Remarks on the Pressure Regularity Criterion of the Micropolar Fluid Equations in Multiplier Spaces

Abstract

This study is devoted to investigating the regularity criterion of weak solutions of the micropolar fluid equations in [superscript] ... 3[/superscript] . The weak solution of micropolar fluid equations is proved to be smooth on (0 ,T ] when the pressure π (x ,t ) satisfies the following growth condition in the multiplier spaces [superscript] X r[/superscript] , [superscript] ∫ 0 T[/superscript] [superscript] || π (s , · ) || [superscript] X ... r[/superscript] 2 / (2 -r )[/superscript] / (1 +ln ( e +[subscript] || π (s , · ) || [superscript] L 2[/superscript] [/subscript] ) ) , ds < ∞ , and 0 ...4;r ...4;1 . The previous results on Lorentz spaces and Morrey spaces are obviously improved.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Beong In Yun

1, School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 26 October 2012; Accepted 14 December 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Consider the Cauchy problem of the three-dimensional (3D) micropolar fluid equations with unit viscosities [figure omitted; refer to PDF] associated with the initial condition: [figure omitted; refer to PDF] where u ( x ,t ) = ( _{u 1} ( x ,t ) , _{u 2} ( x ,t ) , _{u 3} ( x ,t ) ) , and w (x ,t ) = ( _{w 1} (x ,t ) , _{w 2} (x ,t ) , _{w 3} (x ,t ) ) are the unknown velocity vector field and the microrotation vector field. π (x ,t ) is the unknown scalar pressure field. _{u 0} and _{w 0} represent the prescribed initial data for the velocity and microrotation fields.

Micropolar fluid equations introduced by Eringen [ 1] are a special model of the non-Newtonian fluids (see [ 2- 6]) which is coupled with the viscous incompressible Navier-Stokes model, microrotational effects, and microrotational inertia. When the microrotation effects are neglected or w =0 , the micropolar fluid equations ( 1.1) reduce to the incompressible Navier-Stokes flows (see, e.g., [ 7, 8]): [figure omitted; refer to PDF] That is to say, Navier-Stokes equations are viewed as a subclass of the micropolar fluid equations.

Mathematically, there is a large literature on the existence, uniqueness and large time behaviors of solutions of micropolar fluid equations (see [ 9- 15] and references therein); however, the global regularity of the weak solution in the three-dimensional case is still a big open problem. Therefore it is interesting and important to consider the regularity criterion of the weak solutions under some assumptions of certain growth conditions on the velocity or on the pressure.

On one hand, as for the velocity regularity criteria, by means of the Littlewood-Paley decomposition methods, Dong and Chen [ 16] proved the regularity of weak solutions under the velocity condition: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Moreover, the result is further improved by Dong and Zhang [ 17] in the margin case: [figure omitted; refer to PDF]

On the other hand, as for the pressure regularity criteria, Yuan [ 18] investigated the regularity criterion of weak solutions of the micropolar fluid equations in Lebesgue spaces and Lorentz spaces: [figure omitted; refer to PDF] where ^{L p , ∞} ( ^{... 3} ) is the Lorents space (see the definitions in the next section).

Recently, Dong et al. [ 19] improved the pressure regularity of the micropolar fluid equations in Morrey spaces: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Furthermore, Jia et al. [ 20] refined the regularity from Morrey spaces to Besov spaces: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] One may also refer to some interesting results on the regularity criteria of Newtonian and non-Newtonian fluid equations (see [ 21- 27] and references therein).

The aim of the present study is to investigate the pressure regularity criterion of the three-dimensional micropolar fluid equations in the multiplier spaces which are larger than the Lebesgue spaces, Lorentz spaces, and Morrey spaces.

