Introduction In this article, we study then-dimensional Boussinesq equations with fractional dissipation, ∂tu+ (u· ∇)u+Λ2αu+∇Π =ϑen, ∂tϑ+ (u· ∇)ϑ= 0

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GLOBAL REGULARITY CRITERIA FOR THE n-DIMENSIONAL BOUSSINESQ EQUATIONS WITH FRACTIONAL DISSIPATION

ZUJIN ZHANG

Abstract. We consider the n-dimensional Boussinesq equations with frac- tional dissipation, and establish a regularity criterion in terms of the velocity gradient in Besov spaces with negative order.

1. Introduction

In this article, we study then-dimensional Boussinesq equations with fractional dissipation,

∂tu+ (u· ∇)u+Λ^2αu+∇Π =ϑen,

∂tϑ+ (u· ∇)ϑ= 0,

∇ ·u= 0, u(0) =u0, ϑ(0) =ϑ0,

(1.1)

whereu:R⁺×Rⁿ→Rⁿ is the velocity field;ϑ:R⁺×Rⁿ→Ris a scalar function representing the temperature in the content of thermal convection (see [8]) and the density in the modeling of geophysical fluids (see [9]);Π is the the fluid pressure;

en is the unit vector in thexn direction; and Λ:= (−∆)¹²,α≥0 is a real number.

Whenα= 1, Equation (1.1) reduces to the classical Boussinesq equations, which are frequently used in the atmospheric sciences and oceanographic turbulence where rotation and stratification are important (see [8, 9]). Ifϑ= 0, then (1.1) becomes the generalized Navier-Stokes equation, which was first considered by Lions [7], where he showed the global regularity onceα≥ ¹₂+ⁿ₄. One may refer the reader to [5, 10] for recent advances. Xiang-Yan [12], Yamazaki [13] and Ye [14] were able to extend Lions’s result to system (1.1), where there is no diffusion in the ϑ equation. And it remains an open problem for the global-in-time smooth for (1.1) with 0< α < ¹₂+ ⁿ₄. The purpose of the present paper is to establish a blow-up criterion as follows.

Theorem 1.1. Let0< α < ¹₂+ⁿ₄,(u0, ϑ0)∈H^s(Rⁿ)withs >1+ⁿ₂ and∇·u0= 0.

Assume that(u, ϑ)be the smooth local unique solution pair to(1.1)with initial data

2010Mathematics Subject Classification. 35B65, 35Q30, 76D03.

Key words and phrases. Regularity criteria; Generalized Boussinesq equations;

fractional diffusion.

c

2016 Texas State University.

Submitted February 23, 2016. Published April 19, 2016.

1

(u0, ϑ0). If additionally,

∇u∈L^2α−γ2α (0, T; ˙B^−γ_∞,∞(Rⁿ)) (1.2) for some0< γ <2α, then the solution (u, ϑ) can be extended smoothly beyondT.

Here, ˙B^−γ_∞,∞(Rⁿ) is the homogeneous Besov space with negative order, which contains classical Lebesgue space L^nγ(Rⁿ), see [1, Chapter 2]. In the proof of Theorem 1.1 in Section 2, we shall frequently use the following refined Gagliardo- Nirenberg inequality.

Lemma 1.2 ([1, Theorem 2.42]). Let 2< q <∞ andγ be a positive real number.

Then a constant C exists such that

kfkL^q≤Ckfk¹⁻

2 q

B˙^−γ∞,∞kfk^2/q_˙

H^γ(q2−1). (1.3)

Remark 1.3. Our result extends that of Kozono-Shimada [6]. Indeed, the Navier- Stokes equations corresponds to (1.1) withϑ= 0 andα= 1.

Remark 1.4. In [3] (see also the end-point smallness condition in [2]), Geng-Fan proved a regularity criterion

u∈L^1−r2 (0, T; ˙B_∞,∞^−r (R³)) (−1< r <1, r6= 0) (1.4) for system (1.1) with α= 1 andn = 3. Thus our result generalizes (1.4) also, in view of the fact that

C₁k∇fkB˙_∞,∞^−1−r ≤ kfkB˙^−r_∞,∞ ≤C₂k∇fkB˙^−1−r_∞,∞.

