Abstract
We consider the Boussinesq system in the homogeneous spaces of degree −1. To narrow the gap for the existence of small regular solutions in \(\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})\), the biggest homogeneous space of degree −1 among those embedded in the space of tempered distributions, we show small solutions in the homogeneous Besov space \(\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\), with \(n\geq2\), \(n\leq p<\infty\).
Similar content being viewed by others
1 Introduction and main results
The Cauchy problem of the Boussinesq system in \(\mathbb{R}^{n}\) (\(n\geq 2\)) reads
where \(u=u(x,t)\) and \(\theta=\theta(x,t)\) denote the unknown velocity field and the scalar temperature in the content of thermal convection, respectively, and \(\pi=\pi(x,t)\) the scalar density of the geophysical fluids, μ the constant kinematic viscosity, \(\kappa >0\) the thermal diffusivity, and \(e_{n}=(0,0,\ldots,1)^{T}\). While \(u_{0}\), \(\theta_{0}\) are given initial data, with \(\nabla\cdot u_{0}=0\) in the sense of distribution.
The Boussinesq system is extensively used in the atmospheric sciences and oceanographic turbulence (see [1–3] and references therein). The problem of the global regularity of the weak solutions of the 3D Boussinesq equations is a big open problem. It is meaningful to study the regularity of the weak solutions under additional critical growth conditions on the velocity or the pressure. Based on some analysis technique, there are some regularity criteria via the velocity of weak solutions in Besov spaces have been obtained in [4–6]. The pressure criterion is in [7–9]. By the velocity criterion, for the n-dimensional Boussinesq system, Yao et al. in [10] showed the local well-posedness and blow-up criteria in Besov-Morrey spaces \(N_{p,q,r}^{s}(\mathbb{R}^{n})\) in supercritical case \(s > 1 + \frac{n}{p}\), \(1 < q \leq p < \infty\), \(1 \leq r\leq\infty\), and critical case \(s=1+\frac{n}{p}\), \(1 < q \leq p < \infty\), \(r=1\). Zhang et al. [11] got the existence of the 2-dimensional inviscid Boussinesq equations in critical Besov spaces \(B^{\frac{2}{p}+1}_{p,1}(\mathbb{R}^{2})\) and some blow-up criteria.
The global regularity of smooth solution of the 2D Boussinesq equations with the fractional dissipation has been researched recently in [12–18]. Several Beale-Kato-Majada-type regularity criterion have been obtained in [19–25]. There are also some results for the blow-up criteria for the Boussinesq equations (see [26, 27] and the references therein).
For the Navier-Stokes equations, Xin and Chen in [28] researched the small regular solutions in \(\dot{B}^{-1}_{\infty,\infty}\), which is the biggest homogeneous space of degree −1. The authors studied small solutions in the homogeneous Besov space \(\dot{B}^{-1+\frac {n}{p}}_{p,\infty}(\mathbb{R}^{n})\) and a homogeneous space defined by \(\widehat{M}_{n}(\mathbb{R}^{n})\). Here, motivated by the results in [28], our aim is to do some work addressing the small regular solutions of the n-dimensional Boussinesq system in subcritical spaces \(\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\). The corresponding content in the space \(\widehat{M}_{n}(\mathbb{R}^{n})\) is of our further interest. This result partially extends the result in [28] in another system. More precisely, we will prove the following.
Theorem 1.1
Suppose \(n\geq2\), \(n\leq p<\infty\), \(2-\frac{n}{p}<\alpha<2\), \(u_{0}\in\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\), \(\theta _{0}\in\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\cap\dot {B}^{-3+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\) with \(\operatorname{div} u_{0}=0\) in \(\mathbb{R}^{n}\), and \(\Vert u_{0} \Vert _{\dot{B}^{-1+\frac {n}{p}}_{p,\infty}}+ \Vert \theta_{0} \Vert _{\dot{B}^{-1+\frac {n}{p}}_{p,\infty}}+ \Vert \theta_{0} \Vert _{\dot{B}^{-3+\frac {n}{p}}_{p,\infty}}\leq\epsilon\) for some small constant \(\epsilon =\epsilon(n,p,\alpha)\). Then system (1.1) admits a unique regular solution satisfying
This paper is structured as follows. In Section 2, we introduce the Besov spaces and the lemmas used later. In Section 3, we provide the proof of Theorem 1.1.
