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Chinese Annals of Mathematics, Series B

, Volume 31, Issue 6, pp 921–938 | Cite as

On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

  • Fanghua LinEmail author
  • Changyou Wang
Article

Abstract

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N, h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T),W 1,n ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, uL t L x 2 L t 2 H x 1 , ▿ PL t 4/3 L x 4/3 , and ▿dL t L x 2 L t 2 H x 2 ; or (ii) for n = 3, uL t L x 2 L t 2 H x 1 C ([0, T), L n ), PL t n/2 L x n/2 , and ▿dL t 2 L x 2 C ([0, T), L n ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

Keywords

Hydrodynamic flow Harmonic maps Nematic liquid crystals Uniqueness 

2000 MR Subject Classification

35K55 

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Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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