# Some Regularity Criteria for the 3D Boussinesq Equations in the ClassL20,T;B˙∞,∞-1

Authors: Zujin Zhang
Journal: ISRN Applied Mathematics
 (2014)

---

## Abstract

We consider the three-dimensional Boussinesq equations and obtain some regularity criteria via the velocity gradient (or the vorticity, or the deformation tensor) and the temperature gradient.


---

## Body

## 1. Introduction

Consider the following three-dimensional (3D) Boussinesq equations:(1)ut+(u·∇)u-Δu+∇π=θe3,θt+(u·∇)θ-Δθ=0,∇·u=0,u(x,0)=u0,θ(x,0)=θ0.
Here, 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)≥0 is the temperature, while u0 and θ0 are the prescribed initial velocity and temperature, respectively.Whenθ=0, (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are challenging open problems; see [1–3]. Initialed by [4, 5], there have been a lot of literatures devoted to finding sufficient conditions to ensure the smoothness of the solutions; see [6–18] and so forth. Since the convective terms are similar in Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1); see [19–23] and so forth.Motivated by [7], we will consider the regularity criteria for (1) and extend the result of [7] to the case of Boussinesq equations.Before stating the precise result, let us recall the weak formulation of (1).Definition 1.
Letu0∈L2(ℝ3), θ0∈L1∩L∞(ℝ3), and T>0. A measurable pair (u,θ) is said to be a weak solution of (1) in (0,T), provided that (1)
(u,θ)∈L∞(0,T;L2(ℝ3))∩L2(0,T;H1(ℝ3)), θ∈L∞(0, T; L1∩L∞(ℝ3));(2)
(1)1,2,3 are satisfied in the sense of distributions;(3)
the energy inequality(2)∥(u,θ)∥L22+2∫0t‍∥∇(u,θ)∥L22ds≤∫0t‍∫ℝ3‍θu3dxds,for all0≤t≤T.Now, our main result reads the following.Theorem 2.
Letu0∈L2(ℝ3) with ∇·u0=0 in the sense of distributions, θ0∈L1∩L∞(ℝ3). Suppose that (u,θ) is a weak solution of (1) in [0,T), and
(3)(∇u,∇θ)∈L2(0,T;B˙∞,∞-1(ℝ3));
then the solution (u,θ)∈C∞((0,T)×ℝ3).Due to the divergence-free condition of the fluid velocityu, we have
(4)Δui=∑k=13‍∂k(∂kui-∂iuk),Δui=∑k=13‍∂k(∂kui+∂iuk).
Thus,
(5)∂jui=-∑k=13‍ℛjℛk(∂kui-∂iuk),∂jui=-∑k=13‍ℛjℛk(∂kui+∂iuk).
Here, ℛj=∂j/-Δ is the Riesz transformation.Using (5), we can deduce easily from Theorem 2 the following.Corollary 3.
Letu0∈L2(ℝ3) with ∇·u0=0 in the sense of distributions, θ0∈L1∩L∞(ℝ3). Suppose that (u,θ) is a weak solution of (1) in [0,T), and
(6)(∇×u,∇θ)∈L2(0,T;B˙∞,∞-1(ℝ3)),
or
(7)(Defu,∇θ)∈L2(0,T;B˙∞,∞-1(ℝ3));
then the solution (u,θ)∈C∞((0,T)×ℝ3). Here, ∇×u=|ijk∂1∂2∂3u1u2u3| is the vorticity and (Defu)ij=∂iuj + ∂jui is the deformation tensor (the symmetric tensor of the rate of strain).The rest of this paper is organized as follows. In Section2, we recall the definition of Besov spaces and the related interpolation inequalities. Section 3 is devoted to proving Theorem 2.


