the as as in subsection 3.3.

Step 2: Estimation ofkω(t)kL^∞. By using the maximum principle for the transport equation (3.3) we have

kz(t)kL^∞≤ kz(0)kL^∞+ Z t

0

k[Rα, v· ∇]θ(τ)kL^∞dτ, (3.25) By using Lemma 3.4 (ii) withp=∞, we see that (3.2) follows that

kRαθ(t)kL^∞ ≤ kRαθ⁰kL^∞+ Z t

0

k[Rα, v· ∇]θ(τ)kL^∞dτ. (3.26) Forθ⁰∈B_p,1^s−α with 0< α≤1, s≥1 + 2/p, we have B_p,1^s−α,→B^s−α−

2 p

∞,1 ,→B_∞,1^1−α, we haveRαθ⁰∈L^∞. From (3.25) and (3.26) we obtain

kωkL^∞ ≤ kz(t)kL^∞+kRαθ(t)kL^∞

≤ kω⁰kL^∞+kθ⁰k_B^s−α

p,1 + 2 Z t

0

k[Rα, v· ∇]θ(τ)kL^∞dτ. (3.27) Using the classical embeddingB_∞,1⁰ ,→L^∞, and Lemma 3.3 (i), we have

k[Rα, v· ∇]θkL^∞≤Ck[Rα, v· ∇]θk_B⁰

∞,1

≤C(kωkL^∞+kωkL^a)

kθk_B^+1−α

∞,1 +kθkL^a

. (3.28)

Combining (3.27) and (3.28) we have kω(t)kL^∞

≤C+C Z t

0

(kω(τ)kL^∞+kω(τ)kL^a)

kθ(τ)k_B+1−α

∞,1 +kθ(τ)kL^a

dτ

≤C+kωkL^∞_t L^a

kθk_L1

tB^+1−α_∞,1 +tkθ⁰kL^a

+C Z t

0

kω(τ)kL^∞

kθ(τ)k_B^+1−α

∞,1 +kθ⁰kL^a

dτ.

(3.29)

We claim that

kθk_L1

tB_∞,1^+1−α ≤Φ1(t), (3.30) for some suitable >0. In fact that, from step 1 and the Lemma 3.4 (iv) we have

kθkLe¹_tB^α_a,∞ ≤Φ1(t), and then for everyσ < α, we have

kθk_L1

tB_a,1^σ ≤ kθk

Le¹_tB^α_a,∞ ≤Φ1(t).

We choose thatσ=α−1/a, and we keep in mind thatasatisfies 1−2α+ 4/a <0, therefore 2α−1−4/a >0, we choose satisfies 0 < <2α−1−3/a, then we have+ 1−α < α−3/a=σ−2/a(here we note that the selected parameterais make sureα >3/a, and we haveσ−2/a >0). Therefore, we have the embedding B_a,1^α−1/a = B^σ_a,1 ,→ B^σ−2/a_∞,1 ,→ B^+1−α_∞,1 , and we finally obtain (3.30). Thus, by (3.30) and step 1, one has

kω(t)kL^∞ ≤Φ1(t) +C Z t

0

kω(τ)kL^∞

kθ(τ)k_B^+1−α

∞,1 +kθ⁰kL^a

dτ. (3.31)

Again using (3.30) and Gronwall’s inequality, we obtain kω(t)kL^∞ ≤Φ2(t).

Step 3: Estimation ofk∇v(t)kL^∞. By using Lemma 3.1 and Lemma 3.4(iii), from (3.2) and (3.3) we have

kz(t)k_B0

∞,1+kR_αθ(t)k_B0

∞,1 ≤

C+k[R_α, v· ∇]θk_L1 tB⁰_∞,1

(1+k∇vk_L1

tL^∞), (3.32) Thanks to Lemma 3.3, and by step 1, step 2, we have

k[Rα, v· ∇]θk_L¹

tB⁰_∞,1

≤C Z t

0

(kω(τ)kL^∞+kω(τ)kL^a)

kθ(τ)k_B^+1−α

∞,1 +kθ(τ)kL^a

dτ ≤Φ₂(t). (3.33) Therefore, we have

kω(t)k_B0

∞,1 ≤ kz(t)k_B0

∞,1+kR_αθ(t)k_B0

∞,1 ≤Φ₂(t)

1 +k∇vk_L1 tL^∞

. On the other hand, we have

k∇v(t)kL^∞≤ k∇∆₋₁v(t)kL^∞+X

q∈N

k∆q∇v(t)kL^∞

≤ kω(t)kL^a+kω(t)k_B⁰

∞,1

≤Φ1(t) +kω(t)k_B⁰

∞,1.

(3.34)

Using (3.34), we have kω(t)k_B⁰

∞,1≤Φ₂(t) 1 +

Z t

0

kω(τ)k_B⁰

∞,1dτ .

From Gronwall’s inequality we obtain kω(t)k_B0

∞,1 ≤ Φ₃(t). Going back to (3.34), we obtain

k∇vk_L^∞≤Φ₃(t).

Next, we give the estimate ofkv(t)kB_p,1^s ,kθ(t)k_B^s−α

p,1 . Sinces≥1 + 2/p, we have B_p,1^s−α,→B_∞,1^1−α. By Lemma 3.4(i), we have

kθ(t)k

Le^∞_t (B_∞,1^1−α)+ Z t

0

kθ(τ)k_B¹

∞,1dτ ≤Ce^CV(t)(1 +t)kθ⁰k_B^1−α

∞,1

≤Ce^CV(t)(1 +t)kθ⁰k_Bs−α p,1 .