2. Preliminaries and Main Result

Throughout this paper, we use c to denote the constants which may change from line to line. ^{L p} ( ^{... 3} ) , ^{W k ,p} ( ^{... 3} ) with k ∈Z , 1 ...4;p ...4; ∞ denote the usual Lebesgue space and Sobolev space. ^{H s} ( ^{... 3} ) , s ∈ ... denote the fractional Sobolev space with [figure omitted; refer to PDF]

Consider a measurable function f and define for t ...5;0 the Lebesgue measure [figure omitted; refer to PDF] of the set {x ∈ ^{... 3} : |f (x ) | >t } . The Lorentz space is defined by f ∈ ^{L p ,q} ( ^{... 3} ) if and only if [figure omitted; refer to PDF]

We defined _{M p ,q} ( ^{... 3} ) , 1 ...4;p ,q ...4; ∞ the homogeneous Morrey space associated with norm [figure omitted; refer to PDF]

We now recall the definition and some properties of the multiplier space ^{X r} .

Definition 2.1 (see Lemarié-Rieusset [ 28]).

For 0 ...4;r < 3 / 2 , the space ^{X r} is defined as the space of f (x ) ∈ ^{L loc 2} ( ^{... 3} ) such that [figure omitted; refer to PDF]

According the above definition of the multiplier space, it is not difficult to verify the homogeneity properties. For all _{x 0} ∈ ^{... 3} [figure omitted; refer to PDF]

When r =0 , it is clear that (see Lemarié-Rieusset [ 28]) [figure omitted; refer to PDF] where BMO denotes the homogenous space of bounded mean oscillations associated with the norm [figure omitted; refer to PDF]

In particular, the following imbedding (see Lemarié-Rieusset [ 28]) [figure omitted; refer to PDF] holds true.

In order to state our main results, we recall the definition of the weak solution of micropolar flows (see, e.g., Lukaszewicz [ 9]).

Definition 2.2.

Let T >0 , ( _{u 0} , _{w 0} ) ∈ ^{L 2} ( ^{... 3} ) , and ∇ · _{u 0} =0 . (u ,w ) is termed as a weak solution to the 3D micropolar flows ( 1.1) and ( 1.2) on (0 ,T ) , if (u ,w ) satisfies the following properties:

(i) (u ,w ) ∈ ^{L ∞} (0 ,T ; ^{L 2} ( ^{... 3} ) ) ∩ ^{L 2} (0 ,T ; ^{H 1} ( ^{... 3} ) ) ;

(ii) equations ( 1.1) and ( 1.2) are valid in the sense of distributions.

Our main results are now read as follows.

Theorem 2.3.

Suppose T >0 , ( _{u 0} , _{w 0} ) ∈ ^{L 2} ( ^{... 3} ) ∩ ^{L 4} ( ^{... 3} ) , and ∇ · _{u 0} =0 in the sense of distributions. Assume that (u ,w ) is a weak solution of the 3D micropolar fluid flows ( 1.1) and ( 1.2) on (0 ,T ) . If the pressure π (x ,t ) satisfies the logarithmically growth condition: [figure omitted; refer to PDF] then the weak solution (u ,w ) is regular on (0 ,T ] .

Thanks to [figure omitted; refer to PDF] it is easy to deduce the following pressure regularity criterion of the three-dimensional micropolar equations ( 1.1) and ( 1.2).

Corollary 2.4.

On the substitution of the pressure condition ( 2.10) by the following conditions: [figure omitted; refer to PDF] the conclusion of Theorem 2.3holds true.

Remark 2.5.

According to the embedding relation ( 2.9), our results obviously largely improve the previous results ( 1.7) and ( 1.8). Moreover, it seems incomparable with the Besov space ( 1.10).

Remark 2.6.

Furthermore, since we have no additional growth condition on the microrotation vector field w (x ,t ) , Theorem 2.3is also valid for the pressure regularity problem of the three-dimensional Navier-Stokes equations (see, e.g., Zhou [ 29, 30]).

3. Proof of Theorem 2.3

In order to prove our main results, we first recall the following local existence theorem of the three-dimensional micropolar fluid equations ( 1.1) and ( 1.2).

Lemma 3.1 (see Dong et al. [ 19]).