## 2. Preliminaries

We first introduce the Littlewood-Paley decomposition. Let𝒮(ℝ3) be the Schwartz class of rapidly decreasing functions. For f∈𝒮(ℝ3), its Fourier transform ℱf=f^ is defined as
(8)f^(ξ)=∫ℝ3‍f(x)e-ix·ξdx.Let us choose a nonnegative radial functionφ∈𝒮(ℝ3) such that
(9)0≤φ^(ξ)≤1,φ^(ξ)={1,if|ξ|≤1,0,if|ξ|≥2,
and let
(10)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 Δj are, respectively, defined by
(11)Sjf=φj*f,Δjf=ψj*f.
Observe that Δj=Sj-Sj-1. Also, it is easy to check that if f∈L2(ℝ3), then
(12)Sjf⟶0,asj⟶-∞;(13)Sjf⟶f,asj⟶∞,
in the L2 sense. By telescoping the series, we have the following Littlewood-Paley decomposition:
(14)f=∑j=-∞∞‍Δjf,
for all f∈L2(ℝ3), where the summation is in the L2 sense.Lets∈ℝ; p,q∈[1,∞]; the homogeneous Besov space B˙p,qs(ℝ3) is defined by the full dyadic decomposition such as
(15)B˙p,qs={f∈𝒵′(ℝ3);∥f∥B˙p,qs=∥{2js∥Δjf∥Lp}j=-∞∞∥ℓq<∞},
where 𝒵′(ℝ3) is the dual space of
(16)𝒵(ℝ3)={f∈𝒮(ℝ3);Dαf^(0)=0,∀α∈ℕ3}.The following interpolation inequality will be needed in Section3:
(17)∥f∥L4≤C∥f∥H˙11/2∥f∥B˙∞,∞-11/2,∀f∈H˙1(ℝ3)∩B˙∞,∞-1(ℝ3).
See [24] for the proof.


## 3. Proof of Theorem2

In this section, we will prove Theorem2.Multiplying (1)1 by -Δu and (1)2 by -Δθ, integrating over ℝ3, and summing up, we obtain
(18)12ddt∥∇(u,θ)∥L22+∥Δ(u,θ)∥L22=∫ℝ3‍[(u·∇)u]·Δudx-∫ℝ3‍θΔu3dx+∫ℝ3‍[(u·∇)θ]·Δθdx=-∑i,j,k=13‍∫ℝ3‍∂kuj∂jui∂kuidx+∫ℝ3‍∇θ·∇u3dx-∑j,k=13‍∫ℝ3‍∂kuj∂jθ∂kθdx≡I1+I2+I3.Invoking Hölder inequality, (17), and Young inequality, we may bound I1 as
(19)I1≤∥∇u∥L42∥∇u∥L2≤C∥∇u∥B˙∞,∞-1∥Δu∥L2∥∇u∥L2≤C∥∇u∥B˙∞,∞-12∥∇u∥L22+12∥Δu∥L22.
For I2, we use Hölder inequality to dominate as
(20)I2≤12∫ℝ3‍(|∇θ|2+|∇u|2)dx=12(∥∇θ∥L22+∥∇u∥L22).
Finally, I3 can be estimated similarly as I1,
(21)I3≤∥∇θ∥L42∥∇u∥L2≤C∥∇θ∥B˙∞,∞-1∥Δθ∥L2∥∇u∥L2≤C∥∇θ∥B˙∞,∞-12∥∇u∥L22+12∥Δθ∥L22.Substituting (19), (20), and (21) into (18), we gather
(22)ddt∥∇(u,θ)∥L22+∥Δ(u,θ)∥L22≤C(∥∇(u,θ)∥B˙∞,∞-12+1)∥∇(u,θ)∥L22.
Gronwall inequality together with (3) then implies that
(23)(u,θ)∈L∞(0,T;H1(ℝ3))∩L2(0,T;H2(ℝ3)).
Then, we may use standard energy method to drive high-order derivative bounds, which would imply (u,θ)∈C∞((0,T)×ℝ3) by Sobolev embedding theorems, as desired.The proof of Theorem2 is completed.



---

*Source: 564758-2014-02-23.xml*