# An Osgood Type Regularity Criterion for the 3D Boussinesq Equations

Authors: Qiang Wu; Lin Hu; Guili Liu
Journal: The Scientific World Journal
 (2014)

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## Abstract

We consider the three-dimensional Boussinesq equations, and obtain an Osgood type regularity criterion in terms of the velocity gradient.


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## Body

## 1. Introduction

In this paper, we consider the following three-dimensional (3D) Boussinesq equations with the incompressibility condition:(1)ut+(u·∇)u-Δu+∇π=θe3,θt+(u·∇)θ-Δθ=0,∇·u=0,u(x,0)=u0,θ(x,0)=θ0,
where u=(u1(x,t),u2(x,t),u3(x,t)) is the fluid velocity, π=π(x,t) is a scalar pressure, and θ=θ(x,t) is the scalar temperature, while u0 and θ0 are the prescribed initial velocity and temperature, respectively, with ∇·u0=0.In caseθ=0, (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are important open problems; see [1–3]. Starting with [4, 5], there have been a lot of literatures devoted to finding sufficient conditions (which now are called regularity criteria) to ensure the smoothness of the solutions; see [6–16] and so forth. Since the convective terms (u·∇)u are the same in the Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1). In particular, Qiu et al. [17] obtained Serrin type regularity condition:
(2)u∈Lp(0,T;Lq(ℝ3)),2p+3q=1,3<q⩽∞.The extension to the multiplier spaces was established by the same authors in [18]. For the Besov-type regularity criterion, Fan and Zhou [19] and Ishimura and Morimoto [20] showed the following regularity conditions:
(3)∇×u∈L1(0,T;B˙∞,∞0(ℝ3)),∇u∈L1(0,T;L∞(ℝ3)).
Zhang [21, 22] then considers the regularity criterion in terms of the pressure or its gradient. The readers are also referred to [23] for generalized models.Motivated by [24–26], we will improve (3) as in the following.Theorem 1.
Let(u0,θ0)∈H1(ℝ3). Assume that (u,θ) is the smooth solution to (1) with the initial data (u0,θ0) for 0⩽t<T. If
(4)sup⁡2⩽q<∞∫0T‍∥S¯q∇u∥L∞qln⁡q<∞,
then the solution (u,θ) can be extended after time t=T. Here, Δ˙k denotes the Fourier localization operator and ΔSq=∑l=-qq‍Δ˙l.Remark 2.
The Osgood type condition (4) is weaker than (3). Notice that, for q∈[2,∞), we have
(5)∥S¯q∇u∥L∞qln⁡q⩽1qln⁡q∑l=-qq‍∥Δ˙l(∇×u)∥L∞⩽C∥∇×u∥B˙∞,∞0.The rest of this paper is organized as follows. In Section2, we recall the definition of Besov spaces and some interpolation inequalities. Section 3 is devoted to proving Theorem 1.


## 2. Preliminaries

Let𝒮(ℝ3) be the Schwartz class of rapidly decreasing functions. For f∈𝒮(ℝ3), its Fourier transform ℱf=f^ is defined by
(6)f^(ξ)=∫ℝ3f(x)e-ix·ξdx.
Let us choose a nonnegative radial function φ∈𝒮(ℝ3) such that
(7)0⩽φ^(ξ)⩽1,φ^(ξ)={1,if|ξ|⩽1,0,if|ξ|⩾2,
and let
(8)ψ(x)=φ(x)-2-3φ(x2),φj(x)=23jφ(2jx),ψj(x)=23jψ(2jx),j∈ℤ.
For j∈ℤ, the Littlewood-Paley projection operators Sj and Δ˙j are, respectively, defined by
(9)Sjf=φj*f,Δ˙jf=ψj*f.
Observe that Δ˙j=Sj-Sj-1. Also, it is easy to check that if f∈L2(ℝ3), then
(10)Sjf⟶0,asj⟶-∞;Sjf⟶f,asj⟶+∞,
in the L2 sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition:
(11)f=∑j=-∞+∞Δ˙jf,
for all f∈L2(ℝ3), where the summation is the L2 sense. Notice that
(12)Δ˙jf=∑l=j-2j+2Δ˙lΔ˙jf=∑l=j-2j+2ψl*ψj*f;
then from Young's inequality, it readily follows that
(13)∥Δ˙jf∥Lq⩽C23j(1/p-1/q)∥Δ˙jf∥Lp,
where 1⩽p⩽q⩽∞ and C is an absolute constant independent of f and j.Let-∞<s<∞, 1⩽p, q⩽∞; the homogeneous Besov space B˙p,qs is defined by the full-dyadic decomposition such that
(14)B˙p,qs={f∈𝒵′(ℝ3);∥f∥B˙p,qs<∞},
where
(15)∥f∥B˙p,qs=∥{2js∥Δ˙jf∥Lp}j=-∞+∞∥ℓq,
and 𝒵′(ℝ3) is the dual space of
(16)𝒵(ℝ3)={f∈𝒮(ℝ3);Dαf^(0)=0,∀α∈ℕ3}.
Also, it is well known that
(17)H˙s(ℝ3)=B˙2,2s(ℝ3),∀s∈ℝ.
We refer to [27] for more detailed properties.


