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A remark on the regularity criterion of Boussinesq equations with zero heat conductivity
Accepted Manuscript A remark on regularity criterion of the Boussinesq equations with zero heat conductivity Sadek Gala, Zhengguang Guo, Maria Alessandra Ragusa PII: DOI: Reference:
S0893-9659(13)00231-0 http://dx.doi.org/10.1016/j.aml.2013.08.002 AML 4422
To appear in:
Applied Mathematics Letters
Received date: 3 June 2013 Revised date: 12 August 2013 Accepted date: 13 August 2013 Please cite this article as: S. Gala, Z. Guo, M.A. Ragusa, A remark on regularity criterion of the Boussinesq equations with zero heat conductivity, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A remark on regularity criterion of the Boussinesq equations with zero heat conductivity Sadek Gala Department of Mathematics, University of Mostaganem Box 227, Mostaganem 27000, Algeria [email protected]
Zhengguang Guo∗ College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, P. R. China [email protected] and
Maria Alessandra Ragusa Dipartimento di Mathematica e Informatica Universit a di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy.
Abstract: In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by [9]. Mathematics Subject Classification(2000): 35Q35; 76D03 Key words: Regularity criterion; Boussinesq equations with zero heat conductivity ∗
Author to whom correspondence should be addressed
1
1
Introduction and main result
Consider the Boussinesq system with zero heat conductivity in R3 : ∂t u + u · ∇u − ∆u + ∇π = θe3 , for (x, t) ∈ R3 × (0, ∞), ∂ θ + u · ∇θ = 0, for (x, t) ∈ R3 × (0, ∞), t ∇ · u = 0, for (x, t) ∈ R3 × (0, ∞), u(x, 0) = u0 (x), θ(x, 0) = θ0 (x), for x ∈ R3 ,
(1.1)
where u = u(x, t) and θ = θ(x, t) denote the unknown velocity vector field and the scalar temperature, while u0 , θ0 with ∇ · u0 = 0 in the sense of distribution are given initial data. e3 = (0, 0, 1)T . π = π(x, t) the pressure of fluid at the point (x, t) ∈ R3 × (0, ∞). The Boussinesq system has important roles in the atmospheric sciences (See e.g. [12]). Existence and uniqueness theories of solutions to the Boussinesq equations have been studied by many mathematicians and physicists. For 2D Boussinesq problem, the global in time regularity is well-known in [3]. Chae in [2] showed that the 2D Boussinesq system also has a unique smooth global in time solution with zero viscosity. On the other hand, for 3D Boussinesq problem, Fan and Ozawa [5] and Ishimura and Morimoto [14] proved the following regularity criterion, respectively: ³ ´ . 0 u ∈ L2 0, T ; B ∞,∞ (R3 ) , (1.2) ¡ ¢ ∇u ∈ L1 0, T ; L∞ (R3 ) Very recently, Geng and Fan in [9] (see also [11]) established the following criterion for 3D Boussinesq system: ³ ´ . −r 2 u ∈ L 1−r 0, T ; B ∞,∞ (R3 ) with − 1 < r < 1 and r 6= 0, (1.3) . s
where B ∞,∞ denotes the homogeneous Besov space. For the regularity problem on other very related models, we refer readers to the investigations [4, 6, 7, 17]. Motivated by the result in [9], our aim is to consider the limit case r = 1 in (1.3) and we establish a Serrin-type regularity criterion for weak solutions in . −r 2 terms of the velocity in the class L 1−r (0, T ; B ∞,∞ (R3 )), which greatly improves the result in [9]. More precisely, we will prove Theorem 1.1 Let (u0 , θ0 ) ∈ H s (R3 ) with div u0 = 0 in R3 and s > 52 . Let (u, θ, π) be a local smooth solution to (1.1). If there exists a small positive constant 2
δ such that µ ¶ . −1 L∞ 0,T,B ∞,∞ (R3 )
kuk
≤ δ,
(1.4)
then the solution (u, θ, π) to the problem (1.1) remains smooth on [0, T ]. Remark 1.1 This result says that the velocity field of the fluids plays a more dominant role than the temperature θ in the regularity theory of the system (1.1). So our theorem is a complement and improvement of the previous results. .