2 Preliminary lemmas
We denote (cf. [28])
with the heat kernel \(h(x,t)=(4\pi t)^{-n/2}e^{- \vert x \vert ^{2}/(4t)}\). And the Fourier transform f̂ of \(f\in\mathcal {S}\) is defined by
Here \(\mathcal{S}(\mathbb{R}^{n})\) stands for the Schwartz class of rapidly decreasing smooth functions and \(\mathcal{S}'(\mathbb{R}^{n})\) is the space of tempered distributions. The fractional order of the Laplacian is showed by the Fourier transform. For \(\alpha\in\mathbb{R}\),
Due to the homogeneous counterpart Theorem 2.12.2 and the lifting property Theorem 5.2.3/1 in [29], the homogeneous Besov spaces can be given as follows.
Definition 2.1
For \(1\leq p,q\leq\infty\) and \(-\infty<\alpha<\infty\), the homogeneous Besov spaces are defined
where
An important property of the homogeneous spaces is its invariance under the following space scaling:
And furthermore, if \(s<0\), the homogeneous Besov spaces \(\dot {B}^{s}_{p,q}(\mathbb{R}^{n})\) can be equivalently defined as follows (cf. [30]).
Lemma 2.1
Suppose \(1\leq p,q\leq\infty\), \(s<0\). Then \(f\in\dot {B}^{s}_{p,q}(\mathbb{R}^{n})\) if and only if
In particular, for the degree of −1, we have
It is well known that \(\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})\) is the biggest critical homogeneous space of degree −1, and as shown by Frazier, Jaweth and Weiss [31], any critical homogeneous space continuously embedded in \(\mathcal{S}'(\mathbb{R}^{n})\) is also continuously embedded into \(\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})\).
Next we introduce the interpolation theorem in [29, 32].
Lemma 2.2
Suppose \(0<\theta<1\), \(1\leq p,q\leq\infty\), \(-\infty<\alpha<\beta <\infty\), and
Then
where \((\cdot,\cdot)_{\theta,q}\) denotes the real interpolation functor.
We express (1.1)1,2 in the integral form as
where \(\mathbb{P}\) is the Helmholtz-Weyl projection onto a divergence free vector fields defined by
here \(\delta_{j,k}\) is the Kronecker symbol and \(R_{j}=\partial _{j}(-\Delta)^{-\frac{1}{2}}\) are the Riesz transforms. To prove the existence of the regular solution in \(L^{\infty }((0,\infty),L^{n}(\mathbb{R}^{n}))\), we need the \(L^{p}-L^{q}\) type estimate for \(e^{t\Delta}\) in Lebesgue spaces and Besov spaces. See [30] for the proof of the following lemma.
Lemma 2.3
Suppose \(1\leq p,q\leq\infty\). Thus, the following estimates hold:
If \(s_{1},s_{2}<\frac{n}{p}\) and \(s_{1}+s_{2}+n\min\{0,1-\frac{2}{p}\}>0\), then, for a positive constant C, we have
Due to the linearization of the Boussinesq system (1.1), we consider a priori estimates for the Stokes equations. Chen and Xin in [28] gave the estimates in homogeneous spaces. The Stokes equations in \(\mathbb{R}^{n}\), \(n\geq2\) read
We now state the main estimate about the Stokes equations in homogeneous spaces of Chen and Xin in [28], which will be used later.
