Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity
© Wang et al; licensee Springer. 2011
Received: 4 April 2011
Accepted: 15 August 2011
Published: 15 August 2011
In this paper, we investigate the Cauchy problem for the incompressible magneto-micropolar fluid equations with partial viscosity in ℝ n (n = 2, 3). We obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions.
MSC (2010): 76D03; 35Q35.
Keywordsmagneto-micropolar fluid equations smooth solutions; blow-up criterion
where u(t, x), v(t, x), b(t, x) and p(t, x) denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. μ, χ, γ, κ and ν are constants associated with properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities, and is the magnetic Reynold. The incompressible magneto-micropolar fluid equations (1.1) has been studied extensively (see [1–8]). Rojas-Medar  established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. Global existence of strong solution for small initial data was obtained in . Rojas-Medar and Boldrini  proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Wang et al.  obtained a Beale-Kato-Majda type blow-up criterion for smooth solution (u, v, b) to the magneto-micropolar fluid equations with partial viscosity that relies on the vorticity of velocity ∇ × u only (see also ). For regularity results, refer to Yuan  and Gala .
If b = 0, (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first proposed by Eringen . It is a type of fluids which exhibits the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that consisting of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background, we refer to  and references therein. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero  and Yamaguchi , respectively. The global regularity issue has been thoroughly investigated for the 3D micropolar fluid equations and many important regularity criteria have been established (see [13–19]). The convergence of weak solutions of the micropolar fluids in bounded domains of ℝ n was investigated (see ). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.
We obtain a blow-up criterion of smooth solutions to (1.2), which improves our previous result (see ).
In the absence of global well-posedness, the development of blow-up/non-blow-up theory is of major importance for both theoretical and practical purposes. For incompressible Euler and Navier-Stokes equations, the well-known Beale-Kato-Majda's criterion  says that any solution u is smooth up to time T under the assumption that . Beale-Kato-Majda's criterion is slightly improved by Kozono et al.  under the assumption . In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to Cauchy problem for the magneto-micropolar fluid equations (1.2).
Now, we state our results as follows.
then the solution (u, v, b) can be extended beyond t = T.
We have the following corollary immediately.
The plan of the paper is arranged as follows. We first state some preliminary on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.
In what follows, we shall make continuous use of Bernstein inequalities, which comes from .
hold, where c and C are positive constants independent of f and k.
The following inequality is well-known Gagliardo-Nirenberg inequality.
with the following exception: if 1 < r < 1 and is a nonnegative integer, then (2.3) holds only for a satisfying .
The following lemma comes from .
where 1 ≤ α ≤ m and .
holds for all vectors f ∈ H3(ℝ n )(n = 2, 3) with ∇ · f = 0.
It follows from (2.7), (2.8) and Calderon-Zygmand theory that (2.5) holds. Thus, we have completed the proof of lemma. □
In order to prove Theorem 1.1, we need the following interpolation inequalities in two and three space dimensions.
Taking and n = 3 and n = 2, respectively. From (2.14), we immediately get the last inequality in (2.9) and (2.10). Thus, we have completed the proof of Lemma 2.5. □
3 Proof of main results
where we have used ∇ ·· u = 0 and ∇ · b = 0.
where C1 depends on , while C0 is an absolute positive constant.
In what follows, for simplicity, we will set m = 3.
for all T* ≤ t < T.
where C depends on .
Noting that (3.2) and the right-hand side of (3.24) is independent of t for T* ≤ t < T , we know that (u(T, ·), v(T, ·), b(T, ·)) ∈ H3(ℝ n ). Thus, Theorem 1.1 is proved.
The authors would like to thank the referee for his/her pertinent comments and advice. This work was supported in part by Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.
- Gala S: Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space. Nonlinear Differ Equ Appl 2010, 17: 181-194. 10.1007/s00030-009-0047-4View ArticleMathSciNetGoogle Scholar
- Wang Y, Hu L, Wang Y: A Beale-Kato Majda criterion for magneto-micropolar fluid equations with partial viscosity. Bound Value Prob 2011, 2011: 14. Article ID 128614 10.1186/1687-2770-2011-14View ArticleGoogle Scholar
- Ortega-Torres E, Rojas-Medar M: On the uniqueness and regularity of the weak solution for magneto-micropolar fluid equations. Revista de Matemáticas Aplicadas 1996, 17: 75-90.MathSciNetGoogle Scholar
- Ortega-Torres E, Rojas-Medar M: Magneto-micropolar fluid motion: global existence of strong solutions. Abstract Appl Anal 1999, 4: 109-125. 10.1155/S1085337599000287View ArticleMathSciNetGoogle Scholar
- Rojas-Medar M: Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Mathematische Nachrichten 1997, 188: 301-319. 10.1002/mana.19971880116View ArticleMathSciNetGoogle Scholar