. −1
Since the critical Morrey-Campanato space M2,3 ⊂ B ∞,∞ (see e.g. [8, 10]), an immediate corollary is as follows. Corollary 1.2 Let (u0 , θ0 ) ∈ H s (R3 ) with div u0 = 0 in R3 and s > 52 . Let (u, θ, π) be a local smooth solution to (1.1). If there exists a small positive constant δ such that ´ ≤ δ, . kukL∞ ³0,T,M (1.5) (R3 ) 2,3
then the solution (u, θ, π) to the problem (1.1) remains smooth on [0, T ].
Remark 1.2 Corollary 1.2 extends previous results and covers the supercritical case [7, 15]. Next we recall the definition of the homogeneous Besov space (see e.g. [1, 16]). Let et∆ denote the heat semi-group defined by à ! 2 3 |x| et∆ f = Kt ∗ f, Kt (x) = (4πt)− 2 exp − 4t for t > 0 and x ∈ R3 , where ∗ means convolution of functions defined on R3 . We now recall the definition of the homogeneous Besov space with negative . −α indices B ∞,∞ on R3 with α > 0. It is known ([16], page 192) that f ∈ S 0 (R3 ) . −α ° α ° belongs to B ∞,∞ (R3 ) if and only if et∆ ∈ L∞ for all t > 0 and t 2 °et∆ f ° ∈ . −α
L∞ (0, ∞). The norm of B ∞,∞ is defined, up to equivalence, by ° ¢ ¡ α° kf k . −α = sup t 2 °et∆ f °∞ . B ∞,∞
∞
t>0
Here S 0 is the dual of Schwartz space (tempered distribution). The crucial tool in this note is the following lemma which is essentially due to Meyer-Gerard-Oru [13], which plays an important role for the proof of our theorem. 3
¡ ¢ Lemma 1.3 Let 2 < q < ∞ and s = α 2q − 1 > 0. Then there exists a constant C depending only on α and q such that the estimate 1− 2
2
kf kLq ≤ C kf k q. s kf k . −αq H
. s
. −α
B ∞,∞
(1.6)
. s
holds for all f ∈ H (R3 )∩ B ∞,∞ (R3 ), where H denotes the homogeneous Sobolev space. We omit the proof here, the proof can be found in [8, 10]. In this paper the letter C denotes a constant which may vary in different case. Now we are in a position to prove Theorem 1.1. Proof: Let T > 0 be a given fixed time. We assume that u satisfies (1.4). Multiplying the second equation of (1.1) by θ and integrating over R3 , we get immediately 1d kθk2L2 = 0. 2 dt Hence ¡ ¢ θ ∈ L∞ 0, T ; L2 (R3 ) . (1.7) Next, multiplying both sides of the first equation of (1.1) by u, we have after integration by part, Z 1d 2 2 kukL2 + k∇ukL2 = (θe3 ) · udx ≤ kθkL2 kukL2 2 dt R3
≤ C kukL2 , which yields
¡ ¢ ¡ ¢ u ∈ L∞ 0, T ; L2 (R3 ) ∩ L2 0, T ; H 1 (R3 ) ,
where we used (1.7) and Z Z Z 1 1 2 (u · ∇u) · udx = (u · ∇) u dx = − (∇ · u) u2 dx = 0 2 2 R3
R3
R3
by incompressibility of u, that is, ∇ · u = 0. Next, multiplying both sides of (1.1)1 by −∆u and integrating it over R3 , we
4
find, after integration by part, 1d k∇uk2L2 + k∆uk2L2 2 dt Z Z → − = − (θ e 3 ) · ∆udx + (u · ∇u) · ∆udx
R ZR Z ≤ |θ| |∆u| dx + |u| |∇u| |∆u| dx 3
R3
3
R3
≤ kθkL2 k∆ukL2 + kukL6 k∇ukL3 k∆ukL2 µ ¶ 1 2 2 1 1 2 2 ≤ C kθkL2 + k∆ukL2 + C kuk 3. 2 kuk 3. −1 k∇uk 3. 1 k∇uk 3. −2 k∆ukL2 H B ∞,∞ H B ∞,∞ 4 1 ≤ C + k∆uk2L2 + C kuk . −1 k∆uk2L2 , B ∞,∞ 4 where we used 1
2
kukL6 ≤ C kuk 3. 2 kuk 3. −1 , H
k∇ukL3 ≤ C k∇uk Notice that
2 3 . 2
H
B ∞,∞
1
k∇uk 3. −2 . B ∞,∞
¡ 3¢ ¡ 3¢ → − −1 −2 f ∈ B˙ ∞,∞ R ⇐⇒ ∇f ∈ B˙ ∞,∞ R .
