Skip to main content

Advertisement

Persistence of global well-posedness for the 2D Boussinesq equations with fractional dissipation

Article metrics

  • 213 Accesses

Abstract

In this paper, we study the IBVP for the 2D Boussinesq equations with fractional dissipation in the subcritical case, and prove the persistence of global well-posedness of strong solutions. Moreover, we also prove the long time decay of the solutions, and investigate the existence of the solutions in Sobolev spaces \(W^{2,p}({R}^{2})\times W^{1,p}({R}^{2})\) for some \(p>2\).

Introduction

In this paper, we study the 2D Boussinesq equations with fractional dissipation. The model reads

$$\begin{aligned} \begin{aligned} &u_{t}+\nu \varLambda ^{2\alpha } u+u\cdot \nabla u+\nabla P=\theta e_{2}, \\ &\operatorname{div} u=0, \\ &\theta _{t}+\kappa \varLambda ^{2\beta } \theta +u\cdot \nabla \theta =0, \\ &u(x,0)=u_{0}(x), \qquad\theta (x,0)=\theta _{0}(x),\quad x\in \varOmega, \\ &u(x,t)=0, \qquad \theta (x,t)=0, \quad x\in \partial \varOmega, \end{aligned} \end{aligned}$$
(1)

where \(u=(u_{1},u_{2})\) is the velocity vector field, \(u_{i}=u_{i}(x,t) \ (i=1,2), (x,t)\in {R}^{2}\times {R}_{+}\), \(\theta (x,t)\) and \(P(x,t)\) denote the scalar temperature and pressure of the fluid, respectively. The constants \(\nu \geq 0\) and \(\kappa \geq 0\) denote the viscosity and thermal diffusivity; \(e_{2}=(0,1)\) is the unit vector in the vertical direction, and the unknown function \(\theta e_{2}\) is the buoyancy force. For the sake of simplicity, we denote \(\varLambda:=\sqrt{- \Delta }\), the square root of the negative Laplacian, and obviously \(\widehat{\varLambda f}(k)=|k|\hat{f}(k)\), where \(k=(k_{1},k_{2})\) is a tuple consisting two integers, \(|k|=\sqrt{k_{1}^{2}+k_{2}^{2}}\) and the Fourier transform of a tempered distribution \(f(x)\) on Ω is defined as

$$\begin{aligned} \hat{f}(k)=\frac{1}{(2\pi )^{2}} \int _{\varOmega }f(x)e^{-ik\cdot x}\,dx. \end{aligned}$$
(2)

More generally, we will define the fractional Laplacian \(\varLambda ^{s} f\) for \(s\in {R}\) with the Fourier series

$$\begin{aligned} \varLambda ^{s} f:=\sum_{k\in {Z}^{2}} \vert k \vert ^{s}\hat{f}(k)e^{ik\cdot x}. \end{aligned}$$
(3)

As suggested by Jiu, Miao, Wu and Zhang in [33], we classify the parameters α and β into three categories:

  1. (1)

    the subcritical case, \(\alpha +\beta >\frac{1}{2}\);

  2. (2)

    the critical case, \(\alpha +\beta =\frac{1}{2}\);

  3. (3)

    the supercritical case, \(\alpha +\beta <\frac{1}{2}\).

When \(\alpha =\beta =1\), the Boussinesq equations (1) reduce to the standard Boussinesq equation. So far, there has been a lot of literature about the mathematical theory of the standard Boussinesq equation. In the cases when \(\nu,\kappa >0\), \(\nu >0\) and \(\kappa =0\), as well as \(\kappa >0\) and \(\nu =0\), the global regularity has been studied by many authors (see, e.g., [1, 6, 8, 11, 16, 24, 26,27,28, 32, 40, 41, 45, 65, 86, 87]). However, in the case of \(\nu =\kappa =0\), we only have the local well-posedness theory (see, e.g., [12, 13, 23]), the global regularity or singularity question is a rather challenging problem in mathematical fluid mechanics. Recently, the 2D incompressible Boussinesq equations with temperature-dependence or anisotropy dissipation have attracted considerable attention. In the case of temperature-dependent dissipation, the global-in-time regularity is well-known (see, e.g., [4,5,6, 29, 30, 43, 47, 48, 58, 60]). In the case of anisotropy dissipation, many authors have proved the global well-posedness (see, e.g., [2, 3, 9, 17, 42, 44, 61, 80]). For a detailed review on interesting results, we refer the reader to [52, 57].

Our main focus of the research on the 2D Boussinesq equation has been on the global regularity issue when only fractional dissipation is present. Using the Fourier localization method, Fang, Qian, and Zhang [19] obtained the local and global well-posedness and gave some blowup criteria with the velocity or temperature. Hmidi, Keraani, and Rousset [25] proved the global well-posedness results. Jia, Peng, and Li [31] proved that the generalized 2D Boussinesq equation has a global and unique solution. Jiu, Miao, Wu, and Zhang [33, 34] aimed at the global regularity. Jiu, Wu, and Yang [35] studied the solutions in the periodic box. KC, Regmi, Tao, and Wu [38, 39] studied the global (in time) regularity problem. Miao and Xue [49] proved the global well-posedness results for rough initial data. Stefanov and Wu [56] solved the global regularity problem. Wu and Xu [63] were concerned with the global well-posedness and inviscid limits of several systems of Boussinesq equations. Using energy methods, the Fourier localization technique, and Bony’s paraproduct decomposition, Xiang and Yan [64] showed the global existence of the classical solutions. Xu [66] has proved the global existence, uniqueness and regularity of the solution. Xu and Xue [67] considered the Yudovich-type solution and gave a refined blowup criterion in the supercritical case. Yang, Jiu, and Wu [70] examined the global regularity issue and established the global well-posedness. Ye and Xu [83] established the global regularity of the smooth solutions, and in [84] they proved the global regularity of the smooth solutions.

There are many papers dealing with the fractional differential equation [10, 14, 20, 22, 50, 53,54,55, 59, 69, 71, 74,75,76,77,78,79, 81, 82, 85]. For a recent review of the fractional calculus operators, we refer the reader to [72]. In hydrodynamics, Boussinesq equation is a low-dimensional model of fluid dynamics, which plays a very important role in the study of Raleigh–Bernard convection. Boussinesq equation has many applications in modeling fluids and geophysical fluids [15, 21, 51, 70, 73].

The following is the first main result of this paper, which asserts the global well-posedness of the 2D Boussinesq equations (1).

Theorem 1

Let \(\nu >0,\kappa >0\), \(\alpha,\beta \in (\frac{2}{3},1)\). Assume that \((u_{0},\theta _{0})\in H^{1+s}({R}^{2})\times H^{1+s}({R}^{2})\), \(s\in (0,1)\). Then there exists a unique global solution \((u(t), \theta (t) )\) of Boussinesq equations (1) such that, for any \(T>0\),

$$\begin{aligned} &u(t)\in C \bigl([0,T];H^{1+s}\bigl( {R}^{2}\bigr) \bigr)\cap L^{2} \bigl([0,T];H^{1+s+ \alpha }\bigl( {R}^{2}\bigr) \bigr), \end{aligned}$$
(4)
$$\begin{aligned} &\theta (t)\in C \bigl([0,T];H^{1+s}\bigl( {R}^{2}\bigr) \bigr)\cap L^{2} \bigl([0,T];H ^{1+s+\beta }\bigl( {R}^{2} \bigr) \bigr). \end{aligned}$$
(5)

Moreover, there exist positive constants λ and C independent of t and such that

$$\begin{aligned} \bigl\Vert \nabla \theta (t) \bigr\Vert ^{2}\leq C,\qquad \bigl\Vert \nabla u(t) \bigr\Vert ^{2}\leq C. \end{aligned}$$
(6)

And in the case when \(\min \{\nu \lambda ^{2\alpha }, \kappa \lambda ^{2\beta }\}>\frac{1}{2}\), one has

$$\begin{aligned} \bigl\Vert \varLambda ^{1+s}\theta (t) \bigr\Vert ^{2} \leq C,\qquad \bigl\Vert \varLambda ^{1+s} u(t) \bigr\Vert ^{2} \leq C. \end{aligned}$$
(7)

Inspired by the work of [62, 68], the second main result of this paper asserts the existence of the solutions in Sobolev spaces \(W^{2,p}( {R}^{2})\times W^{1,p}({R}^{2})\) for some \(p>2\).

Theorem 2

Let \(\nu >0,\kappa >0\), \(\alpha \geq \frac{1}{2}+\frac{n}{4}\) (we consider \(n=2\)), \(\beta \in (0,1)\). For some \(p\geq 2\), assume that \((u_{0},\theta _{0})\in W^{2,p}( {R}^{2})\times W^{1,p}( {R}^{2})\), with \(\operatorname{div} u_{0}=0\). Then there exists a global solution \((u(t),\theta (t) )\) of Boussinesq equations (1) such that, for any \(T>0\),

$$\begin{aligned} \bigl(u(t),\theta (t) \bigr)\in C \bigl([0,T],W^{2,p}\bigl( {R}^{2}\bigr) \bigr)\times \bigl([0,T],W^{1,p}\bigl( {R}^{2}\bigr) \bigr). \end{aligned}$$
(8)

Remark 1

The same result holds for the case \(n\geq 3\). The persistence of global well-posedness should be true in Sobolev spaces, which is left to a future work.

Remark 2

In the case \(\kappa =0\), our guess is that Theorems 12 remain true.

Preliminaries

In this section, we first introduce Kato–Ponce inequality from [37] (see also [28, 36]) which is important for the proof of Theorem 1, and give a positive inequality from [46] (see also [40, 66]) and Brezis–Wainger inequality from [7] (see also [18]), which are important for the proof of Theorem 2.

Lemma 1

([37])

Suppose that \(f,g\in C_{c}^{\infty }(\varOmega )\). Let \(s> 0\) and \(1< r\leq p_{1},p_{2},q_{1},q_{2} \leq +\infty \) be such that \(\frac{1}{r}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{q_{1}}+\frac{1}{q _{2}}\) with the restriction \(p_{1},q_{2}\neq +\infty \). Then

$$\begin{aligned} \bigl\Vert \varLambda ^{s}(fg) \bigr\Vert _{L^{r}} \leq C \bigl( \bigl\Vert \varLambda ^{s}f \bigr\Vert _{L^{p_{1}}} \Vert g \Vert _{L^{p_{2}}}+ \Vert f \Vert _{L^{q_{1}}} \bigl\Vert \varLambda ^{s}g \bigr\Vert _{L^{q_{2}}} \bigr), \end{aligned}$$
(9)

where \(C>0\) is a constant.

