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Global existence and exponential decay of strong solutions for the three-dimensional Boussinesq equations
Journal of Inequalities and Applications volumeĀ 2020, ArticleĀ number:Ā 50 (2020)
Abstract
In this paper, we consider the global existence of strong solutions to the three-dimensional Boussinesq equations on the smooth bounded domainĀ Ī©. 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.
1 Introduction
In this paper, we consider the following three-dimensional incompressible Boussinesq equations in the Eulerian coordinates:
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. Ī¼ is the kinematic viscosity, Īŗ 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, 3ā5, 10, 17] and the references therein.
Inspired by the results of full compressible NavierāStokes 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.
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 Ī©. Let\(({u},\theta )\)be the strong solution to the initial-boundary value problem (1.1)ā(1.3) on\(\varOmega \times (0,T] \)satisfying
If\(T^{*}<+\infty \)is the maximal time of existence for the strong solution\((u,\theta )\), then
Theorem 1.2
(Global strong solution)
For any given\(K_{i}>0\) (\(i=1,2\)), suppose that the initial data satisfy\((u_{0},\theta _{0})\in H_{0}^{1}\cap H^{2}\), \(\operatorname{div}u_{0}=0\)inĪ©, and
Then the initial-boundary value problem (1.1)ā(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
where\(\varepsilon _{0}\)depends on\(K_{1}\), Ī¼, Īŗ, and some other known constants but is independent of T.
Remark 1.1
When \(\theta =0\), system (1.1) reduces to NavierāStokes equations. Then Theorem 1.2 implies that NavierāStokes 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
where \(\varepsilon _{0}\) depends on \(K_{1}\), Ī¼ 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\(t\in [0, +\infty )\), provided that there exists a constant\(\varepsilon _{0}>0\)such that
where positive constants\({C}_{1}\), \({C}_{2}\), \(\varepsilon _{0}\)depend on\(K_{1}\), \(K_{2}\), Ī¼, Īŗ, 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 āsmallnessā 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\((u_{0},\theta _{0})\in H_{0}^{1}\cap H^{2}\), \(\operatorname{div}u_{0}=0\)inĪ©, and
Then the initial-boundary value problem (1.1)ā(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
Furthermore, whenĪ¼andĪŗare large enough, the global strong solution satisfies the following exponential decay rate:
for any\(t\in [0, +\infty )\), where positive constants\({C}_{1}\), \({C}_{2}\), \(\mu ^{*}\)depend on\(K_{1}\), \(K_{2}\), Ī¼, Īŗ, 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.
2 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
Proof
Multiplying temperature equation (1.1)2 by Īø integration by part and using divergence free property, we get
integrating (2.3) over \((0,t)\), (2.1) is proved.
Multiplying momentum equation (1.1)1 by u and using the PoicarƩ inequality, we obtain
Then we have
Integrating the above inequality over time \((0,t)\) and combining (2.1), one can get (2.2).āā”
Lemma 2.2
Under the conditions of Theorem 1.1, and suppose (1.5) is false. Then it holds that
where constantCdepends on the initial data, Ī¼, Īŗ, 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
By the \(H^{2}\)-theory of Stokes system, we derive that
and thus
Inserting (2.7) into (2.6), we obtain
Then, integrating (2.8) over \((0, t)\), we have
where we have used Lemma 2.1 and constant C depending on \(K_{1}\) and \(K_{2}\).
Next, applying the operator ā on both sides of temperature equation (1.1)2 and multiplying by āĪø, integrating the resulting equation over Ī©, we get
integrating (2.10) over time \((0,t)\) and using Gronwallās inequality, which completes the proof of Lemma 2.2.āā”
Lemma 2.3
Under the conditions of Theorem 1.1, and suppose (1.5) is false. Then it holds that
where constantCdepends on the initial data, Ī¼, Īŗ, 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
On the other hand, multiplying (1.1)2 by \(\theta _{t}\) and integrating the resulting equation over Ī©, we obtain
Then, integrating (2.12), (2.13) over \((0, t)\) and using Gronwallās inequality, we have
Next, applying the operator \(\partial _{t}\) on both sides of temperature equation (1.1)2 and multiplying by \(\theta _{t}\), we get
integrating (2.15) over time \((0,t)\), using Gronwallās inequality, and taking (2.5), (2.14) into consideration, we obtain
which completes the proof of Lemma 2.3.āā”
From Lemmas 2.1ā2.3, we can see that Theorem 1.1 is proved.
