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Global existence and exponential decay of strong solutions for the three-dimensional Boussinesq equations

Abstract

In this paper, we consider the global existence of strong solutions to the three-dimensional Boussinesq equations on the smooth bounded domain Ω. Based on the blow-up criterion and uniform estimates, we prove that the strong solution exists globally in time if the initial \(L^{2}\)-norm of velocity and temperature are small. Moreover, an exponential decay rate of the strong solution is obtained.

Introduction

In this paper, we consider the following three-dimensional incompressible Boussinesq equations in the Eulerian coordinates:

$$ \textstyle\begin{cases} u_{t}-\mu \Delta {u}+u\cdot \nabla {u}+\nabla {P}=\theta {e_{3}} , \\ \theta _{t}-\kappa \Delta {\theta }+u\cdot \nabla {\theta }=0 , \\ \nabla \cdot {u}=0 , \\ u(x,0)=u_{0}(x), \quad\quad \theta (x,0)=\theta _{0}(x) , \end{cases} $$
(1.1)

where \({u}=(u_{1},u_{2}, u_{3})(x,t)\), \(\theta =\theta (x,t)\), \(P(x,t)\) are unknown functions denoting fluid velocity vector field, absolute temperature, and scalar pressure, \(t\geqslant 0\) is time, \(x\in \varOmega \) is spatial coordinate. μ is the kinematic viscosity, κ is the thermal diffusivity, and \(e_{3}=(0,0,1)\) is the unit vector in the \(x_{3}\) direction. The given functions \(u_{0}\) and \(\theta _{0}\) are the initial velocity and initial temperature, respectively.

Boussinesq system (1.1) has been widely used in atmospheric sciences and oceanic fluids [9], and there is a huge amount of literature on the well-posedness theory of strong and week solutions for the three-dimensional Boussinesq equations. In 2004, Sawada and Taniuchi [12] established the local existence and uniqueness of strong solutions in the whole space. In 2008, Danchin and Paicu [6] obtained global small solutions in Lorentz spaces. In 2012, Brandolese and Schonbek [2] showed the polynomial decay rate of week and strong solutions, global smooth solution and its stability were given by Liu and Li [8] in 2014. In 2016, Qin et al. [11] established the global classical solutions for axisymmetric equations with anisotropic initial data. In 2017, Ye [15] proved global existence of smooth solution to a modified Boussinesq model without thermal diffusion. In 2018, Wen and Ye [14] established the regularity and uniqueness of strong solution for the damped Boussinesq equations with zero thermal diffusion. Global well-posedness with fractional partial dissipation can be found in a recent work [16]. For local and global theories of solutions in two-dimensional space, we refer to [1, 35, 10, 17] and the references therein.

Inspired by the results of full compressible Navier–Stokes equations [13] and nonhomogeneous incompressible magnetohydrodynamic equations [7], in this paper we consider global existence and exponential decay of strong solutions to system (1.1) with the following initial-boundary conditions:

$$\begin{aligned}& ({u},\theta ) (x,t)| _{t=0}= \bigl({u}_{0}(x), \theta _{0}(x) \bigr), \end{aligned}$$
(1.2)
$$\begin{aligned}& ({u}, \theta ) (x,t)|_{\partial \varOmega }=0. \end{aligned}$$
(1.3)

Now, we state our main results as follows.

Theorem 1.1

(Blow-up criterion)

Suppose that the initial data satisfy\((u_{0},\theta _{0})\in H_{0} ^{1}\cap H^{2}\)and\(\operatorname{div}u_{0}=0\)in Ω. Let\(({u},\theta )\)be the strong solution to the initial-boundary value problem (1.1)(1.3) on\(\varOmega \times (0,T] \)satisfying

$$ \textstyle\begin{cases} ({u},\theta )\in L^{\infty }(0,T; H^{1})\cap L^{2}(0,T; H^{2}), \\ ( { u}_{t},\theta _{t})\in L^{\infty }(0,T; L^{2})\cap L^{2}(0,T; H ^{1}). \end{cases} $$
(1.4)

If\(T^{*}<+\infty \)is the maximal time of existence for the strong solution\((u,\theta )\), then

$$ \lim_{T\to T^{*}}\sup_{[0,T)} \Vert \nabla u \Vert _{L^{\infty }(0,T;L^{2})}=+ \infty . $$
(1.5)