Lemma 2

([46])

Suppose that \(u\in L^{p}({R}^{n})\) is such that \(\varLambda ^{\alpha }u \in L^{p}({R}^{n})\). Let \(0\leq m\leq 2\). For all \(p>1\), one has

$$\begin{aligned} \frac{4(p-1)}{p^{2}} \int _{{R}^{n}}\bigl(\varLambda ^{\frac{\alpha }{2}} \vert u \vert ^{ \frac{p}{2}}\bigr)^{2}\,dx \leq \int _{{R}^{n}}\varLambda ^{\alpha }u\cdot u \vert u \vert ^{p-2}\,dx. \end{aligned}$$
(10)

Observe that, if \(\alpha =2\), integrating (10) by parts, we obtain

$$\begin{aligned} \int _{{R}^{n}}\bigl(\varLambda \vert u \vert ^{\frac{p}{2}} \bigr)^{2}\,dx = \int _{{R}^{n}}\varLambda ^{2} u\cdot u \vert u \vert ^{p-2}\,dx. \end{aligned}$$
(11)

Lemma 2 is well-known in the theory of sub-Markovian operators, its statement and the proof are given in [46].

Lemma 3

([7])

Suppose that \(u\in L^{2}({R}^{2})\cap W^{1,p}({R}^{2})\). For all \(p>1\), one has

$$\begin{aligned} \Vert u \Vert _{L^{\infty }}\leq C(1+ \Vert \nabla u \Vert _{L^{2}} \bigl(1+\log ^{+}\bigl( \Vert \nabla u \Vert _{L^{p}}\bigr) \bigr)^{\frac{1}{2}}+C \Vert u \Vert _{L^{2}}, \end{aligned}$$
(12)

where \(C>0\) is a constant.

Proof of Theorem 1

The goal of this section is to prove Theorem 1. The proof is divided into two main parts showing global existence and uniqueness.

Global existence

The proof of global existence is based on several steps of careful energy estimates. First, we start with estimates of \(\|u(t)\|\) and \(\|\theta (t)\|\).

Lemma 4

Under the assumptions of Theorem 1, one has

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert \in C \bigl(0,+\infty;L^{2}(\varOmega ) \bigr)\cap L^{2} \bigl(0,+ \infty;H^{\alpha }(\varOmega ) \bigr), \end{aligned}$$
(13)
$$\begin{aligned} & \bigl\Vert \theta (t) \bigr\Vert \in C \bigl(0,+\infty;L^{2}( \varOmega ) \bigr)\cap L^{2} \bigl(0,+\infty;H^{\beta }(\varOmega ) \bigr). \end{aligned}$$
(14)

Moreover, there exist positive constant λ independents of t and such that

$$\begin{aligned} \bigl\Vert \theta (t) \bigr\Vert ^{2}\leq \Vert \theta _{0} \Vert ^{2}e^{-2\kappa \lambda ^{2\beta }t}, \end{aligned}$$
(15)

as well as

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert ^{2}\leq e^{-\nu \lambda ^{2\alpha }t} \Vert u_{0} \Vert ^{2}+\frac{1}{ \nu \lambda ^{2\alpha }} \biggl\vert \frac{e^{-\nu \lambda ^{2\alpha }t}-e ^{-\kappa \lambda ^{2\beta }t}}{\nu \lambda ^{2\alpha }-\kappa \lambda ^{2\beta }} \biggr\vert \Vert \theta _{0} \Vert ^{2}, \\ &\nu \lambda ^{2\alpha }\neq \kappa \lambda ^{2\beta }, \end{aligned}$$
(16)
$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert ^{2}\leq e^{-\nu \lambda ^{2\alpha }t} \Vert u_{0} \Vert ^{2}+\frac{t}{ \nu \lambda ^{2\alpha }} e^{-\nu \lambda ^{2\alpha }t} \Vert \theta _{0} \Vert ^{2}, \\ & \nu \lambda ^{2\alpha }=\kappa \lambda ^{2\beta }. \end{aligned}$$
(17)

Proof

Taking \(L^{2}\)-inner product of (1)3 with θ, and integrating by parts, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \theta \Vert ^{2}+\kappa \bigl\Vert \varLambda ^{\beta } \theta \bigr\Vert ^{2} =0. \end{aligned}$$
(18)

Since \(\theta (x,t)|_{\partial \varOmega }=0\), using Poincaré inequality, we find

$$\begin{aligned} \frac{d}{dt} \Vert \theta \Vert ^{2}+2\kappa \lambda ^{2\beta } \Vert \theta \Vert ^{2}=0, \end{aligned}$$
(19)

where λ is the first eigenvalue of Λ. Then, we can obtain that, for all \(t\in [0,+\infty )\),

$$\begin{aligned} \bigl\Vert \theta (t) \bigr\Vert ^{2}\leq \Vert \theta _{0} \Vert e^{-2\kappa \lambda ^{2\beta }t}. \end{aligned}$$
(20)

Integrating (18) in time gives

$$\begin{aligned} \bigl\Vert \theta (t) \bigr\Vert ^{2}+2\kappa \int _{0}^{t} \bigl\Vert \varLambda ^{\beta }\theta ( \tau ) \bigr\Vert ^{2} \,d\tau \leq \Vert \theta _{0} \Vert ^{2}. \end{aligned}$$
(21)

Similarly, we can also deduce a uniform \(L^{p}\) estimate of θ, for all \(p\in [2,+\infty )\),

$$\begin{aligned} \bigl\Vert \theta (t) \bigr\Vert _{L^{p}}\leq e^{-\frac{\kappa \lambda ^{2\beta }t}{p}} \Vert \theta _{0} \Vert _{L^{p}}. \end{aligned}$$
(22)

Multiplying (1)1 by u and integrating the resulting equation by parts, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u \Vert ^{2}+\nu \bigl\Vert \varLambda ^{\alpha }u \bigr\Vert ^{2} & \leq \int _{\varOmega }\theta e_{2}\cdot u\,dx \\ &\leq \int _{\varOmega } \bigl\vert \varLambda ^{-\alpha }\theta \bigr\vert \cdot \bigl\vert \varLambda ^{\alpha }u \bigr\vert \,dx \\ &\leq \frac{1}{2\nu } \bigl\Vert \varLambda ^{-\alpha }\theta \bigr\Vert ^{2}+ \frac{\nu }{2} \bigl\Vert \varLambda ^{\alpha }u \bigr\Vert ^{2}. \end{aligned}$$
(23)

Hence,

$$\begin{aligned} \frac{d}{dt} \Vert u \Vert ^{2}+\nu \bigl\Vert \varLambda ^{\alpha }u \bigr\Vert ^{2} \leq \frac{1}{ \nu } \bigl\Vert \varLambda ^{-\alpha }\theta \bigr\Vert ^{2}. \end{aligned}$$
(24)

By Poincaré inequality, we have

$$\begin{aligned} \frac{d}{dt} \Vert u \Vert ^{2}+\nu \lambda ^{2\alpha } \Vert u \Vert ^{2} \leq \frac{1}{ \nu \lambda ^{2\alpha }} \Vert \theta \Vert ^{2}. \end{aligned}$$
(25)

Integrating in time and using (20), we have, in the case when \(\nu \lambda ^{2\alpha }\neq \kappa \lambda ^{2\beta }\),

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{2}\leq e^{-\nu \lambda ^{2\alpha }t} \Vert u_{0} \Vert ^{2}+\frac{1}{ \nu \lambda ^{2\alpha }} \biggl\vert \frac{e^{-\nu \lambda ^{2\alpha }t}-e ^{-\kappa \lambda ^{2\beta }t}}{\nu \lambda ^{2\alpha }-\kappa \lambda ^{2\beta }} \biggr\vert \Vert \theta _{0} \Vert ^{2} , \end{aligned}$$
(26)

and in the case when \(\nu \lambda ^{2\alpha }=\kappa \lambda ^{2\beta }\),

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{2}\leq e^{-\nu \lambda ^{2\alpha }t} \Vert u_{0} \Vert ^{2}+\frac{t}{ \nu \lambda ^{2\alpha }} e^{-\nu \lambda ^{2\alpha }t} \Vert \theta _{0} \Vert ^{2}. \end{aligned}$$
(27)

After integration (24) in time and by (20), we obtain

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert ^{2}+\nu \int _{0}^{t} \bigl\Vert \varLambda ^{\alpha }u(\tau ) \bigr\Vert ^{2} \,d \tau \leq \Vert u_{0} \Vert ^{2}+\frac{1}{2\nu \kappa \lambda ^{2(\alpha + \beta )}} \Vert \theta _{0} \Vert ^{2}, \end{aligned}$$
(28)

completing the proof. □

In the next lemma, we shall obtain estimates of \(\|\nabla u(t)\|\) and \(\|\nabla \theta (t)\|\).

Lemma 5

Under the assumptions of Theorem 1, one has

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert \in C \bigl(0,+\infty;H^{1}( \varOmega ) \bigr)\cap L^{2} \bigl(0,+ \infty;H^{1+\alpha }(\varOmega ) \bigr), \end{aligned}$$
(29)
$$\begin{aligned} & \bigl\Vert \theta (t) \bigr\Vert \in C \bigl(0,+\infty;H^{1}( \varOmega ) \bigr)\cap L^{2} \bigl(0,+\infty;H^{1+\beta }(\varOmega ) \bigr). \end{aligned}$$
(30)

Moreover, there exist positive constants λ and C independent of t and such that

$$\begin{aligned} \bigl\Vert \nabla u(t) \bigr\Vert \leq C, \qquad\bigl\Vert \nabla \theta (t) \bigr\Vert \leq C. \end{aligned}$$
(31)

Proof

In order to complete the proof, we need to use vorticity formulation. Taking the curl of (1)1, we have

$$\begin{aligned} \omega _{t}+\nu \varLambda ^{2\alpha }\omega +u\cdot \nabla \omega = \theta _{x_{1}}, \end{aligned}$$
(32)

where \(\omega =\partial _{x_{1}}u_{2}-\partial _{x_{2}}u_{1}\), with the Dirichlet boundary condition

$$\begin{aligned} \omega =0, \quad\text{on } \partial \varOmega. \end{aligned}$$

Taking \(L^{2}\)-inner product of (32) with ω, we obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \omega \Vert ^{2}+\nu \bigl\Vert \varLambda ^{\alpha }\omega \bigr\Vert ^{2} &= \int _{\varOmega }\theta _{x_{1}}\cdot \omega \,dx \\ &\leq \biggl\vert \int _{\varOmega }\varLambda ^{1-\alpha }\theta \cdot \varLambda ^{\alpha }\omega \,dx \biggr\vert \\ &\leq \frac{1}{2\nu } \bigl\Vert \varLambda ^{1-\alpha }\theta \bigr\Vert ^{2}+\frac{ \nu }{2} \bigl\Vert \varLambda ^{\alpha } \omega \bigr\Vert ^{2}, \end{aligned}$$
(33)

from which it follows that

$$\begin{aligned} \frac{d}{dt} \Vert \omega \Vert ^{2}+\nu \bigl\Vert \varLambda ^{\alpha }\omega \bigr\Vert ^{2} \leq \frac{1}{\nu } \bigl\Vert \varLambda ^{1-\alpha }\theta \bigr\Vert ^{2}. \end{aligned}$$
(34)