- Rojas-Medar M, Boldrini J: Magneto-micropolar fluid motion: existence of weak solutions. Rev Mat Complut 1998, 11: 443-460.View ArticleMathSciNetGoogle Scholar
- Yuan B: regularity of weak solutions to magneto-micropolar fluid equations. Acta Mathematica Scientia 2010, 30: 1469-1480. 10.1016/S0252-9602(10)60139-7View ArticleGoogle Scholar
- Yuan J: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations. Math Methods Appl Sci 2008, 31: 1113-1130. 10.1002/mma.967View ArticleMathSciNetGoogle Scholar
- Eringen A: Theory of micropolar fluids. J Math Mech 1966, 16: 1-18.MathSciNetGoogle Scholar
- Lukaszewicz G: Micropolar fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Baston 1999.Google Scholar
- Galdi G, Rionero S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int J Eng Sci 1977, 15: 105-108. 10.1016/0020-7225(77)90025-8View ArticleMathSciNetGoogle Scholar
- Yamaguchi N: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math Methods Appl Sci 2005, 28: 1507-1526. 10.1002/mma.617View ArticleMathSciNetGoogle Scholar
- Yuan B: On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. Proc Am Math Soc 2010, 138: 2025-2036. 10.1090/S0002-9939-10-10232-9View ArticleGoogle Scholar
- Fan J, Zhou Y, Zhu M: A regularity criterion for the 3D micropolar fluid flows with zero angular viscosity. 2010, in press.Google Scholar
- Fan J, He X: A regularity criterion of the 3D micropolar fluid flows. 2011, in press.Google Scholar
- Fan J, Jin L: A regularity criterion of the micropolar fluid flows. 2011, in press.Google Scholar
- Dong B, Chen Z: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J Math Phys 2009, 50: 103525-1-103525-13.View ArticleMathSciNetGoogle Scholar
- Szopa P: Gevrey class regularity for solutions of micropolar fluid equations. J Math Anal Appl 2009, 351: 340-349. 10.1016/j.jmaa.2008.10.026View ArticleMathSciNetGoogle Scholar
- Ortega-Torres E, Rojas-Medar M: On the regularity for solutions of the micropolar fluid equations. Rendiconti del Seminario Matematico della Università de Padova 2009, 122: 27-37.View ArticleMathSciNetGoogle Scholar
- Ortega-Torres E, Rojas-Medar M, Villamizar-Roa EJ: Micropolar fluids with vanishing viscosity. Abstract Appl Anal 2010, 2010: 18. Article ID 843692View ArticleMathSciNetGoogle Scholar
- Sermange M, Temam R: Some mathematical questions related to the MHD equations. Commun Pure Appl Math 1983, 36: 635-666. 10.1002/cpa.3160360506View ArticleMathSciNetGoogle Scholar
- Caisch R, Klapper I, Steele G: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun Math Phys 1997, 184: 443-455. 10.1007/s002200050067View ArticleGoogle Scholar
- Cannone M, Chen Q, Miao C: A losing estimate for the ideal MHD equations with application to blow-up criterion. SIAM J Math Anal 2007, 38: 1847-1859. 10.1137/060652002View ArticleMathSciNetGoogle Scholar
- Cao C, Wu J: Two regularity criteria for the 3D MHD equations. J Diff Equ 2010, 248: 2263-2274. 10.1016/j.jde.2009.09.020View ArticleMathSciNetGoogle Scholar
- He C, Xin Z: Partial regularity of suitable weak sokutions to the incompressible magnetohydrodynamics equations. J Funct Anal 2005, 227: 113-152. 10.1016/j.jfa.2005.06.009View ArticleMathSciNetGoogle Scholar
- Lei Z, Zhou Y: BKM criterion and global weak solutions for Magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst A 2009, 25: 575-583.View ArticleMathSciNetGoogle Scholar
- Wu J: Regularity results for weak solutions of the 3D MHD equations. Discrete Contin Dyn Syst 2004, 10: 543-556.View ArticleMathSciNetGoogle Scholar
- Wu J: Regularity criteria for the generalized MHD equations. Commun Partial Differ Equ 2008, 33: 285-306. 10.1080/03605300701382530View ArticleGoogle Scholar
- Zhou Y: Remarks on regularities for the 3D MHD equations. Discrete Contin Dyn Syst 2005, 12: 881-886.View ArticleMathSciNetGoogle Scholar
- Zhou Y: Regularity criteria for the 3D MHD equations in term of the pressure. Int J Nonlinear Mech 2006, 41: 1174-1180. 10.1016/j.ijnonlinmec.2006.12.001View ArticleGoogle Scholar
- Zhou Y: Regularity criteria for the generalized viscous MHD equations. Ann Inst H Poincaré Anal Non Linéaire 2007, 24: 491-505. 10.1016/j.anihpc.2006.03.014View ArticleGoogle Scholar
- Zhou Y, Gala S: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z Angew Math Phys 2010, 61: 193-199. 10.1007/s00033-009-0023-1View ArticleMathSciNetGoogle Scholar
- Zhou Y, Gala S: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal 2010, 72: 3643-3648. 10.1016/j.na.2009.12.045View ArticleMathSciNetGoogle Scholar
- Zhou Y, Fan J: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math 2010, in press.Google Scholar
- Beale J, Kato T, Majda A: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun Math Phys 1984, 94: 61-66. 10.1007/BF01212349View ArticleMathSciNetGoogle Scholar
- Kozono H, Ogawa T, Taniuchi Y: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math Z 2002, 242: 251-278. 10.1007/s002090100332View ArticleMathSciNetGoogle Scholar
- Triebel H: Theory of Function Spaces. Monograph in Mathematics. Birkhauser, Basel 1983., 78:Google Scholar
- Chemin J: Perfect Incompressible Fluids. In Oxford Lecture Ser Math Appl. Volume 14. The Clarendon Press/Oxford University Press, New York; 1998.Google Scholar
- Majda A, Bertozzi A: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge; 2002.Google Scholar
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