Theorem 1.2

(Global strong solution)

For any given\(K_{i}>0\) (\(i=1,2\)), suppose that the initial data satisfy\((u_{0},\theta _{0})\in H_{0}^{1}\cap H^{2}\), \(\operatorname{div}u_{0}=0\)inΩ, and

$$ \Vert \nabla {{u}}_{0} \Vert _{L^{2}}\le \sqrt{K_{1}},\quad\quad \Vert \theta _{0} \Vert _{L^{2}}\le \sqrt{K_{2}}. $$
(1.6)

Then the initial-boundary value problem (1.1)(1.3) admits a unique global strong solution\((u,\theta )\)on\(\varOmega \times [0,T]\)for any\(T>0\), provided that there exists a constant\(\varepsilon _{0}>0\)such that

$$ \Vert {{u}}_{0} \Vert _{L^{2}}^{2}+ \Vert \theta _{0} \Vert _{L^{2}}^{2} \leqslant \varepsilon _{0}, $$
(1.7)

where\(\varepsilon _{0}\)depends on\(K_{1}\), μ, κ, and some other known constants but is independent of T.

Remark 1.1

When \(\theta =0\), system (1.1) reduces to Navier–Stokes equations. Then Theorem 1.2 implies that Navier–Stokes equations admit a unique global strong solution on \(\varOmega \times [0,T]\) for any \(T>0\), provided that there exists a constant \(\varepsilon _{0}>0\) such that

$$ \Vert {{u}}_{0} \Vert _{L^{2}}^{2} \leqslant \varepsilon _{0}, $$
(1.8)

where \(\varepsilon _{0}\) depends on \(K_{1}\), μ and some other known constants but is independent of T.

Theorem 1.3

(Asymptotic behavior)

Under the conditions of Theorem 1.2, it holds that

$$ \int _{\varOmega } \bigl(\theta ^{2}+ \vert \nabla u \vert ^{2} \bigr)\leqslant {C _{1}}\exp \{-C_{2} t\} $$
(1.9)

for any\(t\in [0, +\infty )\), provided that there exists a constant\(\varepsilon _{0}>0\)such that

$$ \Vert {{u}}_{0} \Vert _{L^{2}}^{2}+ \Vert \theta _{0} \Vert _{L^{2}}^{2} \leqslant \varepsilon _{0}, $$
(1.10)

where positive constants\({C}_{1}\), \({C}_{2}\), \(\varepsilon _{0}\)depend on\(K_{1}\), \(K_{2}\), μ, κ, and some other known constants but are independent of t.

Remark 1.2

In Theorem 1.3, we obtain exponential decay rate (1.9) in a bounded domain, which refines the polynomial decay in [2] and [8].

For the fixed viscosity and heat conduction, we need the “smallness” of initial velocity and temperature. But the velocity and temperature can be large. From the proof of Theorem 1.2 and Theorem 1.3, we have the following corollary.

Corollary 1.4

For any given\(K_{i}>0\) (\(i=1,2\)), suppose that the initial data satisfy\((u_{0},\theta _{0})\in H_{0}^{1}\cap H^{2}\), \(\operatorname{div}u_{0}=0\)inΩ, and

$$ \Vert \nabla {{u}}_{0} \Vert _{L^{2}}\le \sqrt{K_{1}},\quad\quad \Vert \theta _{0} \Vert _{L^{2}}\le \sqrt{K_{2}}. $$
(1.11)

Then the initial-boundary value problem (1.1)(1.3) admits a unique global strong solution\((u,\theta )\)on\(\varOmega \times [0,T]\)for any\(T>0\), provided that there exists a constant\(\mu ^{*}>0\)such that

$$ \mu \geqslant \mu ^{*}. $$

Furthermore, whenμandκare large enough, the global strong solution satisfies the following exponential decay rate:

$$ \int _{\varOmega } \bigl(\theta ^{2}+ \vert \nabla u \vert ^{2} \bigr)\leqslant {C _{1}}\exp \{-C_{2} t\} $$
(1.12)

for any\(t\in [0, +\infty )\), where positive constants\({C}_{1}\), \({C}_{2}\), \(\mu ^{*}\)depend on\(K_{1}\), \(K_{2}\), μ, κ, and some other known constants but are independent of t.

Notations

In the following paragraph, positive generic constants are denoted by C, which may change in different places.