Then Poincaré inequality implies

$$\begin{aligned} \frac{d}{dt} \Vert \omega \Vert ^{2}+\nu \lambda ^{2\alpha } \Vert \omega \Vert ^{2} \leq \frac{1}{\nu } \bigl\Vert \varLambda ^{1-\alpha }\theta \bigr\Vert ^{2}. \end{aligned}$$
(35)

Since \(\alpha,\beta \in (\frac{2}{3},1)\), we know that \(1-\alpha < \beta \), and so, using the interpolation inequality and by (21), we have

$$\begin{aligned} \int _{0}^{t} \bigl\Vert \varLambda ^{1-\alpha }\theta (\tau ) \bigr\Vert ^{2}\,d\tau \leq \int _{0}^{t} \bigl\Vert \varLambda ^{\beta }\theta (\tau ) \bigr\Vert ^{2}\,d\tau \leq C. \end{aligned}$$
(36)

Applying a variant of the uniform Gronwall lemma, and by the Biot–Savart law and (36), we have a uniform estimate \(\|u(t)\|_{H ^{1}}\) for all \(t\in [0,+\infty )\). Furthermore, integrating (34) in time, we can get, for all \(t\in [0,+\infty )\),

$$\begin{aligned} \bigl\Vert \omega (t) \bigr\Vert ^{2}+\nu \int _{0}^{t} \bigl\Vert \varLambda ^{\alpha }\omega (\tau ) \bigr\Vert ^{2}\,d\tau \leq \Vert \omega _{0} \Vert ^{2}+\frac{1}{\nu } \int _{0}^{t} \bigl\Vert \varLambda ^{1-\alpha }\theta (\tau ) \bigr\Vert ^{2}\,d\tau. \end{aligned}$$
(37)

As an immediate consequence, and by Sobolev embedding theorem, we have a uniform \(L^{p}\) estimate for u, that is, for all \(1< p<+\infty \),

$$\begin{aligned} \Vert u \Vert _{L^{p}}\leq C(p) \end{aligned}$$
(38)

and

$$\begin{aligned} \int _{0}^{t} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert \,d\tau \leq C\bigl( \Vert \nabla u_{0} \Vert , \Vert \theta _{0} \Vert \bigr), \end{aligned}$$
(39)

where the constant \(C(p)>0\) only depends on p and \(C(\|\nabla u_{0} \|,\|\theta _{0}\|)\) only depends the initial data \(\|\nabla u_{0}\|\) and \(\|\theta _{0}\|\).

Taking \(L^{2}\)-inner product of (1)3 with Δθ, we obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \nabla \theta \Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+ \beta }\theta \bigr\Vert ^{2} &={-} \int _{\varOmega }(u\cdot \nabla \theta )\cdot \Delta \theta \,dx \\ &\leq \int _{\varOmega } \bigl\vert \varLambda ^{1-\beta }(u\cdot \nabla \theta ) \bigr\vert \bigl\vert \varLambda ^{1+\beta }\theta \bigr\vert \,dx. \end{aligned}$$
(40)

Since u is divergence-free, \(u\cdot \nabla \theta =\nabla \cdot (u \theta )\), and so, using Cauchy–Schwarz inequality, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert \nabla \theta \Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+ \beta }\theta \bigr\Vert ^{2} &\leq \int _{\varOmega } \bigl\vert \varLambda ^{2-\beta }(u \theta ) \bigr\vert \bigl\vert \varLambda ^{1+\beta }\theta \bigr\vert \,dx \\ &\leq \frac{1}{\kappa } \bigl\Vert \varLambda ^{2-\beta }(u\theta ) \bigr\Vert ^{2}+\frac{ \kappa }{4} \bigl\Vert \varLambda ^{1+\beta }\theta \bigr\Vert ^{2}. \end{aligned}$$
(41)

Using Lemma 1, and by (22) and (38), we have

$$\begin{aligned} \bigl\Vert \varLambda ^{2-\beta }(u\theta ) \bigr\Vert ^{2} & \leq C \bigl\Vert \varLambda ^{2-\beta }u \bigr\Vert ^{2}_{L^{4}} \Vert \theta \Vert ^{2}_{L^{4}}+C \Vert u \Vert ^{2}_{L^{6}} \bigl\Vert \varLambda ^{2- \beta }\theta \bigr\Vert ^{2}_{L^{3}} \\ &\leq C \bigl\Vert \varLambda ^{2-\beta }u \bigr\Vert ^{2}_{L^{4}}+C \bigl\Vert \varLambda ^{2-\beta } \theta \bigr\Vert ^{2}_{L^{3}}, \end{aligned}$$
(42)

so by Sobolev embedding theorem, and applying Gagliardo–Nirenberg and Young inequalities, we can obtain

$$\begin{aligned} & \bigl\Vert \varLambda ^{2-\beta }(u\theta ) \bigr\Vert ^{2} \\ &\quad\leq C \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2}+C \bigl\Vert \varLambda ^{-\alpha } \theta \bigr\Vert ^{2(3\alpha +3\beta -4)/3(1+2\alpha )} \bigl\Vert \varLambda ^{1+\alpha }\theta \bigr\Vert ^{2(7+3\alpha -3\beta )/3(1+2\alpha )} \\ &\quad\leq C \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2}+a_{1}C \bigl\Vert \varLambda ^{-\alpha } \theta \bigr\Vert ^{2}+\frac{\kappa }{4} \bigl\Vert \varLambda ^{1+\alpha }\theta \bigr\Vert ^{2}, \end{aligned}$$
(43)

where \(a_{1}=\frac{3\alpha +3\beta -4}{3(1+2\alpha )} (\frac{3 \kappa (1+2\alpha )}{4(7+3\alpha -3\beta )} )^{(7+3\alpha -3\beta )/(4-3\alpha -3\beta )}\). Inserting (43) into (41), we can obtain that

$$\begin{aligned} \frac{d}{dt} \Vert \nabla \theta \Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+\beta }\theta \bigr\Vert ^{2} \leq C \bigl( \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{-\alpha } \theta \bigr\Vert ^{2} \bigr). \end{aligned}$$
(44)

Then Poincaré inequality implies

$$\begin{aligned} \frac{d}{dt} \Vert \nabla \theta \Vert ^{2}+\kappa \lambda ^{2\beta } \Vert \nabla \theta \Vert ^{2} \leq C \biggl( \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2}+ \frac{1}{ \lambda ^{2\alpha }} \Vert \theta \Vert ^{2} \biggr). \end{aligned}$$
(45)

By (20) and (39), we know that

$$\begin{aligned} \int _{0}^{t} \bigl( \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2}+ \Vert \theta \Vert ^{2} \bigr)\,d \tau \leq C. \end{aligned}$$
(46)

Applying a variant of the uniform Gronwall lemma again and (46), we have a uniform estimate of \(\|\nabla \theta (t)\|\) for all \(t\in [0,+ \infty )\). Integrating over \([0,t]\), we obtain, for all \(t\in [0,+ \infty )\),

$$\begin{aligned} \bigl\Vert \nabla \theta (t) \bigr\Vert ^{2}+\kappa \int _{0}^{t} \bigl\Vert \varLambda ^{1+\beta } \theta (\tau ) \bigr\Vert ^{2}\,d\tau \leq \Vert \nabla \theta _{0} \Vert ^{2}+ C, \end{aligned}$$
(47)

where C only depends on p and the initial data. □

Now let us focus on the persistence in \(H^{1+s}({R}^{2})\times H^{1+s}( {R}^{2})\), \(s\in (0,1)\).

Lemma 6

Under the assumptions of Theorem 1, one has

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert \in C \bigl(0,+\infty;H^{1+s}( \varOmega ) \bigr)\cap L^{2} \bigl(0,+ \infty;H^{1+s+\alpha }(\varOmega ) \bigr), \end{aligned}$$
(48)
$$\begin{aligned} & \bigl\Vert \theta (t) \bigr\Vert \in C \bigl(0,+\infty;H^{1+s}( \varOmega ) \bigr)\cap L^{2} \bigl(0,+\infty;H^{1+s+\beta }(\varOmega ) \bigr). \end{aligned}$$
(49)

Moreover, in the case when \(\min \{\nu \lambda ^{2\alpha }, \kappa \lambda ^{2\beta }\}>\frac{1}{2}\), there exist positive constants λ and C independent of t, and it holds that

$$\begin{aligned} \bigl\Vert \varLambda ^{1+s}\theta (t) \bigr\Vert ^{2} \leq C,\qquad \bigl\Vert \varLambda ^{1+s} u(t) \bigr\Vert ^{2} \leq C. \end{aligned}$$
(50)

Proof

Taking \(L^{2}\)-inner product of (1)3 with \(\varLambda ^{2+2s}\theta \), we obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert \varLambda ^{1+s} \theta \bigr\Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+s+\beta } \theta \bigr\Vert ^{2} =- \int _{\varOmega }(u\cdot \nabla \theta )\cdot \varLambda ^{2+2s}\theta \,dx. \end{aligned}$$
(51)

Since u is divergence-free, \(u\cdot \nabla \theta =\nabla \cdot (u \theta )\), using Lemma 1 and (22) with (38), we obtain

$$\begin{aligned} &{-} \int _{\varOmega }(u\cdot \nabla \theta )\cdot \varLambda ^{2+2s}\theta \,dx \\ &\quad\leq \biggl\vert \int _{\varOmega }\varLambda ^{2+s-\beta }(u\theta )\cdot \varLambda ^{1+s+\beta }\theta \,dx \biggr\vert \\ &\quad\leq \bigl\Vert \varLambda ^{2+s-\beta }(u\theta ) \bigr\Vert \bigl\Vert \varLambda ^{1+s+\beta }\theta \bigr\Vert \\ &\quad\leq C \bigl( \bigl\Vert \varLambda ^{2+s-\beta }u \bigr\Vert _{L^{3}} \Vert \theta \Vert _{L^{6}} + \Vert u \Vert _{L^{6}} \bigl\Vert \varLambda ^{2+s-\beta }\theta \bigr\Vert _{L^{3}} \bigr) \bigl\Vert \varLambda ^{1+s+\beta }\theta \bigr\Vert \\ &\quad\leq \frac{C}{\kappa } \bigl( \bigl\Vert \varLambda ^{2+s-\beta }u \bigr\Vert ^{2}_{L^{3}} + \bigl\Vert \varLambda ^{2+s-\beta } \theta \bigr\Vert ^{2}_{L^{3}} \bigr)+\frac{\kappa }{4} \bigl\Vert \varLambda ^{1+s+\beta }\theta \bigr\Vert ^{2}. \end{aligned}$$
(52)