Assume 3 <p < ∞ and ( _{u 0} , _{w 0} ) ∈ ^{L p} ( ^{... 3} ) with ∇ · _{u 0} =0 in the sense of distributions. Then there exist a constant T >0 and a unique strong solution (u ,w ) of the 3D micropolar fluid equations ( 1.1) and ( 1.2) such that [figure omitted; refer to PDF]

By means of the local existence result, ( 1.1) and ( 1.2) with ( _{u 0} , _{w 0} ) ∈ ^{L 2} ( ^{... 3} ) ∩ ^{L 4} ( ^{... 3} ) admit a unique ^{L 4} -strong solution (u ,w ) on a maximal time interval. For the notation simplicity, we may suppose that the maximal time interval is [0 ,T ) . Thus, to prove Theorem 2.3, it remains to show that [figure omitted; refer to PDF] This will lead to a contradiction to the estimates to be derived below. We now begin to follow these arguments.

Taking the inner product of the second equation of ( 1.1) with u |u ^{| 2} and the third equation of ( 1.1) with w |w ^{| 2} , respectively, and integrating by parts, it follows that [figure omitted; refer to PDF] where we have used the following identities due to the divergence free property of the velocity field u : [figure omitted; refer to PDF]

Furthermore, applying Young inequality, Hölder inequality, and integration by parts, we have [figure omitted; refer to PDF]

Combining the above inequalities, it follows that [figure omitted; refer to PDF]

In order to estimate the last term of the right-hand side of ( 3.6), taking the divergence operator ∇ · to the first equation of ( 1.1) produces the expression of the pressure: [figure omitted; refer to PDF] Employing Calderón-Zygmund inequality and the divergence free condition of the velocity derives the estimate of the pressure: [figure omitted; refer to PDF]

Therefore, we estimate the pressure term as [figure omitted; refer to PDF] Now we estimate the integral [figure omitted; refer to PDF] on the right-hand side of ( 3.9). By the Hölder inequality and the Young inequality we have [figure omitted; refer to PDF] where we have used the following interpolation inequality: [figure omitted; refer to PDF]

Hence, combining the above inequalities, we derive [figure omitted; refer to PDF]

Furthermore, we have the second term of the right-hand side of ( 3.13) rewritten as [figure omitted; refer to PDF] Inserting ( 3.14) into ( 3.13) and applying the Gronwall inequality, one shows that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF]

Hence we complete the proof of Theorem 2.3.

References

[1] A. C. Eringen, "Theory of micropolar fluids," Journal of Mathematics and Mechanics , vol. 16, pp. 1-18, 1966.

[2] G. Böhme Non-Newtonian Fluid Mechanics , of Applied Mathematics and Mechanics, North-Holland, Amsterdam, The Netherlands, 1987.

[3] J. Málek, J. Necas, M. Rokyta, M. Ruzicka Weak and Measure-valued Solutions to Evolutionary PDEs , vol. 13, Chapman & Hall, New York, NY, USA, 1996.

[4] B. Q. Dong, Y. Li, "Large time behavior to the system of incompressible non-Newtonian fluids in ^{... 2} ,", vol. 298, no. 2, pp. 667-676, 2004.

[5] C. Zhao, Y. Li, " ^H2 -compact attractor for a non-Newtonian system in two-dimensional unbounded domains," Nonlinear Analysis: Theory, Methods & Applications , vol. 56, no. 7, pp. 1091-1103, 2004.

[6] C. Zhao, S. Zhou, "Pullback attractors for a non-autonomous incompressible non-Newtonian fluid," Journal of Differential Equations , vol. 238, no. 2, pp. 394-425, 2007.

[7] O. Ladyzhenskaya The Mathematical Theory of Viscous Incompressible Fluids , Gorden Brech, New York, NY, USA, 1969.

[8] R. Temam Navier-Stokes Equations. Theory and Numerical Analysis , North-Holland, Amsterdam, The Netherlands, 1977.

[9] G. Lukaszewicz Micropolar Fluids. Theory and Applications , of Modeling and Simulation in Science, Engineering and Technology, Birkhäauser, Boston, Mass, USA, 1999.