The rest of the paper is organized as follows. In Sect. 2, under the assumption of (1.5) is false, we prove that the maximal time of existence for the strong solution is \(T^{*}=+\infty \), where the standard energy estimate is used and it holds uniform in time. In Sect. 3, we show that \(\Vert \nabla u\Vert _{L^{\infty }(0,T;L^{2})}\) will never blow up in finite time, which combines the blow-up criterion in Theorem 1.1, global existence of strong solution is proved in Theorem 1.2 provided the initial data of velocity and temperature are suitably small under the \(L^{2}\)-norm. Finally, in Sect. 4, exponential decay rate of the strong solution is obtained.

Blow-up criterion

Now, we state some uniform a priori estimates to prove Theorem 1.1.

Lemma 2.1

Under the conditions of Theorem 1.1, it holds that

$$\begin{aligned}& \sup_{0\leqslant t\leqslant T} \bigl\Vert \theta (t) \bigr\Vert _{L^{2}}^{2}+ \int _{0} ^{T}\kappa \bigl\Vert \nabla {\theta (s)} \bigr\Vert _{L^{2}}^{2}\,ds= \Vert {\theta _{0}} \Vert _{L^{2}}^{2}\leqslant K_{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \sup_{0\leqslant t\leqslant T} \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{T} \mu \bigl\Vert \nabla {u(s)} \bigr\Vert _{L^{2}}^{2}\,ds\leqslant \Vert u_{0} \Vert _{L^{2}}^{2}+ \frac{4}{ \mu \kappa } \Vert {\theta _{0}} \Vert _{L^{2}}^{2} \leqslant K_{1}+ \frac{4K_{2}}{ \mu \kappa }. \end{aligned}$$
(2.2)

Proof

Multiplying temperature equation (1.1)2 by θ integration by part and using divergence free property, we get

$$ \frac{1}{2}\frac{d}{dt} \bigl\Vert \theta (t) \bigr\Vert _{L^{2}}^{2}+ \kappa \bigl\Vert \nabla { \theta (s)} \bigr\Vert _{L^{2}}^{2} = 0, $$
(2.3)

integrating (2.3) over \((0,t)\), (2.1) is proved.

Multiplying momentum equation (1.1)1 by u and using the Poicaré inequality, we obtain

$$ \frac{1}{2}\frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+ \mu \Vert \nabla {u} \Vert _{L ^{2}}^{2}\leqslant \Vert u \Vert _{L^{2}} \Vert \theta \Vert _{L^{2}}\leqslant \frac{ \mu }{2} \Vert \nabla u \Vert _{L^{2}}^{2} + \frac{2}{\mu } \Vert \nabla \theta \Vert _{L^{2}}^{2}. $$

Then we have

$$ \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+\mu \Vert \nabla {u} \Vert _{L^{2}}^{2} \leqslant \frac{4}{\mu } \Vert \nabla \theta \Vert _{L^{2}}^{2}. $$
(2.4)

Integrating the above inequality over time \((0,t)\) and combining (2.1), one can get (2.2). □

Lemma 2.2

Under the conditions of Theorem 1.1, and suppose (1.5) is false. Then it holds that

$$\begin{aligned} &\sup_{0\leqslant t\leqslant T} \bigl( \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{T} \bigl( \bigl\Vert {u_{t}(s)} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla ^{2}{\theta (s)} \bigr\Vert _{L^{2}}^{2} \bigr)\,ds \leqslant C, \end{aligned}$$
(2.5)

where constantCdepends on the initial data, μ, κ, and some other known constants but is independent of T.

Proof

Multiplying momentum equation (1.1)1 by \(u_{t}\) and integrating by parts, we have

$$\begin{aligned} \frac{\mu }{2}\frac{d}{dt} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \Vert {u}_{t} \Vert _{L^{2}}^{2} &=- \int (u\cdot \nabla u )\cdot u_{t}\,dx+ \int \theta e_{3}\cdot u_{t}\,dx \\ &\leqslant \Vert {u} \Vert _{L^{6}} \Vert \nabla {u} \Vert _{L^{3}} \Vert {u}_{t} \Vert _{L^{2}}+ \Vert \theta \Vert _{L^{2}} \Vert {u}_{t} \Vert _{L^{2}} \\ &\leqslant C \bigl( \Vert \nabla {u} \Vert _{L^{2}}^{\frac{3}{2}} \Vert {u} \Vert _{H ^{2}}^{\frac{1}{2}} \Vert {u}_{t} \Vert _{L^{2}}+ \Vert \theta \Vert _{L^{2}} \Vert {u}_{t} \Vert _{L^{2}} \bigr). \end{aligned}$$
(2.6)