Applying Gagliardo–Nirenberg and Young inequalities, we can get

$$\begin{aligned} \bigl\Vert \varLambda ^{2+s-\beta }u \bigr\Vert ^{2}_{L^{3}} &\leq C \bigl\Vert \varLambda ^{-\beta }u \bigr\Vert ^{2(3\alpha +3\beta -4)/3(1+s+\alpha +\beta )} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2(7+3s)/3(1+s+\alpha +\beta )} \\ &\leq a_{2}C \bigl\Vert \varLambda ^{-\beta }u \bigr\Vert ^{2}+\frac{\nu }{4} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2} \end{aligned}$$
(53)

and

$$\begin{aligned} \bigl\Vert \varLambda ^{2+s-\beta }\theta \bigr\Vert ^{2}_{L^{3}} &\leq C \bigl\Vert \varLambda ^{-\beta }\theta \bigr\Vert ^{4(3\beta -2)/3(1+s+2\beta )} \bigl\Vert \varLambda ^{1+s+\beta }\theta \bigr\Vert ^{2(7+3s)/3(1+s+2\beta )} \\ &\leq a_{3}C \bigl\Vert \varLambda ^{-\beta }\theta \bigr\Vert ^{2}+\frac{\kappa }{4} \bigl\Vert \varLambda ^{1+s+\beta } \theta \bigr\Vert ^{2}, \end{aligned}$$
(54)

where \(a_{2}=\frac{3\alpha +3\beta -4}{3(1+s+\alpha +\beta )} (\frac{3 \nu (1+s+\alpha +\beta )}{4(7+3s)} )^{(7+3s)/(4-3\alpha -3\beta )}\) and \(a_{3}=\frac{2(3\beta -2)}{3(1+s+2\beta )} (\frac{3\kappa (1+s+2 \beta )}{4(7+3s)} )^{(7+3s)/2(2-3\beta )}\). Inserting (52)–(54) into (51), we arrive at

$$\begin{aligned} &\frac{d}{dt} \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+s+ \beta }\theta \bigr\Vert ^{2} \\ &\quad\leq \frac{\nu }{2} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2}+C \bigl\Vert \varLambda ^{- \beta }u \bigr\Vert ^{2}+C \bigl\Vert \varLambda ^{-\beta }\theta \bigr\Vert ^{2}. \end{aligned}$$
(55)

Applying the operator \(\varLambda ^{1+s}\) to (1)1, and taking the scalar product of both sides with \(\varLambda ^{1+s} u\), and then integrating the result by parts, we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl\Vert \varLambda ^{1+s} u \bigr\Vert ^{2}+\nu \bigl\Vert \varLambda ^{1+s+\alpha } u \bigr\Vert ^{2} \\ &\quad = - \int _{\varOmega }\varLambda ^{1+s}(u_{j}\partial _{j} u_{k})\varLambda ^{1+s} u_{k} \,dx+ \int _{\varOmega }\varLambda ^{1+s}(\theta e_{2}) \varLambda ^{1+s} u\,dx. \end{aligned}$$
(56)

Using Lemma 1 and applying fractional embedding theorems together with Young inequality again, we obtain

$$\begin{aligned} &{-} \int _{\varOmega }\varLambda ^{1+s}(u_{j}\partial _{j} u_{k})\varLambda ^{1+s} u _{k} \,dx \\ &\quad\leq \biggl\vert \int _{\varOmega }\varLambda ^{1+s-\alpha }(u_{j}\partial _{j} u _{k})\varLambda ^{1+s+\alpha } u_{k} \,dx \biggr\vert \\ &\quad\leq C \bigl( \bigl\Vert \varLambda ^{1+s-\alpha }u \bigr\Vert _{L^{3}} \Vert \nabla u \Vert _{L^{6}}+ \Vert u \Vert _{L^{6}} \bigl\Vert \varLambda ^{2+s-\alpha }u \bigr\Vert _{L^{3}} \bigr) \bigl\Vert \varLambda ^{1+s+\alpha } u \bigr\Vert \\ &\quad\leq \frac{\nu }{4} \bigl\Vert \varLambda ^{1+s+\alpha } u \bigr\Vert ^{2}+\frac{C}{\nu } \bigl( \bigl\Vert \varLambda ^{1+s-\alpha }u \bigr\Vert ^{2}_{L^{4}} \Vert \nabla u \Vert ^{2}_{L^{4}}+ \bigl\Vert \varLambda ^{2+s-\alpha }u \bigr\Vert ^{2}_{L^{3}} \bigr). \end{aligned}$$
(57)

Applying Gagliardo–Nirenberg and Young inequalities, we can get

$$\begin{aligned} \bigl\Vert \varLambda ^{1+s-\alpha }u \bigr\Vert ^{2}_{L^{3}} \Vert \nabla u \Vert ^{2}_{L^{6}}\leq {}& \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{4(3\alpha -2)/3(1+s+2\alpha )} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2(7+3s)/3(1+s+2\alpha )} \\ &{}\times \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{2(1+3\alpha )/3(1+2\alpha )} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{2(2+3\alpha )/3(1+2\alpha )} \\ \leq{}& a_{4}C \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{b_{2}} +\frac{\nu }{8} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2} \end{aligned}$$
(58)

and

$$\begin{aligned} \bigl\Vert \varLambda ^{2+s-\alpha }u \bigr\Vert ^{2}_{L^{3}} &\leq C \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{4(3\alpha -2)/3(1+s+2\alpha )} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2(7+3s)/3(1+s+2 \alpha )} \\ &\leq a_{5}C \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{2}+\frac{\nu }{8} \bigl\Vert \varLambda ^{1+s+\alpha }u \bigr\Vert ^{2}, \end{aligned}$$
(59)

where \(a_{4}=a_{5}=\frac{2(3\alpha -2)}{3(1+s+2\alpha )} (\frac{3 \nu (1+s+2\alpha )}{8(7+3s)} )^{(7+3s)/2(2-3\alpha )}\), \(b_{1}=2+\frac{(1+3 \alpha )(1+s+2\alpha )}{(1+2\alpha )(3\alpha -2)}\), and \(b_{2}=\frac{(2+3 \alpha )(1+s+2\alpha )}{(1+2\alpha )(3\alpha -2)}\). Using Hölder and Cauchy–Schwarz inequalities, we can get

$$\begin{aligned} \int _{\varOmega }\varLambda ^{1+s}(\theta e_{2}) \varLambda ^{1+s} u\,dx \leq \frac{1}{2} \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+\frac{1}{2} \bigl\Vert \varLambda ^{1+s}u \bigr\Vert ^{2}. \end{aligned}$$
(60)

Inserting (58)–(60) into (56), we arrive at

$$\begin{aligned} &\frac{d}{dt} \bigl\Vert \varLambda ^{1+s} u \bigr\Vert ^{2}+\frac{3\nu }{2} \bigl\Vert \varLambda ^{1+s+\alpha } u \bigr\Vert ^{2} \\ &\quad\leq \frac{1}{2}\bigl( \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}u \bigr\Vert ^{2}\bigr)+C\bigl( \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{b_{2}}\bigr). \end{aligned}$$
(61)

Summing up (55) and (61), we obtain that

$$\begin{aligned} &\frac{d}{dt}\bigl( \bigl\Vert \varLambda ^{1+s} u \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}\bigr)+ \nu \bigl\Vert \varLambda ^{1+s+\alpha } u \bigr\Vert ^{2}+\kappa \bigl\Vert \varLambda ^{1+s+\beta } \theta \bigr\Vert ^{2} \\ &\quad\leq \frac{1}{2}\bigl( \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}u \bigr\Vert ^{2}\bigr) +C\bigl( \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{-\beta }u \bigr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert \varLambda ^{-\beta }\theta \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{-\alpha }u \bigr\Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{b_{2}}\bigr). \end{aligned}$$
(62)

Then Poincaré inequality implies

$$\begin{aligned} &\frac{d}{dt}\bigl( \bigl\Vert \varLambda ^{1+s} u \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}\bigr)+ \nu \lambda ^{2\alpha } \bigl\Vert \varLambda ^{1+s} u \bigr\Vert ^{2}+\kappa \lambda ^{2\beta } \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2} \\ &\quad\leq \frac{1}{2}\bigl( \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}u \bigr\Vert ^{2}\bigr) \\ &\qquad{}+C\biggl(\frac{1}{\lambda ^{2\alpha }} \Vert u \Vert ^{2}+ \frac{1}{\lambda ^{2\beta }} \Vert u \Vert ^{2}+\frac{1}{\lambda ^{2\beta }} \Vert \theta \Vert ^{2} +\frac{1}{\lambda ^{b_{1}}} \Vert u \Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+ \alpha }u \bigr\Vert ^{b_{2}}\biggr). \end{aligned}$$
(63)

Hence

$$\begin{aligned} \frac{d}{dt}X(t)+ \biggl(c_{1}-\frac{1}{2} \biggr)X(t) \leq C\bigl( \Vert u \Vert ^{2}+ \Vert \theta \Vert ^{2} + \Vert u \Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+\alpha }u \bigr\Vert ^{b_{2}}\bigr), \end{aligned}$$
(64)

where \(X(t)=\|\varLambda ^{1+s} u\|^{2}+\|\varLambda ^{1+s}\theta \|^{2}\) and \(c_{1}=\min \{\nu \lambda ^{2\alpha }, \kappa \lambda ^{2\beta }\}\). By (20), (26), (27), and (39), we know that

$$\begin{aligned} \int _{0}^{t}\bigl( \Vert u \Vert ^{2}+ \Vert \theta \Vert ^{2} + \Vert u \Vert ^{b_{1}} \bigl\Vert \varLambda ^{1+ \alpha }u \bigr\Vert ^{b_{2}}\bigr)\,d\tau \leq C. \end{aligned}$$
(65)

Applying a variant of the uniform Gronwall lemma again and (65), in the case \((c_{1}-\frac{1}{2} )>0\), we have uniform estimates of \(\|\varLambda ^{1+s} u\|^{2}\) and \(\|\varLambda ^{1+s}\theta \|^{2}\), for all \(t\in [0,+\infty )\). Integrating (62) over \([0,t]\), we have

$$\begin{aligned} & \bigl\Vert \varLambda ^{1+s} u(t) \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}\theta (t) \bigr\Vert ^{2}+\nu \int _{0}^{t} \bigl\Vert \varLambda ^{1+s+\alpha } u(\tau ) \bigr\Vert ^{2}\,d\tau \\ &\qquad{}+\kappa \int _{0}^{t} \bigl\Vert \varLambda ^{1+s+\beta }\theta (\tau ) \bigr\Vert ^{2}\,d \tau \\ &\quad\leq \bigl\Vert \varLambda ^{1+s} u_{0} \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}\theta _{0} \bigr\Vert ^{2}+C+ \frac{1}{2} \int _{0}^{t}\bigl( \bigl\Vert \varLambda ^{1+s}\theta \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}u \bigr\Vert ^{2}\bigr)\,d\tau. \end{aligned}$$
(66)

Using Gronwall inequality, we find that, for all \(t\in [0,T]\),

$$\begin{aligned} & \bigl\Vert \varLambda ^{1+s} u(t) \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s}\theta (t) \bigr\Vert ^{2}+\nu \int _{0}^{t} \bigl\Vert \varLambda ^{1+s+\alpha } u(\tau ) \bigr\Vert ^{2}\,d\tau \\ &\qquad{}+\kappa \int _{0}^{t} \bigl\Vert \varLambda ^{1+s+\beta }\theta (\tau ) \bigr\Vert ^{2}\,d \tau \\ &\quad\leq e^{\frac{t}{2}} \bigl( \bigl\Vert \varLambda ^{1+s} u_{0} \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s} \theta _{0} \bigr\Vert ^{2}+C \bigr) \\ &\quad\leq e^{\frac{T}{2}} \bigl( \bigl\Vert \varLambda ^{1+s} u_{0} \bigr\Vert ^{2}+ \bigl\Vert \varLambda ^{1+s} \theta _{0} \bigr\Vert ^{2}+C \bigr) \\ &\quad\leq C, \end{aligned}$$
(67)

where \(C=C(\|\varLambda ^{1+s} u_{0}\|,\|\varLambda ^{1+s} \theta _{0}\|, \nu,\kappa,s,T)\) is a positive constant. □

Uniqueness

With the global regularity established in Lemmas 46, we are able to prove the uniqueness of the solution.