Moreover, our result (1.2) is valid for (1.1) with arbitrarily largenand arbitrarily smallα.

Interested readers are referred to [11] for blow-up criterion for (1.1) without diffusion in theuequation.

2. Proof of Theorem 1.1

It is not difficult to prove that there exists aT₀>0 and a unique smooth solution (u, ϑ) to (1.1) on [0, T₀]. We only need to establish the a priori estimates. There- fore, in the following calculations, we assume that the solution (u, ϑ) is sufficiently smooth on [0, T].

First, taking the inner product of (1.1)1and (1.1)2 withu, ϑin L²(Rⁿ) respec- tively, we obtain

1 2

d

dtk(u, ϑ)k²_L2+kΛ^αuk²_L2 = Z

Rⁿ

ϑen·udx≤1

2k(u, ϑ)k²_L2. Applying Gronwall inequality, we deduce

k(u, ϑ)kL^∞(0,t;L²(Rⁿ))+kΛ^αukL²(0,t;L²(Rⁿ)) ≤C. (2.1)

For k > 0, applying Λ^k to (1.1)₁, and testing the resulting equations by Λ^ku respectively, we obtain

1 2

d

dtkΛ^kuk²_L2+kΛ^k+αuk²_L2

=− Z

Rⁿ

Λ^k[(u· ∇)u]·Λ^kudx+ Z

Rⁿ

Λ^k(ϑe_n)·Λ^kudx

=− Z

R³

Λ^k[(u· ∇)u]−(u· ∇)(Λ^ku) ·Λ^kudx+ Z

Rⁿ

Λ^k(ϑe_n)·Λ^kudx

≡I₁^k+I₂^k.

(2.2)

We may use the following commutator estimates of Kato-Ponce [4]:

kΛ^k(f g)−f Λ^kgkL^p≤C

k∇fkL^p1kΛ^k−1gkL^p2 +kΛ^kfkL^p3kgkL^p4

(2.3)

with

1< p, p₂, p₃<∞, 1≤p₁, p₄≤ ∞, 1 p = 1

p1

+ 1 p2

= 1 p3

+ 1 p4

to boundI₁^k as

I₁^k ≤CkΛ^k[(u· ∇)u]−(u· ∇)(Λ^ku)k

L

4(k+γ+α−1) 2k+3γ+2α−2kΛ^kuk

L

4(k+γ+α−1) 2k+γ+2α−2

≤Ck∇uk

L

2(k+γ+α−1) γ kΛ^kuk

L

4(k+γ+α−1)

2k+γ+2α−2 · kΛ^kuk

L

4(k+γ+α−1) 2k+γ+2α−2

≤Ck∇uk

k+α−1 k+γ+α−1

B˙∞,∞^−γ k∇uk

γ k+γ+α−1

H˙^k+α−1

kΛ^kuk

γ 2(k+γ+α−1)

B˙^{−(k−1+γ)}∞,∞

kΛ^kuk

2k+γ+2α−2 2(k+γ+α−1)

H˙

γ(k+γ−1) 2k+γ+2α−2

²

≤Ck∇uk

k+α−1 k+γ+α−1

B˙∞,∞^−γ kΛ^k+αuk

γ k+γ+α−1

L² k∇uk

γ k+γ+α−1

B˙^−γ∞,∞ kΛ^kuk

2k+γ+2α−2 k+γ+α−1

H˙

γ(k+γ−1) 2k+γ+2α−2

≤Ck∇ukB˙_∞,∞^−γ kΛ^k+αuk

γ k+γ+α−1

L²

×

kΛ^kuk¹⁻

γ(k+γ−1) α(2k+γ+2α−2)

L² kΛ^k+αuk

γ(k+γ−1) α(2k+γ+2α−2)

L²

^{2k+γ+2α−2}_k+γ+α−1

≤Ck∇uk_B˙−γ

∞,∞kΛ^kuk

2α−γ α

L² kΛ^k+αuk_L^αγ2

≤Ck∇uk

2α 2α−γ

B˙_∞,∞^−γ kΛ^kuk²_L2+1

2kΛ^k+αuk²_L2.