On the other hand, (1.1)1 can be rewritten as

$$ -\mu \Delta {u}+\nabla {P}=-u_{t}-u\cdot \nabla {u}+\theta {e_{3}}. $$

By the \(H^{2}\)-theory of Stokes system, we derive that

$$\begin{aligned} \mu \Vert {u} \Vert _{H^{2}}\leq C \Vert -u_{t}-u \cdot \nabla {u}+\theta {e_{3}} \Vert _{L^{2}} &\leqslant C \bigl( \Vert u_{t} \Vert _{L^{2}}+ \Vert \nabla {u} \Vert _{L^{2}} ^{\frac{3}{2}} \Vert {u} \Vert _{H^{2}}^{\frac{1}{2}}+ \Vert \theta \Vert _{L^{2}} \bigr), \end{aligned}$$

and thus

$$\begin{aligned} \Vert {u} \Vert _{H^{2}} &\leq \frac{C}{\mu } \bigl( \Vert u_{t} \Vert _{L^{2}}+ \Vert \nabla {u} \Vert _{L^{2}}^{3}+ \Vert \theta \Vert _{L^{2}} \bigr). \end{aligned}$$
(2.7)

Inserting (2.7) into (2.6), we obtain

$$\begin{aligned} &\frac{\mu }{2}\frac{d}{dt} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \Vert {u}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \frac{C}{\mu } \bigl( \Vert \nabla {u} \Vert _{L^{2}}^{\frac{3}{2}} \Vert {u}_{t} \Vert _{L^{2}}^{\frac{3}{2}}+ \Vert \nabla {u} \Vert _{L^{2}}^{3} \Vert {u}_{t} \Vert _{L^{2}}+ \Vert \nabla {u} \Vert _{L^{2}}^{\frac{3}{2}} \Vert \theta \Vert _{L^{2}} ^{\frac{1}{2}} \Vert {u}_{t} \Vert _{L^{2}} \bigr)+C \Vert \theta \Vert _{L^{2}} \Vert {u}_{t} \Vert _{L^{2}} \\ &\quad \leqslant \frac{1}{2} \Vert {u}_{t} \Vert _{L^{2}}^{2}+C \biggl(\frac{1}{\mu ^{4}} \Vert \nabla {u} \Vert _{L^{2}}^{6}+ \frac{1}{\mu ^{2}} \Vert \nabla {u} \Vert _{L ^{2}}^{6}+\frac{1}{\mu ^{2}} \Vert \nabla {u} \Vert _{L^{2}}^{3} \Vert \theta \Vert _{L^{2}}+ \Vert \theta \Vert _{L^{2}}^{2} \biggr) \\ &\quad \leqslant \frac{1}{2} \Vert {u}_{t} \Vert _{L^{2}}^{2}+C \biggl(\frac{1}{\mu ^{4}} \Vert \nabla {u} \Vert _{L^{2}}^{2}+ \frac{1}{\mu ^{2}} \Vert \nabla {u} \Vert _{L ^{2}}^{2}+ \Vert \nabla \theta \Vert _{L^{2}}^{2} \biggr). \end{aligned}$$
(2.8)

Then, integrating (2.8) over \((0, t)\), we have

$$\begin{aligned} \mu \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{t} \bigl\Vert {u_{t}(s)} \bigr\Vert _{L^{2}} ^{2}\,ds\leqslant C \biggl(1+ \frac{1}{\mu ^{5}}+ \frac{1}{\mu ^{3}}+\frac{1}{ \kappa } \biggr), \end{aligned}$$
(2.9)

where we have used Lemma 2.1 and constant C depending on \(K_{1}\) and \(K_{2}\).