Lemma 7

Under the assumptions of Theorem 1, the solution of Boussinesq equations (1) is unique.

Proof

For any fixed \(T>0\), suppose there are two solutions \((u_{1},\theta _{1},P_{1})\) and \((u_{2},\theta _{2},P_{2})\) to Boussinesq equations (1). Setting \(\tilde{u}=u_{1}-u_{2}\), \(\tilde{\theta }= \theta _{1}-\theta _{2}\) and \(\tilde{P}=P_{1}-P_{2}\), we get that \((\tilde{u},\tilde{\theta },\tilde{P})\) satisfies

$$\begin{aligned} &\tilde{u}_{t}+ \nu \varLambda ^{2\alpha } \tilde{u}+u_{1} \cdot \nabla \tilde{u}+\tilde{u}\cdot \nabla u_{2}+\nabla \tilde{P}= \tilde{\theta } e_{2},\quad e_{2}=(0,1), \end{aligned}$$
(68)
$$\begin{aligned} &\operatorname{div} \tilde{u}=0, \end{aligned}$$
(69)
$$\begin{aligned} &\tilde{\theta }_{t}+\kappa \varLambda ^{2\beta } \tilde{\theta }+u_{1} \cdot \nabla \tilde{\theta }+\tilde{u}\cdot \nabla \theta _{2} =0, \end{aligned}$$
(70)
$$\begin{aligned} &\tilde{u}(x,0)=0,\qquad \tilde{\theta }(x,0)=0. \end{aligned}$$
(71)

Taking the \(L^{2}\)-inner product of (68) with ũ and (70) with θ̃, respectively, we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt}\bigl( \Vert \tilde{u} \Vert ^{2}+ \Vert \tilde{\theta } \Vert ^{2}\bigr)+ \nu \Vert \nabla \tilde{u} \Vert ^{2}+\kappa \Vert \nabla \tilde{\theta } \Vert ^{2} \\ &\quad \leq \int _{\varOmega }\tilde{\theta }e_{2}\cdot \tilde{u}\,dx- \int _{ \varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx- \int _{\varOmega } \tilde{u}\cdot \nabla \theta _{2}\tilde{ \theta }\,dx. \end{aligned}$$
(72)

Using Hölder and Cauchy–Schwarz inequalities, a standard calculation gives us the following:

$$\begin{aligned} &\int _{\varOmega }\tilde{\theta }e_{2}\cdot \tilde{u}\,dx \leq \frac{1}{2} \Vert \tilde{\theta } \Vert ^{2}+ \frac{1}{2} \Vert \tilde{u} \Vert ^{2}, \end{aligned}$$
(73)
$$\begin{aligned} &{-} \int _{\varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx\leq \biggl\vert - \int _{\varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx \biggr\vert \\ &\phantom{{-} \int _{\varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx}\leq \Vert \nabla u_{2} \Vert \Vert \tilde{u} \Vert ^{2}_{L^{4}} \\ &\phantom{{-} \int _{\varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx}\leq C \Vert \nabla u_{2} \Vert \Vert \tilde{u} \Vert \Vert \nabla \tilde{u} \Vert \\ &\phantom{{-} \int _{\varOmega }\tilde{u}\cdot \nabla u_{2}\cdot \tilde{u} \,dx}\leq C \Vert \nabla u_{2} \Vert ^{2} \Vert \tilde{u} \Vert ^{2}+\frac{\nu }{4} \Vert \nabla \tilde{u} \Vert , \end{aligned}$$
(74)

and

$$\begin{aligned} &{-} \int _{\varOmega }\tilde{u}\cdot \nabla \theta _{2}\cdot \tilde{\theta }\,dx \\ &\quad\leq \biggl\vert - \int _{\varOmega }\tilde{u}\cdot \nabla \theta _{2}\cdot \tilde{\theta }\,dx \biggr\vert \\ &\quad\leq \Vert \nabla \theta _{2} \Vert \Vert \tilde{\theta } \Vert _{L^{4}} \Vert \tilde{u} \Vert _{L^{4}} \\ &\quad\leq C \Vert \nabla \theta _{2} \Vert \Vert \tilde{\theta } \Vert ^{1/2} \Vert \nabla \tilde{\theta } \Vert ^{1/2} \Vert \tilde{u} \Vert ^{1/2} \Vert \nabla \tilde{u} \Vert ^{1/2} \\ &\quad\leq C\bigl( \Vert \nabla \theta _{2} \Vert \Vert \tilde{ \theta } \Vert \Vert \nabla \tilde{\theta } \Vert + \Vert \nabla \theta _{2} \Vert \Vert \tilde{u} \Vert \Vert \nabla \tilde{u} \Vert \bigr) \\ &\quad\leq C \Vert \nabla \theta _{2} \Vert ^{2} \Vert \tilde{\theta } \Vert ^{2}+ \frac{\nu }{4} \Vert \nabla \tilde{ \theta } \Vert +C \Vert \nabla \theta _{2} \Vert ^{2} \Vert \tilde{u} \Vert ^{2} +\frac{\kappa }{2} \Vert \nabla \tilde{u} \Vert ^{2}. \end{aligned}$$
(75)

Inserting (73)–(75) into (72), we obtain

$$\begin{aligned} &\frac{d}{dt}\bigl( \Vert \tilde{u} \Vert ^{2}+ \Vert \tilde{\theta } \Vert ^{2}\bigr)+\nu \Vert \nabla \tilde{u} \Vert ^{2}+\kappa \Vert \nabla \tilde{\theta } \Vert ^{2} \\ &\quad \leq C\bigl( \Vert \nabla \theta _{2} \Vert ^{2}+ \Vert \nabla u_{2} \Vert ^{2}+1\bigr) \bigl( \Vert \tilde{\theta } \Vert ^{2}+ \Vert \tilde{u} \Vert ^{2} \bigr). \end{aligned}$$
(76)

Using Gronwall inequality and the estimates for \(\theta _{2}\) and \(u_{2}\), (76) implies that, for any \(t\geq 0\),

$$\begin{aligned} e^{-CT}\bigl( \Vert \tilde{u} \Vert ^{2}+ \Vert \tilde{ \theta } \Vert ^{2}\bigr) \leq \bigl\Vert \tilde{u}(0) \bigr\Vert ^{2}+ \bigl\Vert \tilde{\theta }(0) \bigr\Vert ^{2} =0, \end{aligned}$$

i.e., \(\tilde{u}=0, \tilde{\theta }=0, \theta _{1}=\theta _{2}, u _{1}=u_{2}\). So the solution of Boussinesq equations (1) is unique. □

Proof of Theorem 2

The goal of this section is to prove Theorem 2. First of all, we multiply the first equation in (1) with \(u|u|^{p-2}\) (\(p>2\)) and, integrating it over \({R}^{2}\), have

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \Vert u \Vert ^{p}_{L^{p}}+ \int _{\varOmega }\varLambda ^{2}u \cdot u \vert u \vert ^{p-2}\,dx \\ &\quad=- \int _{\varOmega }(u\cdot \nabla u)\cdot u \vert u \vert ^{p-2}\,dx- \int _{\varOmega } \nabla P\cdot u \vert u \vert ^{p-2} \,dx \\ &\qquad{}+ \int _{\varOmega }\theta e_{2}\cdot u \vert u \vert ^{p-2}\,dx. \end{aligned}$$
(77)

Since u is divergence-free, by Lemma 2 and using Hölder inequality, we can get

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert u \Vert ^{p}_{L^{p}}+ \frac{4(p-2)}{p^{2}} \bigl\Vert \nabla \vert u \vert ^{\frac{p}{2}} \bigr\Vert ^{2}_{L^{2}} &\leq \Vert \nabla P \Vert _{L^{p}} \Vert u \Vert ^{p-1} _{L^{p}}+ \Vert \theta \Vert _{L^{p}} \Vert u \Vert ^{p-1}_{L^{p}}. \end{aligned}$$
(78)

Taking the divergence of the first equation in (1), we can obtain

$$\begin{aligned} -\Delta P = \operatorname{div} (u\cdot \nabla u)-\partial _{x_{2}} \theta. \end{aligned}$$
(79)

Hence

$$\begin{aligned} \nabla P = \nabla (-\Delta )^{-1} \bigl(\operatorname{div} (u\cdot \nabla u)-\partial _{x_{2}}\theta \bigr). \end{aligned}$$
(80)

Applying Calderón–Zygmund theorem, we get

$$\begin{aligned} \Vert \nabla P \Vert _{L^{p}} &\leq C\bigl( \Vert u\cdot \nabla u \Vert _{L^{p}}+ \Vert \theta \Vert _{L^{p}}\bigr) \\ &\leq C\bigl( \Vert u \Vert _{L^{\infty }} \Vert \nabla u \Vert _{L^{p}}+ \Vert \theta \Vert _{L^{p}}\bigr) \\ &\leq C\bigl( \Vert u \Vert _{W^{1,p}} \Vert \nabla u \Vert _{L^{p}}+ \Vert \theta \Vert _{L^{p}}\bigr) \\ &\leq C\bigl( \Vert u \Vert _{L^{p}} + \Vert \nabla u \Vert _{L^{p}}\bigr) \Vert \nabla u \Vert _{L^{p}}+C \Vert \theta \Vert _{L^{p}} \\ &\leq C\bigl( \Vert u \Vert _{L^{p}} + \Vert \omega \Vert _{L^{p}}\bigr) \Vert \omega \Vert _{L^{p}}+C \Vert \theta \Vert _{L^{p}}. \end{aligned}$$
(81)

Multiplying the third equation in (1) with \(\theta |\theta |^{p-2}\) (\(p>2\)) and integrating it over \({R}^{2}\), we deduce that