Lemma 2.4
Let \(0<\alpha<2\), \(2\leq n\leq p\leq\infty\), \(1\leq q\leq\infty\), and
Then the solution of equation (2.1) in the following integral formulation:
satisfies the estimates
provided that the right-hand sides of the above inequalities are finite, respectively.
Similar to Lemma 2.4, we have the analogous results.
Lemma 2.5
Under the same conditions with α, p, q in Lemma 2.4, and \(f\in\dot{B}_{p,\infty}^{\alpha-3+\frac{n}{p}}\), the following integral equation:
satisfies the estimation
Proof
By Definition 2.1, we estimate the term \(\Vert \int_{0}^{t} e^{(t-s)\Delta}\mathbb{P}f(s)\,ds \Vert _{\dot{B}^{\alpha-1+\frac {n}{p}}_{p,\infty}(\mathbb{R}^{n})}\) as follows:
For the term \({I=\int_{0}^{\frac{t}{2}}(\tau +t-s)^{-2+(\alpha-1+\frac{n}{p})/2}s^{-\frac{\alpha}{2}}\,ds}\), due to \(0< s<\frac{t}{2}\), thus
and the fact that \(-2+(\alpha-1+\frac{n}{p})/2<0\), we have
As regards the term \(\mathit{II}=\int_{\frac{t}{2}}^{t}(\tau +t-s)^{-2+(\alpha-1+\frac{n}{p})/2}s^{-\frac{\alpha}{2}}\,ds\), since \(\frac{t}{2}< s< t\),
therefore
Thereby, the estimate becomes
Due to the definite integral \(\int_{0}^{t} s^{-\frac{\alpha}{2}}\,ds=\frac {2}{2-\alpha}t^{1-\frac{\alpha}{2}}\), we obtain
According to the fact that
thus
Finally, we get
For the estimate of the initial term \(\Vert e^{t\Delta}\mathbb {P}a \Vert _{\dot{B}^{\alpha-1+\frac{n}{p}}_{p,\infty}}\), we have
because of the inequality
then we obtain
Equations (2.3) and (2.4) imply the second inequality of the lemma. Now we turn to the first inequality. By a similar method, we have
On the other hand, by Lemma 2.3, we get
Therefore, the expressions (2.5) and (2.6) verify the first inequality of the lemma. □
Lemma 2.6
Suppose \(2-\frac{n}{p}<\alpha<2\), \(2\leq n\leq p\leq\infty\), and
Thus equation (2.2) satisfies the estimate
Proof
By the definition in Lemma 2.1 and Lemma 2.3, we have
the way to deal with the above estimate is similar to the one in Lemma 2.4, we omit it here. Finally, we get
For the initial term, according to Lemma 2.3, we obtain
Therefore, we get the proof. □
3 The proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. For simplicity, without loss of generality, we assume \(\mu=\kappa=1\). In order to deal with the convection terms, we observe the interesting interpolation results
It is well known that, for \(s>0\), \(\dot{B}^{s}_{p,q}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})\) is an algebra. Moreover, there exists the following embedding relations (cf. [28, 29]):
Thus by the embedding and interpolation results, we have
and
And then due to the inequality \(a^{1-\beta}b^{\beta}\leq a+b\), \(0<\beta<1\), \(a,b>0\), we have
By a variable transformation, (3.1) implies
Equation (3.2) means
Thus to prove the two equations in Theorem 1.1, we only need to show that
Set
with
and
We will show M and M̄ are the contraction operators mapping a ball of U into itself and a ball of Θ into itself, respectively. Now we deal with the operator M̄. Observing that (3.1) and (3.2) are also true for θ, we have
And using Lemma 2.4, we obtain the estimates of θ. It is more simple for θ here, as the absence of the operator \(\mathbb{P}\), but the estimates are still true, which are
Summing (3.3) and (3.4) up, we have
Define two complete metric spaces by
Suppose \(\theta_{1},\theta_{2}\in\Theta\), by (3.5), we have
According to the contraction mapping principle, equation (2.1) admits a unique solution \(\theta\in\Theta_{\epsilon}\), provided that \(C \Vert \theta_{0} \Vert _{\dot{B}^{-1+\frac {n}{p}}_{p,\infty}}\leq\frac{\epsilon}{2}\) and \(\epsilon>0\) is sufficiently small.
