Regularity criteria of weak solutions to the three-dimensional micropolar flows

Regularity criteria of weak solutions to the three-dimensional micropolar fluid motion equations are discussed. Sufficient conditions for the regularity of weak solutions are presented by imposing Serrin’s type growth conditions on the velocity field in Lorentz spaces, multiplier spaces, bounded mean oscillation spaces, and Besov spaces, respectively. The findings demonstrate that the velocity field plays a dominant role in the regularity problem of micropolar fluid motion equations.

[1]  Sadek Gala,et al.  Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space , 2010 .

[2]  H. Triebel.  Theory Of Function Spaces , 1983 .

[3]  Michael Struwe,et al.  On partial regularity results for the navier‐stokes equations , 1988 .

[4]  Andrew J. Majda,et al.  Vorticity and Incompressible Flow: Index , 2001 .

[5]  W. Price,et al.  Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3-strong solutions , 2006 .

[6]  H.BeirāodaVeiga.  A New Regularity Class for the Navier-Stokes Equations in IR^n , 1995 .

[7]  M. Rojas-Medar,et al.  Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids , 2003 .

[8]  Yong Zhou,et al.  Remarks on regularities for the 3D MHD equations , 2005 .

[9]  S. Edwards,et al.  The Theory of Polymer Dynamics , 1986 .

[10]  S. A. Regirer,et al.  A continuum model of blood flow. , 1974, Biorheology.

[11]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[12]  Richard O’Neil,et al.  Convolution operators and $L(p,q)$ spaces , 1963 .

[13]  Yong Zhou,et al.  Regularity criteria for the 3D MHD equations in terms of the pressure , 2006 .

[14]  Zhouping Xin,et al.  On the regularity of weak solutions to the magnetohydrodynamic equations , 2005 .

[15]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[16]  Hideo Kozono,et al.  Bilinear estimates in BMO and the Navier-Stokes equations , 2000 .

[17]  Takayoshi Ogawa,et al.  The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations , 2002 .

[18]  Bo-Qing Dong,et al.  Pressure regularity criteria of the three‐dimensional micropolar fluid flows , 2011 .

[19]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[20]  Giovanni P. Galdi,et al.  A note on the existence and uniqueness of solutions of the micropolar fluid equations , 1977 .

[21]  A. Eringen,et al.  THEORY OF MICROPOLAR FLUIDS , 1966 .

[22]  Yoshikazu Giga,et al.  Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system , 1986 .

[23]  Yong Zhou,et al.  Regularity criteria for the solutions to the 3D MHD equations in the multiplier space , 2010 .

[24]  J. Serrin.  On the interior regularity of weak solutions of the Navier-Stokes equations , 1962 .

[25]  Marko Antonio Rojas-Medar,et al.  Magneto ‐ Micropolar Fluid Motion: Existence and Uniqueness of Strong Solution , 1997 .

[26]  Pierre Gilles Lemarié-Rieusset,et al.  Recent Developments in the Navier-Stokes Problem , 2002 .

[27]  Qionglei Chen,et al.  The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations , 2007 .