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert \theta \Vert ^{p}_{L^{p}}+ \int _{\varOmega } \varLambda ^{2\beta }\theta \cdot \theta \vert \theta \vert ^{p-2}\,dx =0, \end{aligned}$$
(82)

where we have used the divergence-free condition again. By Lemma 2 and integrating over \([0,t]\), we have, for all \(t\in [0,T]\),

$$\begin{aligned} \Vert \theta \Vert ^{p}_{L^{p}}+\frac{4(p-2)}{p} \int _{0}^{t} \bigl\Vert \varLambda ^{\beta } \vert \theta \vert ^{\frac{p}{2}} \bigr\Vert ^{2}_{L^{2}}\,d\tau = \Vert \theta _{0} \Vert ^{p}_{L ^{p}}. \end{aligned}$$
(83)

Combining (78) with (81) and (83) leads to

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert u \Vert ^{p}_{L^{p}}+ \frac{4(p-2)}{p^{2}} \bigl\Vert \nabla \vert u \vert ^{\frac{p}{2}} \bigr\Vert ^{2}_{L^{2}} \leq C\bigl( \Vert u \Vert _{L^{p}} \Vert \omega \Vert _{L ^{p}}+ \Vert \omega \Vert _{L^{p}}^{2}+1\bigr) \Vert u \Vert _{L^{p}}^{p-1}. \end{aligned}$$
(84)

Taking the \(L^{p}\)-inner product of (27) with \(\omega |\omega |^{p-2}\) (\(p>2\)) and integrating it over \({R}^{2}\), we arrive at

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert \omega \Vert ^{p}_{L^{p}}+ \int _{\varOmega } \varLambda ^{2\alpha }\omega \cdot \omega \vert \omega \vert ^{p-2}\,dx = \int _{ \varOmega }\partial _{x_{1}}\theta \cdot \omega \vert \omega \vert ^{p-2}\,dx. \end{aligned}$$
(85)

Using Lemma 2 again, we know that

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert \omega \Vert ^{p}_{L^{p}}+ \frac{4(p-2)}{p^{2}} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2} \leq \int _{\varOmega }\partial _{x _{1}}\theta \cdot \omega \vert \omega \vert ^{p-2}\,dx. \end{aligned}$$
(86)

By Hölder and Young inequalities, and using Lemma 2 with \(m=2\), we have

$$\begin{aligned} \int _{\varOmega }\partial _{x_{1}}\theta \cdot \omega \vert \omega \vert ^{p-2}\,dx & \leq (p-1) \biggl\vert \int _{\varOmega }\theta \cdot \partial _{x_{1}}\omega \vert \omega \vert ^{p-2}\,dx \biggr\vert \\ &\leq (p-1) \biggl\vert \int _{\varOmega }\theta \cdot \nabla \omega \vert \omega \vert ^{ \frac{p-2}{2}} \vert \omega \vert ^{\frac{p-2}{2}}\,dx \biggr\vert \\ &\leq \frac{2(p-1)}{p} \Vert \theta \Vert _{L^{p}} \bigl\Vert \nabla \vert \omega \vert ^{ \frac{p}{2}} \bigr\Vert _{L^{2}} \Vert \omega \Vert ^{\frac{p-2}{2}}_{L^{p}} \\ &\leq \frac{2(p-1)}{p^{2}} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2}_{L ^{2}}+C \Vert \theta \Vert ^{2}_{L^{p}} \Vert \omega \Vert ^{p-2}_{L^{p}}, \end{aligned}$$
(87)

where constant C depends on p. Inserting (87) into (86), we can obtain

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert \omega \Vert ^{p}_{L^{p}}+ \frac{2(p-2)}{p^{2}} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2} \leq C \Vert \theta \Vert ^{2}_{L^{p}} \Vert \omega \Vert ^{p-2}_{L^{p}}. \end{aligned}$$
(88)

Hence, by Young inequality, we get

$$\begin{aligned} \frac{d}{dt} \Vert \omega \Vert ^{p}_{L^{p}}+ \frac{2(p-2)}{p} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2} \leq C \bigl( \Vert \theta \Vert ^{p}_{L^{p}}+ \Vert \omega \Vert ^{p}_{L^{p}} \bigr). \end{aligned}$$
(89)

Integrating over \([0,t]\), we have, for all \(t\in [0,T]\),

$$\begin{aligned} & \Vert \omega \Vert ^{p}_{L^{p}}+\frac{2(p-2)}{p} \int _{0}^{t} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2}\,d\tau \\ &\quad\leq \Vert \omega _{0} \Vert ^{p}_{L^{p}}+C \int _{0}^{t} \bigl( \Vert \theta \Vert ^{p} _{L^{p}}+ \Vert \omega \Vert ^{p}_{L^{p}} \bigr)\,d\tau \\ &\quad\leq \Vert \omega _{0} \Vert ^{p}_{L^{p}}+CT \Vert \theta _{0} \Vert ^{p}_{L^{p}}+C \int _{0}^{t} \Vert \omega \Vert ^{p}_{L^{p}}\,d\tau. \end{aligned}$$
(90)

Using Gronwall inequality, we find from (90) that, for all \(t\in [0,T]\),

$$\begin{aligned} \Vert \omega \Vert ^{p}_{L^{p}}+\frac{2(p-2)}{p} \int _{0}^{t} \bigl\Vert \nabla \vert \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2}\,d\tau &\leq e^{Ct} \bigl( \Vert \omega _{0} \Vert ^{p}_{L ^{p}}+CT \Vert \theta _{0} \Vert ^{p}_{L^{p}} \bigr) \\ &\leq e^{CT} \bigl( \Vert \omega _{0} \Vert ^{p}_{L^{p}}+CT \Vert \theta _{0} \Vert ^{p} _{L^{p}} \bigr) \\ &\leq C. \end{aligned}$$
(91)

Then inequality (84), together with (91), implies that

$$\begin{aligned} \Vert u \Vert ^{p}_{L^{p}}+\frac{4(p-2)}{p^{2}} \int _{0}^{t} \bigl\Vert \nabla \vert u \vert ^{ \frac{p}{2}} \bigr\Vert ^{2}_{L^{2}}\,d\tau \leq C. \end{aligned}$$
(92)

Taking the derivative \(D=(\partial _{x_{1}},\partial _{x_{2}})\) of both sides of (27), and then multiplying the result equation array by \(D\omega |D\omega |^{p-2}\), after integration by parts, we obtain

$$\begin{aligned} \frac{1}{p}\frac{d}{dt} \Vert D\omega \Vert ^{p}_{L^{p}}+\frac{4(p-2)}{p^{2}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2}\leq{}& {-} \int _{\varOmega }D(u \cdot \nabla \omega )\cdot D\omega \vert D \omega \vert ^{p-2}\,dx \\ &{}{-} \int _{\varOmega }D\theta _{x_{1}}\cdot D\omega \vert D\omega \vert ^{p-2}\,dx. \end{aligned}$$
(93)

Using Hölder and Cauchy–Schwarz inequalities, a standard calculation gives us the following:

$$\begin{aligned} & {-} \int _{\varOmega }D(u\cdot \nabla \omega )\cdot D\omega \vert D\omega \vert ^{p-2}\,dx \\ &\quad\leq C \biggl\vert \int _{\varOmega }u\cdot \nabla \omega \cdot D^{2}\omega \vert D \omega \vert ^{p-2}\,dx \biggr\vert \\ &\quad\leq (p-1) \biggl\vert \int _{\varOmega }u\cdot \nabla \omega \cdot D^{2}\omega \vert D\omega \vert ^{\frac{p-2}{2}} \vert D\omega \vert ^{\frac{p-2}{2}} \,dx \biggr\vert \\ &\quad\leq \frac{2(p-1)}{p} \Vert u \Vert _{L^{p}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert _{L^{2}} \Vert D \omega \Vert _{L^{p}}^{\frac{p-2}{2}} \\ &\quad\leq \frac{p-1}{p^{2}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert _{L^{2}} ^{2}+C \Vert u \Vert _{L^{p}}^{2} \Vert D\omega \Vert _{L^{p}}^{p-2} \end{aligned}$$
(94)

and

$$\begin{aligned} &{-} \int _{\varOmega }D\theta _{x_{1}}\cdot D\omega \vert D\omega \vert ^{p-2}\,dx \\ &\quad\leq \biggl\vert \int _{\varOmega }D\theta _{x_{1}}\cdot D\omega \vert D\omega \vert ^{p-2}\,dx \biggr\vert \\ &\quad\leq (p-1) \biggl\vert \int _{\varOmega }D\theta _{x_{1}}\cdot D\omega \vert D\omega \vert ^{p-2}\,dx \biggr\vert \\ &\quad\leq \frac{2(p-1)}{p} \Vert D\theta \Vert _{L^{p}} \bigl\Vert \partial _{x_{1}} \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert _{L^{2}} \Vert D\omega \Vert _{L^{p}}^{\frac{p-2}{2}} \\ &\quad\leq \frac{p-1}{p^{2}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert _{L^{2}} ^{2}+C \Vert D\theta \Vert _{L^{p}}^{2} \Vert D\omega \Vert _{L^{p}}^{p-2} \\ &\quad\leq \frac{p-1}{p^{2}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert _{L^{2}} ^{2}+C\bigl( \Vert \nabla \theta \Vert _{L^{p}}^{p}+ \Vert D\omega \Vert _{L^{p}}^{p}\bigr). \end{aligned}$$
(95)

Inserting (94) and (95) into (93), we obtain

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \Vert D\omega \Vert ^{p}_{L^{p}}+ \frac{2(p-2)}{p^{2}} \bigl\Vert \nabla \vert D \omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2} \\ &\quad\leq C\bigl( \Vert u \Vert _{L^{p}}^{p}+ \Vert \nabla \theta \Vert _{L^{p}}^{p}+ \Vert D\omega \Vert _{L^{p}}^{p}\bigr). \end{aligned}$$
(96)

Taking the derivative \(\nabla ^{\bot }=(-\partial _{x_{2}},\partial _{x _{1}})\) of both sides of (1)3, we can show that

$$\begin{aligned} \nabla ^{\bot }\theta _{t}+\kappa \varLambda ^{2\beta } \nabla ^{\bot }\theta +\nabla ^{\bot }(u\cdot \nabla \theta ) =0. \end{aligned}$$
(97)

Multiplying (96) by \(\nabla ^{\bot }\theta |\nabla ^{\bot }\theta |^{p-2}\), after integration by parts, we obtain

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \bigl\Vert \nabla ^{\bot }\theta \bigr\Vert ^{p}_{L^{p}}+ \int _{\varOmega }\varLambda ^{2\beta }\nabla ^{\bot }\theta \cdot \nabla ^{ \bot }\theta \bigl\vert \nabla ^{\bot }\theta \bigr\vert ^{p-2}\,dx \\ &\quad =- \int _{\varOmega }\nabla ^{\bot }(u\cdot \nabla \theta )\cdot \nabla ^{\bot }\theta \bigl\vert \nabla ^{\bot }\theta \bigr\vert ^{p-2}\,dx. \end{aligned}$$
(98)

