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

Fanghua Lin, Changyou Wang

Research output: Contribution to journalArticle

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),W1,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, u ∈ LtLx2 ∩ Lt2Hx1, ▽ P ∈ Lt4/3Lx4/3, and ▽d ∈ LtLx2 ∩ Lt2Hx2; or (ii) for n = 3, u ∈ LtLx2 ∩ Lt2Hx1 ∩ C ([0, T), Ln), P ∈ Ltn/2Lxn/2, and ▽d ∈ Lt2Lx2 ∩ C ([0, T), Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

Original languageEnglish (US)
Pages (from-to)921-938
Number of pages18
JournalChinese Annals of Mathematics. Series B
Volume31
Issue number6
DOIs
StatePublished - Nov 2010

Keywords

  • Harmonic maps
  • Hydrodynamic flow
  • Nematic liquid crystals
  • Uniqueness

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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