Remark 3.1
To show the boundedness of the term \({\sup_{s>0}s^{\frac {\alpha}{2}} \Vert \theta \Vert _{\dot{B}^{\alpha-3+\frac{n}{p}}_{p,\infty }}}\), we choose \(f(s)=u(s)\otimes\theta(s)\) in Lemma 2.6, and by Lemma 2.3, we have
thus we get
that is to say, if \(\Vert u \Vert _{U}\) is small enough, the term \({\sup_{s>0}s^{\frac{\alpha}{2}} \Vert \theta(s) \Vert _{\dot{B}^{\alpha-3+\frac{n}{p}}_{p,\infty}}}\) is bounded.
Now we deal with the operator M. Choosing \(f(s)=u(s)\otimes u(s)\) in Lemma 2.4 and \(f(s)=\theta(s)e_{n}\) in Lemma 2.5, respectively, and using (3.2), we have
Adding (3.6) to (3.7), and by the definition of U, we have
Due to Remark 3.1, we know \({C\sup_{0< s< t}s^{\frac{\alpha}{2}} \Vert \theta(s) \Vert _{\dot {B}^{\alpha-3+\frac{n}{p}}_{p,\infty}}}\) is bounded. Similarly, by (3.8), we get
that is, due to \(\mathbb{P}Mu(t)=Mu(t)\), we obtain \(\nabla\cdot Mu(t)=0\). By the definition of \(U_{\epsilon}\), the previous analysis shows that
Therefore, on the basis of the contraction mapping principle, equation (2.1) admits a unique solution \(u\in U_{\epsilon}\), provided that \(C \Vert u_{0} \Vert _{\dot{B}^{-1+\frac {n}{p}}_{p,\infty}}\in\frac{\epsilon}{3}\), \(C \Vert \theta_{0} \Vert _{\dot{B}^{-3+\frac{n}{p}}_{p,\infty}}\in\frac{\epsilon }{3}\) and \(\epsilon>0\) is sufficiently small.
The proof of Theorem 1.1 is done.
References
Majda, A: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematic, vol. 9. Am. Math. Soc., Providence (2003)
Pedlosky, J: Geophysical Fluid Dynamics. Springer, New York (1987)
Shahmurov, R, Hadji, L: Nonlinear stable steady solutions to the Ostroumov problem. Int. J. Heat Mass Transf. 82, 604-612 (2015)
Zhang, Z: Some regularity criteria for the 3D Boussinesq equations in the class \(L^{2}(0, T; \dot{B}^{-1}_{\infty,\infty})\). ISRN Appl. Math. 2014, Article ID 564-758 (2014). doi:10.1155/2014/564758
Xu, F, Zhang, Q, Zheng, X: Regularity criteria of the 3D Boussinesq equations in the Morrey-Campanato space. Acta Appl. Math. 121, 231-240 (2012)
Fan, J, Zhou, Y: A note on regularity criterion for the 3D Boussinesq system with partial viscosity. Appl. Math. Lett. 22(5), 802-805 (2009)
Mechdene, M, Gala, S, Guo, ZG, Ragusa, AM: Logarithmical regularity criterion of the three-dimensional Boussinesq equations in terms of the pressure. Z. Angew. Math. Phys. 67(5), 67-120 (2016). doi:10.1007/s00033-016-0715-2
Jia, Y, Zhang, X, Dong, B: Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the pressure. Abstr. Appl. Anal. 2012, Article ID 395420 (2012). doi:10.1155/2012/395420
Zhang, Z: A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure. Acta Appl. Math. 131, 213-219 (2014)
Yao, ZA, Wang, Q, Bie, Q: On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinet. Relat. Models 8(3), 395-411 (2015)
Liu, X, Wang, M, Zhang, Z: Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces. J. Math. Fluid Mech. 12, 280-292 (2010). doi:10.1007/s00021-008-0286-x
Jiu, Q, Miao, C, Wu, J, Zhang, Z: The 2D incompressible Boussinesq equations with general critical dissipation. SIAM J. Math. Anal. 46, 3426-3454 (2014)
