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

Lei, Zhen; Li, Dong; Zhang, Xiaoyi

doi:10.1090/S0002-9939-2014-12057-0

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.

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