Taking δ > 0 sufficiently small such that
µ ¶ . −1 L∞ 0,T,B ∞,∞ (R3 )
kuk and absorbing the term, following estimate
1 4
≤ δ,
k∆uk2L2 + Cδ k∆uk2L2 to the left hand, we derive the
1d k∇uk2L2 + C k∆uk2L2 ≤ C. 2 dt Thanks to Gronwall’s lemma, we have ¡ ¢ ¡ ¢ u ∈ L∞ 0, T ; H 1 (R3 ) ∩ L2 0, T ; H 2 (R3 ) .
Since it is well-known that the Sobolev space H s (R3 ) with s > embedded into L∞ (R3 ), this yields ¡ ¢ u ∈ L2 0, T ; L∞ (R3 ) .
3 2
is continuously
0 (R3 ) and using (1.2), we obtain From the embedding L∞ (R3 ) ⊂ B˙ ∞,∞ ´ ³ 0 (R3 ) , u ∈ L2 0, T ; B˙ ∞,∞
5
which ensures the continuation of strong solutions (u, θ, π) beyond T > 0. This completes the proof of Theorem 1.1. 2 Acknowledgements. The authors would like to thank the referees for valuable comments and suggestions for improving this paper.
References [1] J. Bergh and J. L¨ofstr¨om, Interpolation spaces, Springer-Verlag, 1976. [2] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), 497–513. [3] J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in Lp , in: Lecture Notes in Mathematics, vol. 771, Springer, Berlin, 1980, pp. 129–144. [4] J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity, Appl. Math. Lett. 22 (2009), 802–805. [5] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity 22 (2009), 553–568. [6] S. Gala, Q. Liu and M. A. Ragusa, Logarithmically improved regularity . −1
criterion for the nematic liquid crystal fows in B ∞,∞ space, Comput. Math. Appl. 65 (2013), 1738–1745. [7] S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, Appl. Anal. 90 (2011), 1829–1835. [8] S. Gala, A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure, Math. Meth. Appl. Sci. 34 (2011), 1945–1953. [9] J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity, Appl. Math. Lett. 25 (2012), 63–66. [10] Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phy. 52 (2011), 063503.
6
[11] Y. Jia, X. Zhang and B. Dong, Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion, Comm. Pure Appl. Anal. 12 (2013), 923–937. [12] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, no. 9, AMS/CIMS, (2003). [13] Y. Meyer, P. Gerard and F. Oru, In´egalit´es de Sobolev pr´ecis´ees, S´eminaire ´ Equations aux d´eriv´ees partielles (Polytechnique) (1996-1997), Exp. No. 4, 8 p. [14] N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Meth. Appl. Sci. 9 (1999), 1323–1332. [15] H. Qiu, Y. Du and Z. Yao, Blow-up criteria for 3D Boussinesq equations in the multiplier space, Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 1820– 1824. [16] H. Triebel, Theory of function spaces II. Birkh¨auser, Basel, 1992. [17] Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-α equations, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 2, 319–327.