(2.4)

Substituting (2.4) in (2.2), we find d

dtkΛ^kuk²_L2+kΛ^k+αuk²_L2 ≤Ck∇uk

2α 2α−γ

B˙^−γ∞,∞kΛ^kuk²_L2+ 2I₂^k. (2.5) Now, we treat 2I₂^k step by step. If 0< k≤α, then

2I₂^k = 2 Z

Rⁿ

ϑe_n·Λ^2kudx

≤2kϑkL²kΛ^2kukL²

≤Ckϑk_L2 kuk_L2+kΛ^k+αuk_L2

H^k+α(Rⁿ)⊂H˙^2k(Rⁿ)

≤C+1

2kΛ^k+αuk²_L2 (by (2.1)).

(2.6)

Substituting (2.6) into (2.5), we apply Gronwall inequality to deduce

kΛ^k(u, ϑ)kL^∞(0,t;L²(Rⁿ))+kΛ^k+αukL²(0,t;L²(Rⁿ)) ≤C (0< k≤α). (2.7)

Suppose we have already the statement for some 0≤l∈N,

kΛ^k(u, ϑ)k_L^∞_(0,t;L2(Rⁿ))+kΛ^k+αuk_L2(0,t;L²(Rⁿ))≤C (∀lα < k≤(l+ 1)α), (2.8) we wish to deduce higher-order estimate

kΛ^k+α(u, ϑ)k_L^∞_(0,t;L2(Rⁿ))+kΛ^k+2αuk_L2(0,t;L²(Rⁿ))≤C. (2.9) Indeed, as long as (2.8) holds, we may dominate 2I₂^k+αas

2I₂^k+α= 2 Z

Rⁿ

Λ^k+α(ϑen)·Λ^k+αudx

= 2 Z

Rⁿ

Λ^k(ϑen)·Λ^k+2αudx

≤2kΛ^kϑk_L2kΛ^k+2αuk_L2

≤2kΛ^kϑk²_L2+1

2kΛ^k+2αuk²_L2.

(2.10)

Putting (2.10) into (2.5) withkreplaced byk+α, and using (2.8), we deduce (2.9) as desired.

Now prove that (2.7) and (2.8) imply (2.9), we see readily that

kΛ^suk_L∞(0,t;L²(Rⁿ))+kΛ^s+αuk_L2(0,t;L²(Rⁿ))≤C. (2.11) With this good estimate of the velocity field, we are now in a position to treat that ofϑ. ApplyingΛ^sto (1.1)₂, and testing the resultant equation byΛ^sϑ, we obtain

1 2

d

dtkΛ^sϑk²_L2

=− Z

Rⁿ

Λ^s[(u· ∇)ϑ]·Λ^sϑdx

=− Z

Rⁿ

{Λ^s[(u· ∇)ϑ]−(u· ∇)Λ^sϑ} ·Λ^sϑdx

≤C

k∇ukL^∞kΛ^sϑk_L2+k∇ϑkL^∞kΛ^suk_L2

kΛ^sϑk_L2 (by (2.3))

≤C

kukL²+kΛ^sukL²

kΛ^sϑk²_L2+

kϑkL²+kΛ^sϑkL²

kΛ^sukL²kΛ^sϑkL²

(byH^s(Rⁿ)⊂W^1,∞(Rⁿ))

≤C+CkΛ^sϑk²_L2 (by (2.1) and (2.11)).

(2.12)

Applying Gronwall inequality, we obtain

kΛ^sϑk_L∞(0,t;L²(Rⁿ))≤C.

With this estimate and (2.11), we complete the proof.

Acknowledgements. This work is supported by the Natural Science Foundation of Jiangxi (grant no. 20151BAB201010), the National Natural Science Foundation of China (grant nos. 11501125, 11361004) and the Supporting the Development for Local Colleges and Universities Foundation of China – Applied Mathematics Innovative Team Building.

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Zujin Zhang

School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, Jiangxi, China

E-mail address:[email protected], phone (86) 07978393663