Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions
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- by Zhen Lei, Dong Li and Xiaoyi Zhang PDF
- Proc. Amer. Math. Soc. 142 (2014), 3801-3810 Request permission
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.References
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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
- MR Author ID: 714906
- Email: zh.xiaoyi@gmail.com
- 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
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 3801-3810
- MSC (2010): Primary 35Q35
- DOI: https://doi.org/10.1090/S0002-9939-2014-12057-0
- MathSciNet review: 3251721