7
S0893-9659(13)00231-0 http://dx.doi.org/10.1016/j.aml.2013.08.002 AML 4422
To appear in:
Applied Mathematics Letters
Received date: 3 June 2013 Revised date: 12 August 2013 Accepted date: 13 August 2013 Please cite this article as: S. Gala, Z. Guo, M.A. Ragusa, A remark on regularity criterion of the Boussinesq equations with zero heat conductivity, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.08.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A remark on regularity criterion of the Boussinesq equations with zero heat conductivity Sadek Gala Department of Mathematics, University of Mostaganem Box 227, Mostaganem 27000, Algeria [email protected]
Zhengguang Guo∗ College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, P. R. China [email protected] and
Maria Alessandra Ragusa Dipartimento di Mathematica e Informatica Universit a di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy.
Abstract: In this note, we consider the regularity problem under the critical condition to the Boussinesq equations with zero heat conductivity. The Serrin type regularity criteria are established in terms of the critical Besov spaces. This improves a result established in a recent work by [9]. Mathematics Subject Classification(2000): 35Q35; 76D03 Key words: Regularity criterion; Boussinesq equations with zero heat conductivity ∗
Author to whom correspondence should be addressed
1
1
Introduction and main result
Consider the Boussinesq system with zero heat conductivity in R3 : ∂t u + u · ∇u − ∆u + ∇π = θe3 , for (x, t) ∈ R3 × (0, ∞), ∂ θ + u · ∇θ = 0, for (x, t) ∈ R3 × (0, ∞), t ∇ · u = 0, for (x, t) ∈ R3 × (0, ∞), u(x, 0) = u0 (x), θ(x, 0) = θ0 (x), for x ∈ R3 ,
(1.1)
where u = u(x, t) and θ = θ(x, t) denote the unknown velocity vector field and the scalar temperature, while u0 , θ0 with ∇ · u0 = 0 in the sense of distribution are given initial data. e3 = (0, 0, 1)T . π = π(x, t) the pressure of fluid at the point (x, t) ∈ R3 × (0, ∞). The Boussinesq system has important roles in the atmospheric sciences (See e.g. [12]). Existence and uniqueness theories of solutions to the Boussinesq equations have been studied by many mathematicians and physicists. For 2D Boussinesq problem, the global in time regularity is well-known in [3]. Chae in [2] showed that the 2D Boussinesq system also has a unique smooth global in time solution with zero viscosity. On the other hand, for 3D Boussinesq problem, Fan and Ozawa [5] and Ishimura and Morimoto [14] proved the following regularity criterion, respectively: ³ ´ . 0 u ∈ L2 0, T ; B ∞,∞ (R3 ) , (1.2) ¡ ¢ ∇u ∈ L1 0, T ; L∞ (R3 ) Very recently, Geng and Fan in [9] (see also [11]) established the following criterion for 3D Boussinesq system: ³ ´ . −r 2 u ∈ L 1−r 0, T ; B ∞,∞ (R3 ) with − 1 < r < 1 and r 6= 0, (1.3) . s
where B ∞,∞ denotes the homogeneous Besov space. For the regularity problem on other very related models, we refer readers to the investigations [4, 6, 7, 17]. Motivated by the result in [9], our aim is to consider the limit case r = 1 in (1.3) and we establish a Serrin-type regularity criterion for weak solutions in . −r 2 terms of the velocity in the class L 1−r (0, T ; B ∞,∞ (R3 )), which greatly improves the result in [9]. More precisely, we will prove Theorem 1.1 Let (u0 , θ0 ) ∈ H s (R3 ) with div u0 = 0 in R3 and s > 52 . Let (u, θ, π) be a local smooth solution to (1.1). If there exists a small positive constant 2
δ such that µ ¶ . −1 L∞ 0,T,B ∞,∞ (R3 )
kuk
≤ δ,
(1.4)
then the solution (u, θ, π) to the problem (1.1) remains smooth on [0, T ]. Remark 1.1 This result says that the velocity field of the fluids plays a more dominant role than the temperature θ in the regularity theory of the system (1.1). So our theorem is a complement and improvement of the previous results. .
