Acta mathematica scientia, Series B >
REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS
Received date: 2008-09-24
Online published: 2010-09-20
Supported by
The author was partially supported by the National Natural Science Foundation of China (10771052), Program for Science \& Technology Innovation Talents in Universities of Henan Province (2009HASTIT007), Doctor Fund of Henan Polytechnic University (B2008-62), and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.
In this article, we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid
equations in R3. We obtain the classical blow-up criteria for smooth solutions (u, ω,b), i.e., Lq(0,T; Lp(R3) )for 2/q+3/p≤1 with 3<p≤∞, u ∈ C([0, T); L3(R3)) or $\nabla u\in Lq(0, T; Lp) for 3/2< p ≤∞ satisfying 2/q+3/p≤2. Moreover, our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid. In the end-point case p=∞, the blow-up criteria can be extended to more general spaces $\nabla u\in L1(0, T; B0∞, ∞ (R3)).
YUAN Bao-Quan . REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1469 -1480 . DOI: 10.1016/S0252-9602(10)60139-7
[1] Bergh J, L\"ofstrom J. Inerpolation Spaces, an Introduction. New York: Springer-Verlag, 1976
[2] Caflish R E, Klapper I, Steel G. Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm Math Phys, 1997, 184: 443--455
[3] Cannone M, Miao C X, Prioux N, Yuan B Q. The Cauchy problem for the Magneto-hydrodynamic system. Banach Center Publications, 2006, 74: 59--93
[4] Chemin J Y. Perfect Incompressible Fluids. Oxford: Clarendon press, 1998
[5] Chen Q L, Miao C X, Zhang Z F. The Beale-Kato-Majda criterioa for the 3D magneto-hydrodynamics equations. Comm Math Phys, 2007, 275: 861--872
[6] Duvant G, Lions J L. Inequations en thermoelasticite et magnetohydrodynamique. Arch Ration Mech Anal, 1972, 46: 241--279
[7] Eringen A C. Theory of micropolar fluids. J Math Mech, 1996, 16: 1--18
[8] Ferreira L C F, Villamizar-Roa E J. On the existence and stability of solutions for the micropolar fluids in exterior domains. Math Meth Appl Sci, 2007, 30: 1185--1208
[9] Gao S Q, Duan H Y. Negative norm least-squares methods for the incompressible magnetohydrodynamic equations. Acta Math Sci, 2008, 28B: 675--684
[10] Galdi G P, Rionero S. A note on the existence and uniqueness of solutions of the micropolar fluid equations. Internat J Engrg Sci, 1997, 15: 105--108
[11] He C, Wang Y. On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J Differential Equations, 2007, 238: 1--17
[12] He C, Xin Z P. On the regularity of weak solutions to the magnetohydrodynamic equations. J Differential Equations, 2005, 213: 235--254
[13] Jiu Q S, Niu D J. Mathematical results related to a two-dimensional magnetohydrodynamic equations. Acta Math Sci, 2006, 26B: 744--756
[14] Ladyzhenskaya O. The Mathematical Theory of Viscous Incompressible Flows. New York: Gordon and Breach, 1969
[15] Lemari\'{e}-Rieusset P G. Recent Developments in the Navier-Stokes Problem. London: Chapman & Hall/CRC, 2002
[16] Lions P L. Mathematical Topics in Fluid Mechanics. New York: Oxford University Press Inc, 1996
[17] Miao C X. Harmonic Analysis and Application to Partial Differential Equations (2nd ed). Beijing: Science Press, 2004
[18] Ortega-Torres E E, Rojas-Medar M A. Magneto-Micropolar fluid motion: Global existence of strong solutions. Abstr Appl Anal, 1999, 4: 109--125
[19] Rojas-Medar M A. Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Math Nachr, 1997, 188: 301--319
[20] Rojas-Medar M A, Boldrini J L. Magneto-micropolar fluid motion: existence of weak solutions. Rev Mat Complut, 1998, 11: 443--460
[21] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36: 635--664
[22] Triebel H. Theory of Function Spaces, Monograph in Mathematics, Vol 78. Basel: Birkhäuser Verlag, 1983
[23] Villamizar-Roa E J, Rodrìguez-Bellido M A. Global existence and exponential stability for the micropolar fluid system. Z Angew Math Phys, 2008, 59: 790--809
[24] Yamaguchi N. Existence of global strong solution to the micropolar fluid system in a bounded domain. Math Meth Appl Sci, 2005, 28: 1507--1526
[25] Li F P, Yuan B Q. On some explicit blow-up solutions to 3D incompressible magnethohydrodynamic equations. Acta Math Sci, 2009, 29A(6): 1651--1656
[26] Zhou Y. Remarks on regularities for the 3D MHD equations. Discrete Contin Dyn Syst, 2005, 12: 881--886
[27] Zhou Y. Regularity criteria for the 3D MHD equations in terms of the pressure. Internat J Non-Linear Mech, 2006, 41: 1174--1180
[28] Zhou Y. Regularity criteria for the generalized viscous MHD equations. Ann Inst H Poincarè Anal Non Linèaire, 2007, 24: 491--505
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