Next, applying the operator on both sides of temperature equation (1.1)2 and multiplying by θ, integrating the resulting equation over Ω, we get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2}+ \kappa \bigl\Vert \nabla ^{2}{\theta } \bigr\Vert _{L^{2}}^{2} &= - \int \nabla (u\cdot \nabla \theta )\cdot \nabla \theta \,dx \\ &\leqslant \Vert \nabla {u} \Vert _{L^{3}} \Vert \nabla \theta \Vert _{L^{6}} \Vert \nabla \theta \Vert _{L^{2}} \\ &\leqslant C \Vert {u} \Vert _{H^{2}} \bigl\Vert \nabla ^{2}\theta \bigr\Vert _{L^{2}} \Vert \nabla \theta \Vert _{L^{2}} \\ &\leqslant \frac{\kappa }{2} \bigl\Vert \nabla ^{2}\theta \bigr\Vert _{L^{2}}^{2}+\frac{C}{ \kappa } \Vert {u} \Vert _{H^{2}}^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}, \end{aligned}$$
(2.10)

integrating (2.10) over time \((0,t)\) and using Gronwall’s inequality, which completes the proof of Lemma 2.2. □

Lemma 2.3

Under the conditions of Theorem 1.1, and suppose (1.5) is false. Then it holds that

$$\begin{aligned} &\sup_{0\leqslant t\leqslant T} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \theta _{t}(t) \bigr\Vert _{L^{2}}^{2} \bigr)+ \int _{0}^{T} \bigl( \bigl\Vert {\nabla u _{t}(s)} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \nabla {\theta _{t}(s)} \bigr\Vert _{L^{2}}^{2} \bigr)\,ds \leqslant C, \end{aligned}$$
(2.11)

where constantCdepends on the initial data, μ, κ, and some other known constants but is independent of T.

Proof

Taking the operator \(\partial _{t}\) to (1.1)1, multiplying the resulting equation by \(u_{t}\) and integrating by parts, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert u_{t}(t) \bigr\Vert _{L^{2}}^{2}+ \mu \Vert \nabla {u}_{t} \Vert _{L^{2}}^{2} &=- \int (u_{t}\cdot \nabla u )\cdot u_{t}\,dx+ \int \theta _{t} e_{3}\cdot u_{t} \,dx \\ &\leqslant \Vert \nabla {u} \Vert _{L^{3}} \Vert {u}_{t} \Vert _{L^{6}} \Vert {u}_{t} \Vert _{L ^{2}}+ \Vert \theta _{t} \Vert _{L^{2}} \Vert {u}_{t} \Vert _{L^{2}} \\ &\leqslant C \bigl( \Vert {u} \Vert _{H^{2}} \Vert \nabla {u}_{t} \Vert _{L^{2}} \Vert {u} _{t} \Vert _{L^{2}}+ \Vert \theta _{t} \Vert _{L^{2}} \Vert {u}_{t} \Vert _{L^{2}} \bigr) \\ &\leqslant \frac{\mu }{2} \Vert \nabla {u}_{t} \Vert _{L^{2}}^{2}+ \frac{C}{ \mu } \bigl( \Vert {u} \Vert _{H^{2}}^{2} \Vert {u}_{t} \Vert _{L^{2}}^{2}+ \Vert \theta _{t} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(2.12)

On the other hand, multiplying (1.1)2 by \(\theta _{t}\) and integrating the resulting equation over Ω, we obtain

$$\begin{aligned} \frac{\kappa }{2}\frac{d}{dt} \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2}+ \Vert \theta _{t} \Vert _{L^{2}}^{2} &=- \int (u\cdot \nabla \theta )\theta _{t}\,dx \leqslant \Vert {u} \Vert _{L^{\infty }} \Vert \nabla \theta \Vert _{L^{2}} \Vert \theta _{t} \Vert _{L^{2}} \\ &\leqslant \frac{1}{2} \Vert \theta _{t} \Vert _{L^{2}}^{2}+C \Vert {u} \Vert _{H^{2}} ^{2} \Vert \nabla \theta \Vert _{L^{2}}^{2}. \end{aligned}$$
(2.13)

Then, integrating (2.12), (2.13) over \((0, t)\) and using Gronwall’s inequality, we have

$$\begin{aligned} \mu \Vert u_{t} \Vert _{L^{2}}^{2}+ \kappa \Vert \nabla \theta \Vert _{L^{2}}^{2}+ \int _{0}^{t} \bigl(\mu ^{2} \bigl\Vert \nabla {u_{t}(s)} \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert {\theta _{t}(s)} \bigr\Vert _{L^{2}}^{2} \bigr)\,ds\leqslant C. \end{aligned}$$
(2.14)

