An Osgood Type Regularity Criterion for the 3D Boussinesq Equations

Abstract

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

1. Introduction

In this paper, we consider the following three-dimensional (3D) Boussinesq equations with the incompressibility condition:

$$\begin{array}{c}{\mathbf{u}}_t+(\mathbf{u}·\nabla )\mathbf{u}-\mathrm{\Delta }\mathbf{u}+\nabla \pi =\theta {\mathbf{e}}_{\mathrm{3}},\\ {\theta }_t+(\mathbf{u}·\nabla )\theta -\mathrm{\Delta }\theta =\mathrm{0},\\ \nabla ·\mathbf{u}=\mathrm{0},\\ \mathbf{u}(x,\mathrm{0})={\mathbf{u}}_{\mathrm{0}},\hspace{1em}\theta (x,\mathrm{0})={\theta }_{\mathrm{0}},\end{array}$$ (1)

where u = (u ₁(x, t), u ₂(x, t), u ₃(x, t)) is the fluid velocity, π = π(x, t) is a scalar pressure, and θ = θ(x, t) is the scalar temperature, while u ₀ and θ ₀ are the prescribed initial velocity and temperature, respectively, with ∇·u ₀ = 0.

In case θ = 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:

$$\begin{array}{c}\mathbf{u}\in L^p\left(\mathrm{0},T;L^q\left({ℝ}^{\mathrm{3}}\right)\right),\hspace{1em}\hspace{1em}\frac{\mathrm{2}}{p}+\frac{\mathrm{3}}{q}=\mathrm{1},\hspace{1em}\mathrm{3}<q⩽\infty .\end{array}$$ (2)

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:

$$\begin{array}{c}\nabla \times \mathbf{u}\in L^{\mathrm{1}}\left(\mathrm{0},T;{\dot{B}}_{\infty ,\infty }^{\mathrm{0}}\left({ℝ}^{\mathrm{3}}\right)\right),\\ \nabla \mathbf{u}\in L^{\mathrm{1}}(\mathrm{0},T;L^{\infty }\left({ℝ}^{\mathrm{3}}\right)).\end{array}$$ (3)

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 (u ₀, θ ₀) ∈ H ¹(ℝ³). Assume that (u, θ) is the smooth solution to (1) with the initial data (u ₀, θ ₀) for 0 ⩽ t < T. If

$$\begin{array}{c}\underset{\mathrm{2}⩽q<\infty }{\sup }\underset{\mathrm{0}}{\overset{T}{{\displaystyle \int }}}\frac{{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}}{q\ln q}<\infty ,\end{array}$$ (4)

then the solution (u, θ) can be extended after time t = T. Here, ${\dot{\Delta }}_k$ denotes the Fourier localization operator and $\Delta S_q={\sum }_{l=-q}^q{\dot{\Delta }}_l$.

Remark 2 —

The Osgood type condition (4) is weaker than (3). Notice that, for q ∈ [2, ∞), we have

$$\begin{array}{c}\frac{{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}}{q\ln q}⩽\frac{\mathrm{1}}{q\ln q}{\displaystyle \sum }_{l=-q}^q{||{\dot{\mathrm{\Delta }}}_l(\nabla \times \mathbf{u})||}_{L^{\infty }}⩽C{||\nabla \times \mathbf{u}||}_{{\dot{B}}_{\infty ,\infty }^{\mathrm{0}}}.\end{array}$$ (5)

The rest of this paper is organized as follows. In Section 2, we recall the definition of Besov spaces and some interpolation inequalities. Section 3 is devoted to proving Theorem 1.

2. Preliminaries

Let 𝒮(ℝ³) be the Schwartz class of rapidly decreasing functions. For f ∈ 𝒮(ℝ³), its Fourier transform $ℱf=\widehat{f}$ is defined by

$$\begin{array}{c}\widehat{f}\left(\xi \right)={{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}f\left(x\right)e^{-ix·\xi }dx.\end{array}$$ (6)