. −1
Since the critical Morrey-Campanato space M2,3 ⊂ B ∞,∞ (see e.g. [8, 10]), an immediate corollary is as follows. Corollary 1.2 Let (u0 , θ0 ) ∈ H s (R3 ) with div u0 = 0 in R3 and s > 52 . Let (u, θ, π) be a local smooth solution to (1.1). If there exists a small positive constant δ such that ´ ≤ δ, . kukL∞ ³0,T,M (1.5) (R3 ) 2,3
then the solution (u, θ, π) to the problem (1.1) remains smooth on [0, T ].
Remark 1.2 Corollary 1.2 extends previous results and covers the supercritical case [7, 15]. Next we recall the definition of the homogeneous Besov space (see e.g. [1, 16]). Let et∆ denote the heat semi-group defined by à ! 2 3 |x| et∆ f = Kt ∗ f, Kt (x) = (4πt)− 2 exp − 4t for t > 0 and x ∈ R3 , where ∗ means convolution of functions defined on R3 . We now recall the definition of the homogeneous Besov space with negative . −α indices B ∞,∞ on R3 with α > 0. It is known ([16], page 192) that f ∈ S 0 (R3 ) . −α ° α ° belongs to B ∞,∞ (R3 ) if and only if et∆ ∈ L∞ for all t > 0 and t 2 °et∆ f ° ∈ . −α
L∞ (0, ∞). The norm of B ∞,∞ is defined, up to equivalence, by ° ¢ ¡ α° kf k . −α = sup t 2 °et∆ f °∞ . B ∞,∞
∞
t>0
Here S 0 is the dual of Schwartz space (tempered distribution). The crucial tool in this note is the following lemma which is essentially due to Meyer-Gerard-Oru [13], which plays an important role for the proof of our theorem. 3
¡ ¢ Lemma 1.3 Let 2 < q < ∞ and s = α 2q − 1 > 0. Then there exists a constant C depending only on α and q such that the estimate 1− 2
2
kf kLq ≤ C kf k q. s kf k . −αq H
. s
. −α
B ∞,∞
(1.6)
. s
holds for all f ∈ H (R3 )∩ B ∞,∞ (R3 ), where H denotes the homogeneous Sobolev space. We omit the proof here, the proof can be found in [8, 10]. In this paper the letter C denotes a constant which may vary in different case. Now we are in a position to prove Theorem 1.1. Proof: Let T > 0 be a given fixed time. We assume that u satisfies (1.4). Multiplying the second equation of (1.1) by θ and integrating over R3 , we get immediately 1d kθk2L2 = 0. 2 dt Hence ¡ ¢ θ ∈ L∞ 0, T ; L2 (R3 ) . (1.7) Next, multiplying both sides of the first equation of (1.1) by u, we have after integration by part, Z 1d 2 2 kukL2 + k∇ukL2 = (θe3 ) · udx ≤ kθkL2 kukL2 2 dt R3
≤ C kukL2 , which yields
¡ ¢ ¡ ¢ u ∈ L∞ 0, T ; L2 (R3 ) ∩ L2 0, T ; H 1 (R3 ) ,
where we used (1.7) and Z Z Z 1 1 2 (u · ∇u) · udx = (u · ∇) u dx = − (∇ · u) u2 dx = 0 2 2 R3
R3
R3
by incompressibility of u, that is, ∇ · u = 0. Next, multiplying both sides of (1.1)1 by −∆u and integrating it over R3 , we
4
find, after integration by part, 1d k∇uk2L2 + k∆uk2L2 2 dt Z Z → − = − (θ e 3 ) · ∆udx + (u · ∇u) · ∆udx
R ZR Z ≤ |θ| |∆u| dx + |u| |∇u| |∆u| dx 3
R3
3
R3
≤ kθkL2 k∆ukL2 + kukL6 k∇ukL3 k∆ukL2 µ ¶ 1 2 2 1 1 2 2 ≤ C kθkL2 + k∆ukL2 + C kuk 3. 2 kuk 3. −1 k∇uk 3. 1 k∇uk 3. −2 k∆ukL2 H B ∞,∞ H B ∞,∞ 4 1 ≤ C + k∆uk2L2 + C kuk . −1 k∆uk2L2 , B ∞,∞ 4 where we used 1
2
kukL6 ≤ C kuk 3. 2 kuk 3. −1 , H
k∇ukL3 ≤ C k∇uk Notice that
2 3 . 2
H
B ∞,∞
1
k∇uk 3. −2 . B ∞,∞
¡ 3¢ ¡ 3¢ → − −1 −2 f ∈ B˙ ∞,∞ R ⇐⇒ ∇f ∈ B˙ ∞,∞ R .