Stefanov, A, Wu, J: A global regularity result for the 2D Boussinesq equations with critical dissipation. J. Anal. Math., accepted for publication; arXiv:1411.1362v3 [math.AP]
Wu, J, Xu, X, Xue, L, Ye, Z: Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation. Commun. Math. Sci. 14, 1963-1997 (2016)
Ye, Z: Global smooth solution to the 2D Boussinesq equations with fractional dissipation. arXiv:1510.03237v2 [math.AP]
Ye, Z, Xu, X: Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation. J. Differ. Equ. 260, 6716-6744 (2016)
Abidi, H, Hmidi, T: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233(1), 199-220 (2007)
Hou, T, Li, C: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1-12 (2005)
Ye, Z: Blow-up criterion of smooth solutions for the Boussinesq equations. Nonlinear Anal. 110, 97-103 (2014)
Ye, Z: Regularity criteria for 3D Boussinesq equations with zero thermal diffusion. Electron. J. Differ. Equ. 2015, 97 (2015)
Ye, Z: A logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesq equations. Osaka J. Math. 53, 417-423 (2016)
Yang, X, Qin, Y: A Beale-Kato-Majda criterion for the 3D viscous magnetohydrodynamic equations. Math. Methods Appl. Sci. 38(4), 701-707 (2015)
Yang, X, Zhang, L: BKM’s criterion of weak solutions for the 3D Boussinesq equations. J. Partial Differ. Equ. 27(1), 64-73 (2014)
Tian, C, Yang, X: A Beale-Kato-Majda regularity criteria to the 2D viscous MHD equations in BMO space. Int. J. Contemp. Math. Sci. 8(1-4), 117-123 (2013)
Gala, S, Guo, Z, Ragusa, A: A remark on the regularity criterion of Boussinesq equations with zero heat conductivity. Appl. Math. Lett. 27, 70-73 (2014)
Qin, Y, Yang, X, Wang, Y, Liu, X: Blow-up criteria of smooth solutions to the 3D Boussinesq equations. Math. Methods Appl. Sci. 35(3), 278-285 (2012)
Cui, X, Dou, C, Jiu, Q: Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces. J. Partial Differ. Equ. 25(3), 220-238 (2012)
Chen, ZM, Xin, ZP: Homogeneity criterion for the Navier-Stokes equations in the whole spaces. J. Math. Fluid Mech. 3, 152-182 (2001)
Triebel, H: Theory of Functions Spaces II. Birkhäuser, Basel (1992)
Miao, C, Yuan, B, Zhang, B: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. 68, 461-484 (2008)
Frazier, M, Jawerth, B, Weiss, G: Littlewood-Paley Theory and the Study of Function Spaces. Am. Math. Soc., New Jork (1991)
Bergh, J, Löfström, J: Interpolation Spaces: An Introduction. Grundle der Mathematischen Wissenschaften, vol. 233. Springer, Berlin (1976)
Acknowledgements
This research is supported by the Foundation for Ph.D. of Henan Normal University of China under Grant 5101019170156.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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.
About this article
Cite this article
Cui, X. The regularity criterion for weak solutions to the n-dimensional Boussinesq system. Bound Value Probl 2017, 44 (2017). https://doi.org/10.1186/s13661-017-0778-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-017-0778-9