Let us choose a nonnegative radial function φ ∈ 𝒮(ℝ³) such that

$$\begin{array}{c}\mathrm{0}⩽\widehat{\phi }\left(\xi \right)⩽\mathrm{1},\hspace{1em}\hspace{1em}\widehat{\phi }\left(\xi \right)=\{\begin{array}{cc}\mathrm{1},\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\left|\xi \right|⩽\mathrm{1},\hfill \\ \mathrm{0},\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\left|\xi \right|⩾\mathrm{2},\hfill \end{array}\end{array}$$ (7)

and let

$$\begin{array}{c}\psi \left(x\right)=\phi \left(x\right)-{\mathrm{2}}^{-\mathrm{3}}\phi \left(\frac{x}{\mathrm{2}}\right),\\ {\phi }_j\left(x\right)={\mathrm{2}}^{\mathrm{3}j}\phi \left({\mathrm{2}}^{j}x\right),\hspace{1em}\hspace{1em}{\psi }_j\left(x\right)={\mathrm{2}}^{\mathrm{3}j}\psi \left({\mathrm{2}}^{j}x\right),\hspace{1em}j\in \mathbb{Z}.\end{array}$$ (8)

For j ∈ ℤ, the Littlewood-Paley projection operators S _j and ${\dot{\Delta }}_j$ are, respectively, defined by

$$\begin{array}{c}{S}_{j}f={\phi }_j\ast f,\hspace{1em}\hspace{1em}{\dot{\mathrm{\Delta }}}_{j}f={\psi }_j\ast f.\end{array}$$ (9)

Observe that ${\dot{\Delta }}_j=S_j-S_{j-\mathrm{1}}$. Also, it is easy to check that if f ∈ L ²(ℝ³), then

$$\begin{array}{c}{S}_{j}f⟶\mathrm{0},\hspace{1em}\text{as}\hspace{0.17em}\hspace{0.17em}j⟶-\infty ;\hspace{1em}\hspace{1em}{S}_{j}f⟶f,\hspace{1em}\text{as}\hspace{0.17em}\hspace{0.17em}j⟶+\infty ,\end{array}$$ (10)

in the L ² sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition:

$$\begin{array}{c}f={\displaystyle \sum }_{j=-\infty }^{+\infty }{\dot{\mathrm{\Delta }}}_{j}f,\end{array}$$ (11)

for all f ∈ L ²(ℝ³), where the summation is the L ² sense. Notice that

$$\begin{array}{c}{\dot{\mathrm{\Delta }}}_{j}f={\displaystyle \sum }_{l=j-\mathrm{2}}^{j+\mathrm{2}}{\dot{\mathrm{\Delta }}}_l{\dot{\mathrm{\Delta }}}_{j}f={\displaystyle \sum }_{l=j-\mathrm{2}}^{j+\mathrm{2}}{\psi }_l\ast {\psi }_j\ast f;\end{array}$$ (12)

then from Young's inequality, it readily follows that

$$\begin{array}{c}{||{\dot{\mathrm{\Delta }}}_{j}f||}_{L^q}⩽C{\mathrm{2}}^{\mathrm{3}j(\mathrm{1}/p-\mathrm{1}/q)}{||{\dot{\mathrm{\Delta }}}_{j}f||}_{L^p},\end{array}$$ (13)

where 1 ⩽ p ⩽ q ⩽ ∞ and C is an absolute constant independent of f and j.

Let −∞ < s < ∞, 1 ⩽ p, q ⩽ ∞; the homogeneous Besov space ${\dot{B}}_{p,q}^s$ is defined by the full-dyadic decomposition such that

$$\begin{array}{c}{\dot{B}}_{p,q}^s=\left\{f\in {𝒵}^{\prime }\left({ℝ}^{\mathrm{3}}\right);{||f||}_{{\dot{B}}_{p,q}^s}<\infty \right\},\end{array}$$ (14)

where

$$\begin{array}{c}{||f||}_{{\dot{B}}_{p,q}^s}={||{\left\{{\mathrm{2}}^{js}{||{\dot{\mathrm{\Delta }}}_{j}f||}_{L^p}\right\}}_{j=-\infty }^{+\infty }||}_{{\ell }^q},\end{array}$$ (15)

and 𝒵′(ℝ³) is the dual space of

$$\begin{array}{c}𝒵\left({ℝ}^{\mathrm{3}}\right)=\left\{f\in 𝒮\left({ℝ}^{\mathrm{3}}\right);\hspace{0.17em}\hspace{0.17em}{D}^{\alpha }\widehat{f}\left(\mathrm{0}\right)=\mathrm{0},\hspace{0.17em}\forall \alpha \in {\mathbb{N}}^{\mathrm{3}}\right\}.\end{array}$$ (16)

Also, it is well known that

$$\begin{array}{c}{\dot{H}}^s\left({ℝ}^{\mathrm{3}}\right)={\dot{B}}_{\mathrm{2,2}}^s\left({ℝ}^{\mathrm{3}}\right),\hspace{1em}\forall s\in ℝ.\end{array}$$ (17)

We refer to [27] for more detailed properties.