Taking δ > 0 sufficiently small such that
µ ¶ . −1 L∞ 0,T,B ∞,∞ (R3 )
kuk and absorbing the term, following estimate
1 4
≤ δ,
k∆uk2L2 + Cδ k∆uk2L2 to the left hand, we derive the
1d k∇uk2L2 + C k∆uk2L2 ≤ C. 2 dt Thanks to Gronwall’s lemma, we have ¡ ¢ ¡ ¢ u ∈ L∞ 0, T ; H 1 (R3 ) ∩ L2 0, T ; H 2 (R3 ) .
Since it is well-known that the Sobolev space H s (R3 ) with s > embedded into L∞ (R3 ), this yields ¡ ¢ u ∈ L2 0, T ; L∞ (R3 ) .
3 2
is continuously
0 (R3 ) and using (1.2), we obtain From the embedding L∞ (R3 ) ⊂ B˙ ∞,∞ ´ ³ 0 (R3 ) , u ∈ L2 0, T ; B˙ ∞,∞
5
which ensures the continuation of strong solutions (u, θ, π) beyond T > 0. This completes the proof of Theorem 1.1. 2 Acknowledgements. The authors would like to thank the referees for valuable comments and suggestions for improving this paper.
References [1] J. Bergh and J. L¨ofstr¨om, Interpolation spaces, Springer-Verlag, 1976. [2] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), 497–513. [3] J. R. Cannon and E. Dibenedetto, The initial problem for the Boussinesq equation with data in Lp , in: Lecture Notes in Mathematics, vol. 771, Springer, Berlin, 1980, pp. 129–144. [4] J. Fan and Y. Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity, Appl. Math. Lett. 22 (2009), 802–805. [5] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity 22 (2009), 553–568. [6] S. Gala, Q. Liu and M. A. Ragusa, Logarithmically improved regularity . −1
criterion for the nematic liquid crystal fows in B ∞,∞ space, Comput. Math. Appl. 65 (2013), 1738–1745. [7] S. Gala, On the regularity criterion of strong solutions to the 3D Boussinesq equations, Appl. Anal. 90 (2011), 1829–1835. [8] S. Gala, A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure, Math. Meth. Appl. Sci. 34 (2011), 1945–1953. [9] J. Geng and J. Fan, A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity, Appl. Math. Lett. 25 (2012), 63–66. [10] Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phy. 52 (2011), 063503.
6
[11] Y. Jia, X. Zhang and B. Dong, Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion, Comm. Pure Appl. Anal. 12 (2013), 923–937. [12] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, no. 9, AMS/CIMS, (2003). [13] Y. Meyer, P. Gerard and F. Oru, In´egalit´es de Sobolev pr´ecis´ees, S´eminaire ´ Equations aux d´eriv´ees partielles (Polytechnique) (1996-1997), Exp. No. 4, 8 p. [14] N. Ishimura and H. Morimoto, Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Meth. Appl. Sci. 9 (1999), 1323–1332. [15] H. Qiu, Y. Du and Z. Yao, Blow-up criteria for 3D Boussinesq equations in the multiplier space, Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 1820– 1824. [16] H. Triebel, Theory of function spaces II. Birkh¨auser, Basel, 1992. [17] Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-α equations, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 2, 319–327.
7