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

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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}-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: θ tκ θ + u · ∇θ = 0, θ (x, 0) = θ 0 (x), (1.1) where u = (u 1 , u 2 , u 3 )(x, t), θ = θ (x, t), P(x, t) are unknown functions denoting fluid velocity vector field, absolute temperature, and scalar pressure, t ≥ 0 is time, x ∈ Ω 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 θ 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, 3-5, 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: Now, we state our main results as follows.
Remark 1.1 When θ = 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 Ω × [0, T] for any T > 0, provided that there exists a constant ε 0 > 0 such that where ε 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
for any t ∈ [0, +∞), provided that there exists a constant ε 0 > 0 such that where positive constants C 1 , C 2 , ε 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
Then the initial-boundary value problem (1.1)-(1.3) admits a unique global strong solution (u, θ ) on Ω × [0, T] for any T > 0, provided that there exists a constant μ * > 0 such that Furthermore, when μ and κ are large enough, the global strong solution satisfies the following exponential decay rate: for any t ∈ [0, +∞), where positive constants C 1 , C 2 , μ * 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 * = +∞, where the standard energy estimate is used and it holds uniform in time. In Sect. 3, we show that ∇u L ∞ (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
Proof Multiplying temperature equation (1.1) 2 by θ integration by part and using divergence free property, we get Then we have

5)
where constant C depends 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 On the other hand, (1.1) 1 can be rewritten as -μ u + ∇P = -u tu · ∇u + θ e 3 .
By the H 2 -theory of Stokes system, we derive that and thus Inserting (2.7) into (2.6), we obtain 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

11)
where constant C depends on the initial data, μ, κ, and some other known constants but is independent of T.
Proof Taking the operator ∂ t to (1.1) 1 , multiplying the resulting equation by u t and integrating by parts, we have On the other hand, multiplying (1.1) 2 by θ t and integrating the resulting equation over Ω, we obtain Then, integrating (2.12), (2.13) over (0, t) and using Gronwall's inequality, we have (2.14) Next, applying the operator ∂ t on both sides of temperature equation (1.1) 2 and multiplying by θ t , we get integrating (2.15) over time (0, t), using Gronwall's inequality, and taking (2.5), (2.14) into consideration, we obtain

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 * = +∞ by using contradiction arguments. If T * < +∞, 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 Proposition 3.1 Assume that the initial data satisfy the conditions in Theorem 1.2, and the local strong solution satisfies

2)
provided that there exists ε 0 depending on K 1 , μ, κ, and some other known constants but independent of T such that u 0 2 L 2 + θ 0 2 L 2 ≤ ε 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 of T, it holds that Proof Recalling Lemma 2.2, from (2.8) we can see that Then, integrating (3.4) over (0, t), we obtain where we have used Lemma 2.1. Then we have 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

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
holds for any t ∈ [0, +∞), provided that ε 0 ≤ C * for some positive constants C 1 , C 2 , and C * depending on μ, κ, K 1 , K 2 , and some other known constants but independent of t.
Proof From (2.4) we can see that At last, from (4.4), we have which deduces Y (t) ≤ Y (0) exp{-t M }. This completes the proof of Theorem 1.3.