3. Proof of Theorem 1

This section is devoted to proving Theorem 1. From standard continuity arguments, we need to only provide the uniform H ¹ 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

$$\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}\frac{d}{dt}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}+{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ \hspace{1em}={{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left[\left(\mathbf{u}·\nabla \right)\mathbf{u}\right]·\mathrm{\Delta }\mathbf{u}\hspace{0.17em}dx-{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\theta \mathrm{\Delta }{u}_{\mathrm{3}}\hspace{0.17em}\text{d}x\\ \hspace{1em}\hspace{1em}+{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left[(\mathbf{u}·\nabla )\theta \right]·\mathrm{\Delta }\theta \hspace{0.17em}\text{d}x\\ \hspace{1em}={{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}{\partial }_k\theta {\partial }_k{u}_{\mathrm{3}}\hspace{0.17em}\text{d}x\\ \hspace{1em}\hspace{1em}-{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}{\partial }_k{u}_j\left({\partial }_j{u}_i{\partial }_k{u}_i+{\partial }_j\theta {\partial }_k\theta \right)\text{d}x\\ \hspace{1em}\equiv I+J.\end{array}$$ (18)

For I, we use Hölder's inequality to get

$$\begin{array}{c}{I}_{\mathrm{1}}⩽\frac{\mathrm{1}}{\mathrm{2}}{||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}^{\mathrm{2}}.\end{array}$$ (19)

For J, applying the Littilewood-Paley decomposition as in (11), we get

$$\begin{array}{c}\nabla \mathbf{u}={\displaystyle \sum }_{l<-q}\dot{\mathrm{\Delta }}\nabla \mathbf{u}+{\displaystyle \sum }_{l=-q}^q\dot{\mathrm{\Delta }}\nabla \mathbf{u}+{\displaystyle \sum }_{l>q}\dot{\mathrm{\Delta }}\nabla \mathbf{u},\end{array}$$ (20)

where q is positive integral to be determined later on. Plugging (20) into J, we see that

$$\begin{array}{c}J⩽{\displaystyle \sum }_{l<-q}{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left|{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}\right|‍‍·{\left|\nabla \left(\mathbf{u},\theta \right)\right|}^{\mathrm{2}}\hspace{0.17em}\text{d}x\\ \hspace{1em}+{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left|{\displaystyle \sum }_{l=-q}^q{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}‍\right|‍·{\left|\nabla \left(\mathbf{u},\theta \right)\right|}^{\mathrm{2}}\hspace{0.17em}\text{d}x\\ \hspace{1em}+{\displaystyle \sum }_{l>q}{{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left|{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}\right|‍‍·{\left|\nabla \left(\mathbf{u},\theta \right)\right|}^{\mathrm{2}}\hspace{0.17em}\text{d}x\\ \equiv J_{\mathrm{1}}+J_{\mathrm{2}}+J_{\mathrm{3}}.\end{array}$$ (21)

For J ₁, we dominate as

$$\begin{array}{c}{J}_{\mathrm{1}}⩽{\displaystyle \sum }_{l<-q}{||{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}||}_{L^{\infty }}‍{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ ⩽C{\displaystyle \sum }_{l<-q}{\mathrm{2}}^{\mathrm{3}l/\mathrm{2}}{||{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}||}_{L^{\mathrm{2}}}‍{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\hspace{1em}\left(\text{by}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{13}\right)\right)\\ ⩽C{\left({\displaystyle \sum }_{l<-q}{\mathrm{2}}^{\left(\mathrm{3}l/\mathrm{2}\right)·\mathrm{2}}‍\right)}^{\mathrm{1}/\mathrm{2}}·{\left({\displaystyle \sum }_{l<-q}\hspace{0.17em}{||{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}||}_{L^{\mathrm{2}}}^{\mathrm{2}}‍\right)}^{\mathrm{1}/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ ⩽C{\mathrm{2}}^{-\mathrm{3}q/\mathrm{2}}{||\nabla \mathbf{u}||}_{L^{\mathrm{2}}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\hspace{1em}\left(\text{by}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{17}\right)\right)\\ ={\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right]}^{\mathrm{3}}.\end{array}$$ (22)

For J ₂, we have

$$\begin{array}{c}{J}_{\mathrm{2}}={{\displaystyle \int }}_{{ℝ}^{\mathrm{3}}}\left|{\overline{S}}_q\nabla \mathbf{u}\right|‍·{\left|\nabla \left(\mathbf{u},\theta \right)\right|}^{\mathrm{2}}\hspace{0.17em}\text{d}x\\ ⩽{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}.\end{array}$$ (23)