Next, applying the operator \(\partial _{t}\) on both sides of temperature equation (1.1)2 and multiplying by \(\theta _{t}\), we get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl\Vert \theta _{t}(t) \bigr\Vert _{L^{2}}^{2}+ \kappa \Vert \nabla {\theta }_{t} \Vert _{L^{2}}^{2} &= - \int (u_{t}\cdot \nabla \theta +u\cdot \nabla \theta _{t} )\theta _{t} \,dx \\ &\leq \Vert {u}_{t} \Vert _{L^{3}} \Vert \nabla \theta \Vert _{L^{6}} \Vert \theta _{t} \Vert _{L^{2}} \\ &\leq \Vert {u}_{t} \Vert _{H^{1}}^{2}+C \bigl\Vert \nabla ^{2}\theta \bigr\Vert _{L^{2}}^{2} \Vert \theta _{t} \Vert _{L^{2}}^{2}, \end{aligned}$$
(2.15)

integrating (2.15) over time \((0,t)\), using Gronwall’s inequality, and taking (2.5), (2.14) into consideration, we obtain

$$ \bigl\Vert \theta _{t}(t) \bigr\Vert _{L^{2}}^{2}+ \int _{0}^{t} \bigl\Vert \nabla {\theta _{t}(s)} \bigr\Vert _{L^{2}}^{2}\,ds\leqslant C, $$
(2.16)

which completes the proof of Lemma 2.3. □

From Lemmas 2.12.3, we can see that Theorem 1.1 is proved.

Global strong solution

In this section, we prove the global existence and uniqueness of the strong solution. Assume that \(T^{*}>0\) is the maximal existence time of the strong solution. We prove \(T^{*}=+\infty \) by using contradiction arguments. If \(T^{*}<+\infty \), our aim is to prove that (1.5) is not true under the conditions of Theorem 1.2, which is the desired contradiction.

We define

$$ M(T)=\sup_{0\leq t\leq T} \Vert \nabla {u} \Vert _{L^{2}}^{2}. $$

Proposition 3.1

Assume that the initial data satisfy the conditions in Theorem 1.2, and the local strong solution satisfies

$$ M(T)\leqslant 2K_{1}, \quad (x,t)\in \varOmega \times [0,T], $$
(3.1)

then

$$ M(T)\leqslant \frac{3}{2}K_{1},\quad (x,t)\in \varOmega \times [0,T], $$
(3.2)

provided that there exists\(\varepsilon _{0}\)depending on\(K_{1}\), μ, κ, and some other known constants but independent ofTsuch that\(\Vert {{u}}_{0}\Vert _{L^{2}}^{2}+\Vert \theta _{0}\Vert _{L^{2}}^{2} \leqslant \varepsilon _{0}\)is suitably small.

Lemma 3.2

Under the condition of Proposition 3.1, there exists\(C^{*}\)depending on\(K_{1}\), μ, κ, and some other known constants but independent ofT, it holds that

$$\begin{aligned} M(T)\leqslant \frac{3}{2}K_{1}, \end{aligned}$$
(3.3)

provided that\(\varepsilon _{0} \leqslant C^{*}\).

Proof

Recalling Lemma 2.2, from (2.8) we can see that

$$\begin{aligned} &\frac{\mu }{2}\frac{d}{dt} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \Vert {u}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \frac{1}{2} \Vert {u}_{t} \Vert _{L^{2}}^{2}+C \biggl(\frac{1}{\mu ^{4}} \Vert \nabla {u} \Vert _{L^{2}}^{6}+ \frac{1}{\mu ^{2}} \Vert \nabla {u} \Vert _{L ^{2}}^{6}+\frac{1}{\mu ^{2}} \Vert \nabla {u} \Vert _{L^{2}}^{3} \Vert \theta \Vert _{L^{2}}+ \Vert \theta \Vert _{L^{2}}^{2} \biggr) \\ &\quad \leqslant \frac{1}{2} \Vert {u}_{t} \Vert _{L^{2}}^{2}+C \biggl(\frac{K_{1} ^{2}}{\mu ^{4}} \Vert \nabla {u} \Vert _{L^{2}}^{2}+ \frac{K_{1}^{2}}{\mu ^{2}} \Vert \nabla {u} \Vert _{L^{2}}^{2}+ \Vert \nabla \theta \Vert _{L^{2}}^{2} \biggr). \end{aligned}$$
(3.4)