[10] B.-Q. Dong, Z. Zhang, "Global regularity of the 2D micropolar fluid flows with zero angular viscosity," Journal of Differential Equations , vol. 249, no. 1, pp. 200-213, 2010.

[11] N. Yamaguchi, "Existence of global strong solution to the micropolar fluid system in a bounded domain," Mathematical Methods in the Applied Sciences , vol. 28, no. 13, pp. 1507-1526, 2005.

[12] B.-Q. Dong, Z.-M. Chen, "Global attractors of two-dimensional micropolar fluid flows in some unbounded domains," Applied Mathematics and Computation , vol. 182, no. 1, pp. 610-620, 2006.

[13] B.-Q. Dong, Z.-M. Chen, "On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations," Journal of Mathematical Analysis and Applications , vol. 334, no. 2, pp. 1386-1399, 2007.

[14] G. P. Galdi, S. Rionero, "A note on the existence and uniqueness of solutions of the micropolar fluid equations," International Journal of Engineering Science , vol. 15, no. 2, pp. 105-108, 1977.

[15] B.-Q. Dong, Z.-M. Chen, "Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows," Discrete and Continuous Dynamical Systems A , vol. 23, no. 3, pp. 765-784, 2009.

[16] B.-Q. Dong, Z.-M. Chen, "Regularity criteria of weak solutions to the three-dimensional micropolar flows," Journal of Mathematical Physics , vol. 50, no. 10, article 103525, 2009.

[17] B.-Q. Dong, W. Zhang, "On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 73, no. 7, pp. 2334-2341, 2010.

[18] B. Yuan, "On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space," Proceedings of the American Mathematical Society , vol. 138, no. 6, pp. 2025-2036, 2010.

[19] B.-Q. Dong, Y. Jia, Z.-M. Chen, "Pressure regularity criteria of the three-dimensional micropolar fluid flows," Mathematical Methods in the Applied Sciences , vol. 34, no. 5, pp. 595-606, 2011.

[20] Y. Jia, W. Zhang, B.-Q. Dong, "Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure," Applied Mathematics Letters , vol. 24, no. 2, pp. 199-203, 2011.

[21] Q. Chen, C. Miao, Z. Zhang, "On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations," Communications in Mathematical Physics , vol. 284, no. 3, pp. 919-930, 2008.

[22] B.-Q. Dong, Y. Jia, W. Zhang, "An improved regularity criterion of three-dimensional magnetohydrodynamic equations," Nonlinear Analysis: Real World Applications , vol. 13, no. 3, pp. 1159-1169, 2012.

[23] C. He, Z. Xin, "On the regularity of weak solutions to the magnetohydrodynamic equations," Journal of Differential Equations , vol. 213, no. 2, pp. 235-254, 2005.

[24] B.-Q. Dong, Z. Zhang, "The BKM criterion for the 3D Navier-Stokes equations via two velocity components," Nonlinear Analysis: Real World Applications , vol. 11, no. 4, pp. 2415-2421, 2010.

[25] C. Cao, J. Wu, "Two regularity criteria for the 3D MHD equations," Journal of Differential Equations , vol. 248, no. 9, pp. 2263-2274, 2010.

[26] Y. Zhou, "A new regularity criterion for weak solutions to the Navier-Stokes equations," Journal de Mathématiques Pures et Appliquées , vol. 84, no. 11, pp. 1496-1514, 2005.

[27] B. Dong, G. Sadek, Z. Chen, "On the regularity criteria of the 3D Navier-Stokes equations in critical spaces," Acta Mathematica Scientia B , vol. 31, no. 2, pp. 591-600, 2011.

[28] P. G. Lemarié-Rieusset Recent Developments in the Navier-Stokes Problem , Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.

[29] Y. Zhou, "On regularity criteria in terms of pressure for the Navier-Stokes equations in ^{... 3} ,", vol. 134, no. 1, pp. 149-156, 2006.

[30] Y. Zhou, "On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in ^{... n} ,", vol. 57, no. 3, pp. 384-392, 2006.

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