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Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions


Authors: Zhen Lei, Dong Li and Xiaoyi Zhang
Journal: Proc. Amer. Math. Soc. 142 (2014), 3801-3810
MSC (2010): Primary 35Q35
DOI: https://doi.org/10.1090/S0002-9939-2014-12057-0
Published electronically: July 29, 2014
MathSciNet review: 3251721
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Abstract: We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of the harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global wellposedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin and Lin-Lin-Wang. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under a certain angle condition. Our proof is based on a frequency localization argument combined with the concentration-compactness approach which can be of independent interest.


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Additional Information

Zhen Lei
Affiliation: School of Mathematical Sciences, LMNS, and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
Email: leizhn@gmail.com

Dong Li
Affiliation: Department of Mathematics, University of British Columbia, 1984 Mathematical Road, Vancouver, BC V6T 1Z2, Canada
Email: mpdongli@gmail.com

Xiaoyi Zhang
Affiliation: Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, Iowa 52242
Email: zh.xiaoyi@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12057-0
Received by editor(s): May 11, 2012
Received by editor(s) in revised form: October 5, 2012
Published electronically: July 29, 2014
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society