Then, integrating (3.4) over \((0, t)\), we obtain

$$\begin{aligned} & \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \frac{1}{\mu } \int _{0}^{t} \bigl\Vert {u_{t}(s)} \bigr\Vert _{L^{2}}^{2}\,ds \\ &\quad \leqslant \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+C \biggl(\frac{K_{1}^{2}}{\mu ^{5}}+\frac{K_{1}^{2}}{\mu ^{3}} \biggr) \int _{0}^{t} \bigl\Vert \nabla {u}(s) \bigr\Vert _{L^{2}}^{2}\,ds+\frac{C}{\mu } \int _{0}^{t} \bigl\Vert \nabla \theta (s) \bigr\Vert _{L ^{2}}^{2}\,ds \\ &\quad \leqslant \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \frac{CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \biggl( \frac{1}{\mu } \Vert {u}_{0} \Vert _{L^{2}}^{2}+ \frac{1}{\mu ^{2}\kappa } \Vert \theta _{0} \Vert _{L^{2}}^{2} \biggr)+\frac{C}{ \mu \kappa } \Vert \theta _{0} \Vert _{L^{2}}^{2}, \end{aligned}$$
(3.5)

where we have used Lemma 2.1. Then we have

$$\begin{aligned} & \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2} \\ &\quad \leqslant \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \frac{CK_{1}^{2}}{\mu ^{4}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \Vert {u}_{0} \Vert _{L^{2}}^{2}+ \frac{CK _{1}^{2}}{\mu ^{5}\kappa } \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \Vert \theta _{0} \Vert _{L^{2}}^{2}+\frac{C}{\mu \kappa } \Vert \theta _{0} \Vert _{L^{2}}^{2} \\ &\quad \leqslant \Vert \nabla u_{0} \Vert _{L^{2}}^{2}+ \frac{CK_{1}^{2}}{\mu ^{4}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \Vert {u}_{0} \Vert _{L^{2}}^{2}+ \frac{C}{ \mu \kappa } \biggl( \frac{K_{1}^{2}}{\mu ^{6}}+ \frac{K_{1}^{2}}{\mu ^{4}}+1 \biggr) \Vert \theta _{0} \Vert _{L^{2}}^{2} \\ &\quad \leqslant K_{1}+\frac{1}{2}K_{1}= \frac{3}{2}K_{1}, \end{aligned}$$

provided \(C^{*}\leqslant \frac{1}{2}K_{1}\delta ^{-1}\), \(\delta = \max \Bigl\{ \frac{CK_{1}^{2}}{\mu ^{4}} \Bigl(\frac{1}{\mu ^{2}}+1 \Bigr), \frac{C}{\mu \kappa } \Bigl( \frac{K_{1}^{2}}{\mu ^{6}}+\frac{K_{1}^{2}}{\mu ^{4}}+1 \Bigr) \Bigr\} \). This completes the proof of Lemma 3.2. □

Hence, from Theorem 1.1 and Proposition 3.1, we can see that Theorem 1.2 is proved provided

$$ \Vert {{u}}_{0} \Vert _{L^{2}}^{2}+ \Vert \theta _{0} \Vert _{L^{2}}^{2} \leqslant \varepsilon _{0}\leqslant C^{*}. $$

Decay estimates

Finally, in this section, based on the global in time strong solution, we have the following exponential decay rate.

Lemma 4.1

Under the conditions of Theorem 1.2, we get that

$$ \int _{\varOmega } \bigl(\theta ^{2}+ \vert \nabla u \vert ^{2} \bigr)\leqslant {C _{1}}\exp \{-C_{2} t\} $$
(4.1)

holds for any\(t\in [0, +\infty )\), provided that

$$ \varepsilon _{0} \leqslant C^{*} $$

for some positive constants\(C_{1}\), \(C_{2}\), and\(C^{*}\)depending onμ, κ, \(K_{1}\), \(K_{2}\), and some other known constants but independent of t.

