Logarithmically improved regularity criterion for the nematic liquid crystal flows in Ḃ∞, ∞-1 space

Abstract In this work, we study the regularity criterion of the three-dimensional nematic liquid crystal flows. It is proved that if the vorticity satisfies ∫ 0 T ‖ ω ( t , ⋅ ) ‖ B . ∞ , ∞ − 1 2 1 + log ( e + ‖ ω ( t , ⋅ ) ‖ B . ∞ , ∞ − 1 ) d t ∞ , where B ∞ , ∞ − 1 denotes the critical Besov space, then the solution ( u , d ) becomes a regular solution on ( 0 , T ] . This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11] .

[1]  Yong Zhou,et al.  Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces , 2009 .

[2]  Fanghua Lin,et al.  Partial regularity of the dynamic system modeling the flow of liquid crystals , 1995 .

[3]  Yong Zhou,et al.  A Regularity Criterion for the Nematic Liquid Crystal Flows , 2010 .

[4]  Qiao Liu,et al.  Regularity of solutions to 3-D nematic liquid crystal flows , 2010 .

[5]  Yong Zhou,et al.  Logarithmically Improved Regularity Criteria for the Navier–Stokes and MHD Equations , 2011 .

[6]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[7]  Stephen Montgomery-Smith.  Conditions Implying Regularity of the Three Dimensional Navier-Stokes Equation , 2003 .

[8]  Qiao Liu,et al.  Logarithmically Improved Criteria for the 3D Nematic Liquid Crystal Flows in the Multiplier Spaces , 2012 .

[9]  J. Ericksen.  Continuum theory of nematic liquid crystals , 1987 .

[10]  F. Lin.  Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena , 1989 .

[11]  Jishan Fan,et al.  Regularity Criterion for the 3D Nematic Liquid Crystal Flows , 2012 .

[12]  Tosio Kato,et al.  Commutator estimates and the euler and navier‐stokes equations , 1988 .

[13]  Björn Jawerth,et al.  Some observations on Besov and Lizorkin-Triebel spaces. , 1977 .

[14]  Song Jiang,et al.  On logarithmically improved regularity criteria for the Navier–Stokes equations in ℝn , 2011 .

[15]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[16]  Qiao Liu,et al.  A new regularity criterion for the nematic liquid crystal flows , 2012 .