(3.35) On the other hand, by Lemma 3.2 we have

kv(t)k

Le^∞_t (B^s−α_p,1 )≤Ce^CV(t)

kv⁰kB^s_p,1+ Z t

0

kθ(τ)k_B^s−α

p,1 dτ .

Therefore, by the embeddingB_∞,1⁰ ,→L^∞we have Z t

0

k∇θ(τ)kL^∞kv(τ)k_Bs−α p,1 dτ

≤ sup

0≤τ≤t

kv(τ)k_B^s−α

p,1

Z t

0

kθ(τ)k_B1

∞,1dτ

≤Ce^CV(t)(1 +t)kθ⁰k_B^s−α

p,1

kv⁰kB_p,1^s + Z t

0

kθ(τ)k_B^s−α

p,1 dτ .

(3.36)

Applying (3.36) and (3.11), the estimate ofθ reads as follows kθ(t)k

Le^∞_t (B_p,1^s−α)+ Z t

0

kθ(τ)k_B^s

p,1dτ

≤Ce^CV(t)

kθ⁰k_Bs−α

p,1 (1 +t) + Z t

0

k∇θ(τ)kL^∞kv(τ)k_Bs−α p,1 dτ

≤Ce^CV(t)(1 +t)kθ⁰k_B^s−α

p,1

1 +kv⁰kB^s_p,1+ Z t

0

kθ(τ)k_B^s−α

p,1 dτ .

(3.37)

Combining the estimates ofv (applying Lemma 3.2 again) and (3.37), we have kv(t)kB^s_p,1+kθ(t)k_B^s−α

p,1 + Z t

0

kθ(τ)kB_p,1^s dτ

≤Ce^CV(t)(1 +t)kθ⁰k_B^s−α

p,1

1 +kv⁰kB^s_p,1+ Z t

0

kθ(τ)k_B^s−α

p,1 dτ

≤Φ₄(t) 1 +

Z t

0

kθ(τ)k_Bs−α p,1 dτ

.

(3.38)

Therefore, by Gronwall’s inequality, we finally obtain kv(t)kB_p,1^s +kθ(t)k_B^s−α

p,1 + Z t

0

kθ(τ)kB^s_p,1dτ ≤Φ5(t). (3.39) This completes the first part (when assumption (1.4) holds) of this theorem.

The proof of the second part (when assumption (1.5) holds) of this theorem is quite similar to the one in the first part. The main difference is the estimates of kωk_La in step 1. We begin with (3.18), and by the embedding B_a,2⁰ ,→ L^a and Lemma 3.3 (i), we have

k[R_α, v· ∇]θ(t)k_La≤Ck[R_α, v· ∇]θ(t)k_B0 a,2

≤Ck∇v(t)kL^a

kθ(t)k_B^1−α

∞,2 +kθ(t)kL^a

. (3.40)

It follows that

kω(t)k_La ≤C

kω⁰k_Lp∩L^∞+kθ⁰k_Bs−α p,1

+C Z t

0

kω(τ)kL^a

kθ(τ)k_B1−α

∞,2 +kθ⁰kL^a

dτ.

(3.41)

According to Gronwall’s inequality we obtain

kωkL^a ≤Ce^Cte^{CkθkL1tB1−α∞,2}. (3.42) The estimate of kθk_L1

tB^1−α_∞,2 is as follows, let N ∈N, by the Littlewood-Paley de- composition, by condition (1.5) we see that

kθk_L1 tB^1−α_∞,2

≤ kSNθk_L1

tB^1−α_∞,2 +k(Id− SN)θk_L1 tB^1−α_∞,1

≤ kSNθk_L1

tB^1−α_∞,2 +X

q≥N

2^q(1−α)k∆qθk_L¹

tL^∞

≤Ctkθ⁰kL^∞+C Z t

0

X

0≤q<N

k∆qΛ^1−αθ(τ)k²_L∞

1/2

dτ +X

q≥N

2^q(1−α)k∆qθk_L1 tL^∞

≤C√ N

Z t

0

kΛ^1−αθ(τ)kB˙⁰_∞,∞dτ+Ctkθ⁰kL^∞+C X

q≥N

2^q(1−α+2a)k∆qθk_L1 tL^a

≤C√

N+Ctkθ⁰kL^∞+CX

q≥N

2^q(1−α+2a)k∆qθk_L1 tL^a. Then, as for (3.21), we can choose suitableN such that

kθk_L1

tB^1−α_∞,2 ≤C(t+ 1)kθ⁰kL^∞∩L^a+Ch log

e+ Z t

0

kω(τ)kL^adτi^1/2

. (3.43) Combining (3.42) with (3.43), we have

kθk_L1

tB_∞,2^1−α ≤Φ1(t).

For the rest, we can follow the same process as above, and complete the proof.

Acknowledgements. I would like to thank the referee for a careful reading of the work and many valuable comments. The research is supported by the Natural Science Foundation of Zhejiang Province (LQ16A010001) and NSFC 11501517.

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Chenyin Qian

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China E-mail address:[email protected]