## 3. Proof of Theorem1

This section is devoted to proving Theorem1. From standard continuity arguments, we need to only provide the uniform H1 bounds of the solution (u,θ).Taking the inner products of (1)1 with -Δu, (1)2 with -Δθ, we obtain by adding together that
(18)12ddt∥∇(u,θ)∥L22+∥Δ(u,θ)∥L22=∫ℝ3[(u·∇)u]·Δudx-∫ℝ3θΔu3dx+∫ℝ3‍[(u·∇)θ]·Δθdx=∫ℝ3∂kθ∂ku3dx-∫ℝ3∂kuj(∂jui∂kui+∂jθ∂kθ)dx≡I+J.
For I, we use Hölder’s inequality to get
(19)I1⩽12∥∇(u,θ)∥L22.
For J, applying the Littilewood-Paley decomposition as in (11), we get
(20)∇u=∑l<-qΔ˙∇u+∑l=-qqΔ˙∇u+∑l>qΔ˙∇u,
where q is positive integral to be determined later on. Plugging (20) into J, we see that
(21)J⩽∑l<-q∫ℝ3|Δ˙l∇u|‍‍·|∇(u,θ)|2dx+∫ℝ3|∑l=-qqΔ˙l∇u‍|‍·|∇(u,θ)|2dx+∑l>q∫ℝ3|Δ˙l∇u|‍‍·|∇(u,θ)|2dx≡J1+J2+J3.
For J1, we dominate as
(22)J1⩽∑l<-q∥Δ˙l∇u∥L∞‍∥∇(u,θ)∥L22⩽C∑l<-q23l/2∥Δ˙l∇u∥L2‍∥∇(u,θ)∥L22(by(13))⩽C(∑l<-q2(3l/2)·2‍)1/2·(∑l<-q∥Δ˙l∇u∥L22‍)1/2∥∇(u,θ)∥L22⩽C2-3q/2∥∇u∥L2∥∇(u,θ)∥L22(by(17))=[C2-q/2∥∇(u,θ)∥L2]3.
For J2, we have
(23)J2=∫ℝ3|S¯q∇u|‍·|∇(u,θ)|2dx⩽∥S¯q∇u∥L∞∥∇(u,θ)∥L22.
Finally, for J3, we estimate as
(24)J3⩽∑l>q∥Δl∇u∥L3‍∥∇(u,θ)∥L32⩽C∑l>q2l/2∥Δl∇u∥L2‍∥∇(u,θ)∥L2∥Δ(u,θ)∥L2(by(13)andGagliardo-Nireberginequality)⩽C(∑l>q2-(l/2)·2‍)1/2·(∑l>q2l·2∥Δ˙l∇u∥L22‍)1/2×∥∇(u,θ)∥L2∥Δ(u,θ)∥L2⩽[C2-q/2∥∇(u,θ)∥L2]∥Δ(u,θ)∥L22(by(17)).
Gathering (22), (23), and (24) together and plugging them into (21), we deduce
(25)J⩽[C2-q/2∥∇(u,θ)∥L2]3+∥S¯q∇u∥L∞∥∇(u,θ)∥L22+[C2-q/2∥∇(u,θ)∥L2]∥Δ(u,θ)∥L22.
Substituting (19) and (25) into (18), we find
(26)12ddt∥∇(u,θ)∥L22+∥Δ(u,θ)∥L22⩽12∥∇(u,θ)∥L22+[C2-q/2∥∇(u,θ)∥L2]3+∥S¯q∇u∥L∞qln⁡q·qln⁡q∥∇(u,θ)∥L22+[C2-q/2∥∇(u,θ)∥L2]∥Δ(u,θ)∥L22.
Taking
(27)q=[2ln⁡2ln⁡+(C∥∇(u,θ)∥L2)]+1,
where [t] is the largest integer smaller than t∈ℝ and ln⁡+t=ln⁡(e+t), then (26) implies that
(28)ddt∥∇(u,θ)∥L22⩽∥∇(u,θ)∥L22+C+∥S¯q∇u∥L∞qln⁡qln⁡+(∥∇(u,θ)∥L2)ln⁡+ln⁡+(∥∇(u,θ)∥L2)×∥∇(u,θ)∥L22.
Applying Gronwall inequality three times, we deduce
(29)∥∇(u,θ)∥L22+∫0t∥Δ(u,θ)∥L2dτ⩽Cexp⁡exp exp(∫0t‍∥S¯q∇u∥L∞qln⁡qdτ).
Recalling (4), we see the solution (u,θ) is uniformly bounded in L∞(0,T;H1(ℝ3)). This completes the proof of Theorem 1.



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*Source: 563084-2014-03-11.xml*