Proof

From (2.4) we can see that

$$ \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+\mu \Vert \nabla {u} \Vert _{L^{2}}^{2} \leqslant \frac{4}{\mu } \Vert \nabla \theta \Vert _{L^{2}}^{2}. $$
(4.2)

Multiplying \(\frac{2CK_{1}^{2}}{\mu ^{3}} \Bigl(\frac{1}{\mu ^{2}}+1 \Bigr)\) on both sides of inequality (4.2) and adding the resulting equation into (3.4), we obtain

$$\begin{aligned} &\frac{d}{dt} \biggl(\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+ \frac{\mu }{2} \bigl\Vert \nabla u(t) \bigr\Vert _{L ^{2}}^{2} \biggr) \\ &\quad\quad{} +\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl( \frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert \nabla {u(t)} \bigr\Vert _{L^{2}}^{2}+ \frac{1}{2} \Vert {u}_{t} \Vert _{L^{2}}^{2} \\ &\quad \leqslant C \biggl(\frac{8K_{1}^{2}(1+\mu ^{2})}{\mu ^{6}}+1 \biggr) \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2}. \end{aligned}$$
(4.3)

Next, multiplying \(2C \Bigl(\frac{8K_{1}^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+\frac{1}{\kappa } \Bigr)\) on both sides of inequality (2.3) and adding the resulting equation into (4.3), we have

$$\begin{aligned} \begin{aligned}[b] &\frac{d}{dt} \biggl(\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl( \frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+\frac{\mu }{2} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+2C \biggl(\frac{8K_{1}^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+ \frac{1}{\kappa } \biggr) \bigl\Vert \theta (t) \bigr\Vert _{L^{2}}^{2} \biggr) \\ &\quad{} +\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert \nabla {u(t)} \bigr\Vert _{L^{2}}^{2}+C \biggl( \frac{8K_{1}^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+\frac{1}{\kappa } \biggr) \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}} ^{2}\leqslant 0. \end{aligned} \end{aligned}$$
(4.4)

Taking

$$ Y(t)=\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert u(t) \bigr\Vert _{L^{2}}^{2}+\frac{\mu }{2} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+2C \biggl( \frac{8K _{1}^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+\frac{1}{\kappa } \biggr) \bigl\Vert \theta (t) \bigr\Vert _{L^{2}}^{2}, $$

it can be estimated as

$$\begin{aligned} Y(t) &\leqslant \frac{2CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+ \frac{\mu }{2} \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+2C \biggl(\frac{8K_{1}^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+\frac{1}{\kappa } \biggr) \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2} \\ &\leqslant \biggl(\frac{2CK_{1}^{2}(1+\mu ^{2})}{\mu ^{5}}+\frac{ \mu }{2} \biggr) \bigl\Vert \nabla u(t) \bigr\Vert _{L^{2}}^{2}+2C \biggl( \frac{8K_{1}^{2}(1+ \mu ^{2})}{\mu ^{6}\kappa }+\frac{1}{\kappa } \biggr) \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(4.5)

where we have used the Poincaré inequality.

Then there exists a constant M depending on C, \(K_{1}\), μ, κ such that

$$ Y(t)\leqslant M \biggl(\frac{2CK_{1}^{2}}{\mu ^{3}} \biggl(\frac{1}{\mu ^{2}}+1 \biggr) \bigl\Vert \nabla {u(t)} \bigr\Vert _{L^{2}}^{2}+C \biggl(\frac{8K_{1} ^{2}(1+\mu ^{2})}{\mu ^{6}\kappa }+ \frac{1}{\kappa } \biggr) \bigl\Vert \nabla \theta (t) \bigr\Vert _{L^{2}}^{2} \biggr). $$

At last, from (4.4), we have

$$ Y^{\prime }(t)+\frac{1}{M}Y(t)\leqslant 0, $$

which deduces \(Y(t)\leqslant Y(0)\exp \{ -\frac{t}{M} \} \). This completes the proof of Theorem 1.3. □

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Acknowledgements

The authors would like to thank the anonymous referees for helpful suggestions and comments which improved our original paper, and the authors would also like to thank Journal of Inequalities and Applications for considering this paper to be published.

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The authors’ research was supported in part by Chinese National Natural Science Foundation under grants 11571232 and 11831011.

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Shang, Z., Tang, F. Global existence and exponential decay of strong solutions for the three-dimensional Boussinesq equations. J Inequal Appl 2020, 50 (2020). https://doi.org/10.1186/s13660-020-02315-x

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Keywords

  • Boussinesq equations
  • Blow-up criterion
  • Global strong solutions
  • Exponential decay rate