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Volume 28, Issue 4
A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows

Meng Bai, Qiao Liu & Jihong Zhao

DOI: 10.4208/jpde.v28.n4.5

J. Part. Diff. Eq., 28 (2015), pp. 358-369.

Published online: 2015-12

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  • Abstract
In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$∫^T_0\frac{||u||^{\frac{2}{1-r}}_{\dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.
  • Keywords

Ericksen-Leslie system Navier-Stokes equations blow-up criterion

  • AMS Subject Headings

76A15, 35B65, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

124204602@qq.com (Meng Bai)

liuqiao2005@163.com (Qiao Liu)

jihzhao@163.com (Jihong Zhao)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-28-358, author = {Bai , MengLiu , Qiao and Zhao , Jihong}, title = {A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows}, journal = {Journal of Partial Differential Equations}, year = {2015}, volume = {28}, number = {4}, pages = {358--369}, abstract = { In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$∫^T_0\frac{||u||^{\frac{2}{1-r}}_{\dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v28.n4.5}, url = {http://global-sci.org/intro/article_detail/jpde/5121.html} }
TY - JOUR T1 - A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows AU - Bai , Meng AU - Liu , Qiao AU - Zhao , Jihong JO - Journal of Partial Differential Equations VL - 4 SP - 358 EP - 369 PY - 2015 DA - 2015/12 SN - 28 DO - http://doi.org/10.4208/jpde.v28.n4.5 UR - https://global-sci.org/intro/article_detail/jpde/5121.html KW - Ericksen-Leslie system KW - Navier-Stokes equations KW - blow-up criterion AB - In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$∫^T_0\frac{||u||^{\frac{2}{1-r}}_{\dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.
Bai , MengLiu , Qiao and Zhao , Jihong. (2015). A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows. Journal of Partial Differential Equations. 28 (4). 358-369. doi:10.4208/jpde.v28.n4.5
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