Finally, for J ₃, we estimate as

$$\begin{array}{c}{J}_{\mathrm{3}}⩽{\displaystyle \sum }_{l>q}{||{\mathrm{\Delta }}_l\nabla \mathbf{u}||}_{L^{\mathrm{3}}}‍{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{3}}}^{\mathrm{2}}\\ ⩽C\hspace{0.17em}{\displaystyle \sum }_{l>q}{\mathrm{2}}^{l/\mathrm{2}}{||{\mathrm{\Delta }}_l\nabla \mathbf{u}||}_{L^{\mathrm{2}}}‍{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\\ \hspace{1em}\left(\text{by}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{13}\right)\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\text{Gagliardo-Nireberg}\hspace{0.17em}\hspace{0.17em}\text{inequality}\right)\\ ⩽C{\left({\displaystyle \sum }_{l>q}{\mathrm{2}}^{-\left(l/\mathrm{2}\right)·\mathrm{2}}‍\right)}^{\mathrm{1}/\mathrm{2}}·{\left({\displaystyle \sum }_{l>q}{\mathrm{2}}^{l·\mathrm{2}}{||{\dot{\mathrm{\Delta }}}_l\nabla \mathbf{u}||}_{L^{\mathrm{2}}}^{\mathrm{2}}‍\right)}^{\mathrm{1}/\mathrm{2}}\\ \hspace{1em}\times {||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\\ ⩽\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right]{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\hspace{1em}\left(\text{by}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{17}\right)\right).\end{array}$$ (24)

Gathering (22), (23), and (24) together and plugging them into (21), we deduce

$$\begin{array}{c}J⩽{\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right]}^{\mathrm{3}}\\ +{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}{||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ +\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}\right]{||\mathrm{\Delta }(\mathbf{u},\theta )||}_{L^{\mathrm{2}}}^{\mathrm{2}}.\end{array}$$ (25)

Substituting (19) and (25) into (18), we find

$$\begin{array}{c}\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}\frac{d}{dt}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}+{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ \hspace{1em}⩽\frac{\mathrm{1}}{\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}+{\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right]}^{\mathrm{3}}\\ \hspace{1em}\hspace{1em}+\frac{{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}}{q\ln q}·q\ln q{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ \hspace{1em}\hspace{1em}+\left[C{\mathrm{2}}^{-q/\mathrm{2}}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right]{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}.\end{array}\end{array}$$ (26)

Taking

$$\begin{array}{c}q=\left[\frac{\mathrm{2}}{\ln \hspace{0.17em}\mathrm{2}\hspace{0.17em}{\ln }^{+}\left(C{||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}\right)}\right]+\mathrm{1},\end{array}$$ (27)

where [t] is the largest integer smaller than t ∈ ℝ and ln⁡⁺ t = ln⁡(e + t), then (26) implies that

$$\begin{array}{c}\frac{d}{dt}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}\\ \hspace{1em}⩽{||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}^{\mathrm{2}}+C\\ \hspace{1em}\hspace{1em}+\frac{{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}}{q\ln q}{\ln }^{+}\left({||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right){\ln }^{+}{\ln }^{+}\left({||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\right)\\ \hspace{1em}\hspace{1em}\times {||\nabla (\mathbf{u},\theta )||}_{L^{\mathrm{2}}}^{\mathrm{2}}.\end{array}$$ (28)

Applying Gronwall inequality three times, we deduce

$$\begin{array}{c}{||\nabla \left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}^{\mathrm{2}}+\underset{\mathrm{0}}{\overset{t}{{\displaystyle \int }}}{||\mathrm{\Delta }\left(\mathbf{u},\theta \right)||}_{L^{\mathrm{2}}}\hspace{0.17em}\text{d}\tau \\ \hspace{1em}⩽C\hspace{0.17em}\exp \hspace{0.17em}\text{exp\hspace{0.17em}exp}\left(\underset{\mathrm{0}}{\overset{t}{{\displaystyle \int }}}\frac{{||{\overline{S}}_q\nabla \mathbf{u}||}_{L^{\infty }}}{q\ln q}\hspace{0.17em}\text{d}\tau \right).\end{array}$$ (29)

Recalling (4), we see the solution (u, θ) is uniformly bounded in L ^∞(0, T; H ¹(ℝ³)). This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.