Since u is divergence-free, using Lemma 2 again, we know that

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \bigl\Vert \nabla ^{\bot }\theta \bigr\Vert ^{p}_{L^{p}}+\frac{4(p-2)}{p ^{2}} \bigl\Vert \varLambda ^{\beta } \bigl\vert \nabla ^{\bot }\theta \bigr\vert ^{\frac{p}{2}} \bigr\Vert ^{2} _{L^{2}} \\ &\quad\leq - \int _{\varOmega }\nabla u\cdot \nabla ^{\bot }\theta \cdot \nabla ^{\bot }\theta \bigl\vert \nabla ^{\bot }\theta \bigr\vert ^{p-2}\,dx \\ &\quad\leq \Vert \nabla u \Vert _{L^{\infty }} \bigl\Vert \nabla ^{\bot }\theta \bigr\Vert ^{p}_{L ^{p}}. \end{aligned}$$
(99)

By Lemma 3, we know that

$$\begin{aligned} \Vert \nabla u \Vert _{L^{\infty }} &\leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) \bigl(1+ \log ^{+}\bigl( \Vert D\omega \Vert ^{p}_{L^{p}}\bigr) \bigr)+C \Vert \omega \Vert _{L^{2}}. \end{aligned}$$
(100)

By (91) for \(p=2\), and inserting (100) into (99), we obtain

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \bigl\Vert \nabla ^{\bot }\theta \bigr\Vert ^{p}_{L^{p}}+\frac{4(p-2)}{p ^{2}} \bigl\Vert \varLambda ^{\beta } \bigl\vert \nabla ^{\bot }\theta \bigr\vert ^{\frac{p}{2}} \bigr\Vert ^{2} _{L^{2}} \\ &\quad\leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) \bigl(1+\log ^{+}\bigl( \Vert D\omega \Vert ^{p} _{L^{p}}\bigr) \bigr) \bigl\Vert \nabla ^{\bot }\theta \bigr\Vert ^{p}_{L^{p}}. \end{aligned}$$
(101)

Using the obvious identity \(\|\nabla \theta \|_{L^{p}}=\|\nabla ^{ \bot }\theta \|_{L^{p}}\), and summing up (96) and (101), we obtain that

$$\begin{aligned} &\frac{1}{p}\frac{d}{dt}\bigl( \Vert D\omega \Vert ^{p}_{L^{p}}+ \Vert \nabla \theta \Vert ^{p}_{L^{p}}\bigr)+\frac{2(p-2)}{p^{2}} \bigl\Vert \nabla \vert D\omega \vert ^{\frac{p}{2}} \bigr\Vert ^{2} + \frac{4(p-2)}{p^{2}} \bigl\Vert \varLambda ^{\beta } \vert \nabla \theta \vert ^{ \frac{p}{2}} \bigr\Vert ^{2}_{L^{2}} \\ &\quad\leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) \bigl(1+\log ^{+}\bigl( \Vert D\omega \Vert ^{p} _{L^{p}}\bigr) \bigr) \bigl( \Vert \nabla \theta \Vert _{L^{p}}^{p}+ \Vert D\omega \Vert _{L^{p}} ^{p}\bigr) \\ &\quad\leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) \bigl(1+\log ^{+}\bigl( \Vert D\omega \Vert ^{p} _{L^{p}}+ \Vert \nabla \theta \Vert _{L^{p}}^{p} \bigr) \bigr) \bigl( \Vert D\omega \Vert _{L^{p}} ^{p}+ \Vert \nabla \theta \Vert _{L^{p}}^{p}\bigr). \end{aligned}$$
(102)

Setting \(X(t)=\|D\omega \|^{p}_{L^{p}}+\|\nabla \theta \|^{p}_{L^{p}}\), we easily show that

$$\begin{aligned} \frac{d}{dt}X \leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) \bigl(1+\log ^{+}X \bigr)X. \end{aligned}$$
(103)

Setting \(Y=\log ^{+}X\), we know that

$$\begin{aligned} \frac{d}{dt}X=X\frac{d}{dt}Y. \end{aligned}$$
(104)

So, inequality (103), along with (104), implies that

$$\begin{aligned} \frac{d}{dt}Y \leq C\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr) (1+Y). \end{aligned}$$
(105)

By (91) for \(p=2\), and integrating over \([0,t]\), we have, for all \(t\in [0,T]\),

$$\begin{aligned} Y(t) \leq Y(0)+C \int _{0}^{t}\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr)\,d\tau +C \int _{0}^{t}\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr)Y\,d\tau. \end{aligned}$$
(106)

Using Gronwall inequality, from (105) we find that, for all \(t\in [0,T]\),

$$\begin{aligned} Y(t) &\leq C \biggl(Y(0)+ \int _{0}^{t}\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr)\,d \tau \biggr) e^{C\int _{0}^{t}(1+ \Vert D\omega \Vert _{L^{2}}^{2})\,d\tau } \\ &\leq C \biggl(Y(0)+ \int _{0}^{T}\bigl(1+ \Vert D\omega \Vert _{L^{2}}^{2}\bigr)\,d\tau \biggr) e^{C\int _{0}^{T}(1+ \Vert D\omega \Vert _{L^{2}}^{2})\,d\tau } \\ &\leq C \bigl(Y(0)+CT \bigr)e^{CT}, \end{aligned}$$
(107)

which implies

$$\begin{aligned} X(t) \leq e^{C (\log ^{+}X(0)+CT )e^{CT}}. \end{aligned}$$
(108)

This thus completes the proof of Theorem 2.

Conclusions

In this paper, we study the well-posedness and related problem on Boussinesq equations with fractional dissipation which have recently attracted considerable interest. This paper proves the persistence of global well-posedness of strong solutions and their long-time decay, as well as investigates the existence of the solutions in Sobolev spaces. The obtained results will not only further improve the theory of fractional nonlinear evolution equations, but also provide support for the innovation on research methods and the related properties of fluid dynamics models.

References

  1. 1.

    Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233(1), 199–220 (2007)

  2. 2.

    Adhikari, D., Cao, C., Wu, J.: The 2D Boussinesq equation with vertical viscosity and vertical diffusivity. J. Differ. Equ. 249, 1078–1088 (2010)

  3. 3.

    Adhikari, D., Cao, C., Wu, J.: Global regularity results for the 2D Boussinesq equation with vertical dissipation. J. Differ. Equ. 251, 1637–1655 (2011)

  4. 4.

    Boldrini, J.L., Climent-Ezquerra, B., Rojas-Medar, M.D., Rojas-Medar, M.A.: On an iterative method for approximate solutions of a generalized Boussinesq model. J. Math. Fluid Mech. 13(1), 33–53 (2011)

  5. 5.

    Boldrini, J.L., Fernandez-Cara, E., Rojas-Medar, M.A.: An optimal control problem for a generalized Boussinesq model: the time dependent case. Rev. Mat. Complut. 20(2), 339–366 (2007)

  6. 6.

    Boldrini, J.L., Rojas-Medar, M.A., Rocha, M.S.D.: Existence of relaxed weak solutions of a generalized Boussinesq system with restriction on the state variables. SeMA J. 47, 63–72 (2009)

  7. 7.

    Brezis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Differ. Equ. 5(7), 773–789 (1980)

  8. 8.

    Cannon, J.R., DiBenedetto, E.: The initial value problem for the boussinesq equation with data in \(L^{p}\). In: Approximation Methods for Navier–Stokes Problems, pp. 129–144. Springer, Berlin (1980)

  9. 9.

    Cao, C., Wu, J.: Global regularity for the 2D anisotropic Boussinesq equation with vertical dissipation. Arch. Ration. Mech. Anal. 208(3), 985–1004 (2013)

  10. 10.

    Cattani, C.: Sinc-fractional operator on Shannon wavelet space. Front. Phys. (2018)

  11. 11.

    Chae, D.: Global regularity for the 2D Boussinesq equation with partial viscosity terms. Adv. Math. 203(2), 497–513 (2006)

  12. 12.

    Chae, D., Kim, S.K., Nam, H.S.: Local existence and blow-up criterion of Holder continuous solutions of the Boussinesq equation. Nagoya Math. J. 155, 55–80 (1999)

  13. 13.

    Chae, D., Nam, H.S.: Local existence and blow-up criterion for the Boussinesq equation. Proc. R. Soc. Edinb. 127A, 935–946 (1997)

  14. 14.

    Choi, J., Agarwal, P.: A note on fractional integral operator associated with multiindex Mittag-Leffler functions. Filomat 30(7), 1931–1939 (2016)

  15. 15.

    Constantin, P., Doering, C.R.: Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172 (1999)

  16. 16.

    Danchin, R., Paicu, M.: Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces. Phys. D, Nonlinear Phenom. 237, 1444–1460 (2008)

  17. 17.

    Danchin, R., Paicu, M.: Global existence results for the anisotropic Boussinesq system in dimension two. Math. Models Methods Appl. Sci. 21(3), 421–457 (2011)

  18. 18.

    Engler, H.: An alternative proof of the Brezis–Wainger inequality. Commun. Partial Differ. Equ. 14(4), 541–544 (1989)

  19. 19.

    Fang, D., Qian, C., Zhang, T.: Global well-posedness for 2D Boussinesq system with general supercritical dissipation. Nonlinear Anal., Real World Appl. 27, 326–349 (2016)

  20. 20.

    Gao, F.: General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems. Therm. Sci. 21(s1), 11–18 (2017)

  21. 21.

    Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, London (1982)

  22. 22.

    Hammouch, Z., Mekkaoui, T., Agarwal, P.: Optical solitons for the Calogero–Bogoyavlenskii–Schiff equation narray in (\(2+1\)) dimensions with time-fractional conformable derivative. Eur. Phys. J. Plus 133, 248 (2018)

  23. 23.

    Hassainia, Z., Hmidi, T.: On the inviscid Boussinesq system with rough initial data. J. Math. Anal. Appl. 430, 777–809 (2015)

  24. 24.

    Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity. Indiana Univ. Math. J. 58(4), 1591–1618 (2009)

  25. 25.

    Hmidi, T., Keraani, S., Rousset, F.: Global well-posedness for Euler–Boussinesq system with critical dissipation. Commun. Partial Differ. Equ. 36, 420–445 (2011)

  26. 26.

    Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equation. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)

  27. 27.

    Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equation in a bounded domain. J. Math. Phys. 54(8), 081507 (2013)

  28. 28.

    Hu, W., Kukavica, I., Ziane, M.: Persistence of regularity for the viscous Boussinesq equation with zero diffusivity. Asymptot. Anal. 91, 111–134 (2015)

  29. 29.