3 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
Proposition 3.1
Assume that the initial data satisfy the conditions in Theorem 1.2, and the local strong solution satisfies
then
provided that there exists\(\varepsilon _{0}\)depending on\(K_{1}\), Ī¼, Īŗ, 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\(K_{1}\), Ī¼, Īŗ, and some other known constants but independent ofT, it holds that
provided that\(\varepsilon _{0} \leqslant C^{*}\).
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
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.āā”
Hence, from Theorem 1.1 and Proposition 3.1, we can see that Theorem 1.2 is proved provided
4 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\(t\in [0, +\infty )\), provided that
for some positive constants\(C_{1}\), \(C_{2}\), and\(C^{*}\)depending onĪ¼, Īŗ, \(K_{1}\), \(K_{2}\), and some other known constants but independent of t.
Proof
From (2.4) we can see that
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
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
Taking
it can be estimated as
where we have used the PoincarƩ inequality.
Then there exists a constant M depending on C, \(K_{1}\), Ī¼, Īŗ such that
At last, from (4.4), we have
which deduces \(Y(t)\leqslant Y(0)\exp \{ -\frac{t}{M} \} \). This completes the proof of Theorem 1.3.āā”
References
Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J.Ā Differ. Equ. 233(1), 199ā220 (2007)
Brandolese, L., Schonbek, M.E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364(10), 5057ā5090 (2012)
Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203(2), 497ā513 (2006)
Chae, D., Kim, S.-K., Nam, H.-S.: Local existence and blow-up criterion of Hƶlder continuous solutions of the Boussinesq equations. Nagoya Math.Ā J. 155, 55ā80 (1999)
Chae, D., Nam, H.-S.: Local existence and blow-up criterion for the Boussinesq equations. Proc. R. Soc. Edinb., Sect.Ā A 127(5), 935ā946 (1997)
Danchin, R., Paicu, M.: Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces. PhysicaĀ D 237(10ā12), 1444ā1460 (2008)
Fan, J., Li, F.: Global strong solutions to the nonhomogeneous incompressible MHD equations in a bounded domain. Nonlinear Anal., Real World Appl. 46, 1ā11 (2019)
Liu, X., Li, Y.: On the stability of global solutions to the 3D Boussinesq system. Nonlinear Anal. 95, 580ā591 (2014)
Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol.Ā 9. New York University, Courant Institute of Mathematical Sciences, New York (2003)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, vol.Ā 27. Cambridge University Press, Cambridge (2002)
Qin, Y., Wang, Y., Su, X., Zhang, J.: Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data. Discrete Contin. Dyn. Syst. 36(3), 1563ā1581 (2016)
Sawada, O., Taniuchi, Y.: On the Boussinesq flow with nondecaying initial data. Funkc. Ekvacioj 47(2), 225ā250 (2004)
Wen, H., Zhu, C.: Global solutions to the three-dimensional full compressible NavierāStokes equations with vacuum at infinity in some classes of large data. SIAM J. Math. Anal. 49(1), 162ā221 (2017)
Wen, Z., Ye, Z.: On the global existence of strong solution to the 3D damped Boussinesq equations with zero thermal diffusion. Z.Ā Anal. Anwend. 37(3), 341ā348 (2018)
Ye, Z.: Global regularity for a 3D Boussinesq model without thermal diffusion. Z.Ā Angew. Math. Phys. 68(4), Article ID 83 (2017)
Ye, Z.: On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation. Appl. Math. Lett. 90, 1ā7 (2019)
Ye, Z.: An alternative approach to global regularity for the 2d EulerāBoussinesq equations with critical dissipation. Nonlinear Anal. 190, 111591 (2020)
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 authorsā 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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DOI: https://doi.org/10.1186/s13660-020-02315-x