    Huang, A.: The 2D Euler–Boussinesq equation in planar polygonal domains with Yudovich’s type data. Commun. Math. Stat. 2(3–4), 369–391 (2014). arXiv:1405.2631

  30. 30.

    Huang, A.: The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity. Nonlinear Anal. TMA 113, 401–429 (2015)

  31. 31.

    Jia, J., Peng, J., Li, K.: On the global well-posedness of a generalized 2D Boussinesq equation. NoDEA Nonlinear Differ. Equ. Appl. 22, 911–945 (2015)

  32. 32.

    Jin, L., Fan, J.: Uniform regularity for the 2D Boussinesq system with a slip boundary condition. J. Math. Anal. Appl. 400(1), 96–99 (2013)

  33. 33.

    Jiu, Q., Miao, C., Wu, J., Zhang, Z.: The 2D incompressible Boussinesq equation with general dissipation. Soc. Sci. Electron. Publ. 17(4), 1132–1157 (2012) arXiv:1212.3227v1

  34. 34.

    Jiu, Q., Miao, C., Wu, J., Zhang, Z.: The two-dimensional incompressible Boussinesq equation with general critical dissipation. SIAM J. Math. Anal. 46(5), 3426–3454 (2014)

  35. 35.

    Jiu, Q., Wu, J., Yang, W.: Eventual regularity of the two-dimensional Boussinesq equation with supercritical dissipation. J. Nonlinear Sci. 25, 37–58 (2015)

  36. 36.

    Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equation. Commun. Math. Phys. 255(1), 161–181 (2005)

  37. 37.

    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equation. Commun. Pure Appl. Math. 41, 891–907 (1988)

  38. 38.

    KC, D.: A study on the global well-posedness for the two-dimensional Boussinesq and Lans-Alpha magnetohydrodynamics equation, Dissertations and Theses-Gradworks, Oklahoma State University, 2014

  39. 39.

    KC, D., Regmi, D., Tao, L., Wu, J.: Generalized 2D Euler–Boussinesq equation with a singular velocity. J. Differ. Equ. 257, 82–108 (2014)

  40. 40.

    Kukavica, I., Wang, F., Ziane, M.: Persistence of regularity for solutions of the Boussinesq equation in Sobolev spaces. Adv. Differ. Equ. 21, 1/2 (2016)

  41. 41.

    Lai, M., Pan, R., Zhao, K.: Initial boundary value problem for two-dimensional viscous Boussinesq equation. Arch. Ration. Mech. Anal. 199(3) 739–760 (2011)

  42. 42.

    Larios, A., Lunasin, E., Titi, E.S.: Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. J. Differ. Equ. 255(9), 2636–2654 (2013)

  43. 43.

    Li, H., Pan, R., Zhang, W.: Initial boundary value problem for 2D Boussinesq equation with temperature-dependent heat diffusion. J. Hyperbolic Differ. Equ. 12(3), 469–488 (2015)

  44. 44.

    Li, J., Titi, E.S.: Global well-posedness of the 2D Boussinesq equation with vertical dissipation. Arch. Ration. Mech. Anal. 220(3), 983–1001 (2016)

  45. 45.

    Li, Y.: Global regularity for the viscous Boussinesq equation. Math. Methods Appl. Sci. 27(3), 363–369 (2004)

  46. 46.

    Liskevich, V.A., Semenov, Y.A.: Some problems on Markov semigroups. Schrodinger operators, Markov semigroups, wavelet analysis, operator algebras. In: Math. Top., vol. 11, pp. 163–217. Akademie Verlag, Berlin (1996)

  47. 47.

    Lorca, S.A., Boldrini, J.L.: The initial value problem for a generalized Boussinesq model: regularity and global existence of strong solutions. Mat. Contemp. 11, 71–94 (1996)

  48. 48.

    Lorca, S.A., Boldrini, J.L.: The initial value problem for a generalized Boussinesq model. Nonlinear Anal. TMA 36(4), 457–480 (1999)

  49. 49.

    Miao, C., Xue, L.: On the global well-posedness of a class of Boussinesq–Navier–Stokes systems. NoDEA Nonlinear Differ. Equ. Appl. 18, 707–735 (2011)

  50. 50.

    Morales-Delgado, V.F., Gomez-Aguilar, J.F., Saad, K.M., AltafKhan, M., Agarwal, P.: Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach. Phys. A, Stat. Mech. Appl. 523(1), 48–65 (2019)

  51. 51.

    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)

  52. 52.

    Qin, Y., Su, X., Wang, Y., Zhang, J.: Global regularity for a two-dimensional nonlinear Boussinesq system. Math. Methods Appl. Sci. (2016). https://doi.org/10.1002/mma.4118

  53. 53.

    Ruzhansky, M., Je, C.Y., Cho, Y.J., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Applications. Springer, Singapore (2017)

  54. 54.

    Saad, K.M., Iyiola, O.S., Agarwal, P.: An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system. AIMS Math. 3(1), 183–194 (2018)

  55. 55.

    Shi, Q., Wang, S.: Nonrelativistic approximation in the energy space for KGS system. J. Math. Anal. Appl. 462(2), 1242–1253 (2018)

  56. 56.

    Stefanov, A., Wu, J.: A global regularity result for the 2D Boussinesq equation with critical dissipation. Mathematics 29(1), 195–205 (2014)

  57. 57.

    Su, X.: The global attractor of the 2D Boussinesq system with fractional vertical dissipation. Bound. Value Probl. 2016(1), 1 (2016)

  58. 58.

    Sun, Y., Zhang, Z.: Global regularity for the initial-boundary value problem of the 2D Boussinesq system with variable viscosity and thermal diffusivity. J. Differ. Equ. 255(6), 1069–1085 (2013)

  59. 59.

    Tariboon, J., Ntouyas, S.K., Agarwal, P.: New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equation. Adv. Differ. Equ. 2015, 18 (2015)

  60. 60.

    Wang, C., Zhang, Z.: Global well-posedness for the 2D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. Adv. Math. 228(1), 43–62 (2011)

  61. 61.

    Wu, G., Zheng, X.: Golbal well-posedness for the two-dimensional nonlinear Boussinesq equation with vertical dissipation. J. Differ. Equ. 255, 2891–2926 (2013)

  62. 62.

    Wu, J.: Generalized MHD equation. J. Differ. Equ. 195, 284–312 (2003)

  63. 63.

    Wu, J., Xu, X.: Well-posedness and inviscid limits of the Boussinesq equation with fractional Laplacian dissipation. Nonlinearity 27, 2215–2232 (2014)

  64. 64.

    Xiang, Z., Yan, W.: Global regularity of solutions to the Boussinesq equation with fractional diffusion. Adv. Differ. Equ. 18, 11/12 (2013)

  65. 65.

    Xu, F., Yuan, J.: On the global well-posedness for the 2D Euler–Boussinesq system. Nonlinear Anal., Real World Appl. 17, 137–146 (2014)

  66. 66.

    Xu, X.: Global regularity of solutions of 2D Boussinesq equation with fractional diffusion. Nonlinear Anal. 72, 677–681 (2010)

  67. 67.

    Xu, X., Xue, L.: Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation. J. Differ. Equ. 256, 3179–3207 (2014)

  68. 68.

    Yamazaki, K.: On the global regularity of N-dimensional generalized Boussinesq system. Appl. Math. 60(2), 109–133 (2015)

  69. 69.

    Yang, A., Yang, H., Li, J., Liu, W.: On steady heat flow problem involving Yang–Srivastava–Machado fractional derivative without singular kernel. Therm. Sci. 20(suppl. 3), 717–721 (2016)

  70. 70.

    Yang, W., Jiu, Q., Wu, J.: Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation. J. Differ. Equ. 257, 4188–4213 (2014)

  71. 71.

    Yang, X.: New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 19(1), 45–52 (2018)

  72. 72.

    Yang, X.: General Fractional Derivatives: Theory, Methods and Applications. CRC Press, New York (2019)

  73. 73.

    Yang, X., Feng, Y., Cattani, C., Inc, M.: Fundamental solutions of anomalous diffusion equations with the decay exponential kernel. Math. Methods Appl. Sci. 42(11), 4054–4060 (2019)

  74. 74.

    Yang, X., Gao, F., Ju, Y., Zhou, H.: Fundamental solutions of the general fractional-order diffusion equations. Math. Methods Appl. Sci. 41(18), 9312–9320 (2018)

  75. 75.

    Yang, X., Gao, F., Srivastava, H.: Non-differentiable exact solutions for the nonlinear ODEs defined on fractal sets. Fractals 25(04), 1740002 (2007)

  76. 76.

    Yang, X., Gao, F., Srivastava, H.: New rheological models within local fractional derivative. Rom. Rep. Phys. 69(3), 113 (2017)

  77. 77.

    Yang, X., Gao, F., Srivastava, H.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2017)

  78. 78.

    Yang, X., Gao, F., Tenreiro, M., Dumitru, B.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Spec. Top. 226(16–18), 3567–3575 (2017)

  79. 79.

    Yang, X., Machado, J.: A new fractional operator of variable order: application in the description of anomalous diffusion. Phys. A, Stat. Mech. Appl. 481, 276–283 (2017)

  80. 80.

    Yang, X., Machado, J., Baleanu, D.: On exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(4), 1740006 (2017)

  81. 81.

    Yang, X., Mahmoud, A., Cattani, C.: A new general fractional-order derivative with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer. Therm. Sci. 23 1677–1681 (2019)

  82. 82.

    Yang, X., Srivastava, H., Tenreiro, J.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20(2), 753–756 (2015)

  83. 83.

    Ye, Z., Xu, X.: Remarks on global regularity of the 2D Boussinesq equation with fractional dissipation. Nonlinear Anal. TMA 125, 715–724 (2015)

  84. 84.

    Ye, Z., Xu, X.: Global regularity results of the 2D Boussinesq equation with fractional Laplacian dissipation. J. Math. Fluid Mech. 260(8), 1–20 (2015)

  85. 85.

    Zhang, X., Agarwal, P., Liu, Z., Peng, H.: The general solution for impulsive differential equation with Riemann–Liouville fractional-order q \(\in (1,2)\). Open Math. 13(1), 2391–5455 (2015)

  86. 86.

    Zhao, K.: 2D inviscid heat conductive Boussinesq equation on a bounded domain. Mich. Math. J. 59, 329–352 (2010)

  87. 87.

    Zhou, D., Li, Z.: Global well-posedness for the 2D Boussinesq Equation with Zero Viscosity (2016). arXiv:1603.08301v2 [math.AP]

Download references

Author information

All the authors contributed equally to this work. All authors read and approved the final manuscript.

Correspondence to Gangwei Wang.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Su, X., Wang, G. & Wang, Y. Persistence of global well-posedness for the 2D Boussinesq equations with fractional dissipation. Adv Differ Equ 2019, 420 (2019) doi:10.1186/s13662-019-2348-1

Download citation

Keywords

  • Boussinesq equations
  • Fractional dissipation
  • Global well-posedness