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

Fang-Hua 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),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, u ∈ L t L x 2 ∩ L t 2H x 1, ▽ P ∈ L t 4/3L x 4/3, and ▽d ∈ L t L x 2 ∩ L t 2H x 2; or (ii) for n = 3, u ∈ L t L x 2 ∩ L t 2H x 1 ∩ C ([0, T), L n), P ∈ L t n/2L x n/2, and ▽d ∈ L t 2L x 2 ∩ C ([0, T), L n). 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

Fingerprint

Nematic liquid crystals
Harmonic Maps
Heat Flow
Nematic Liquid Crystal
Hydrodynamics
Uniqueness
Heat transfer
Compact Manifold
Riemannian Manifold
Weak Solution
n-dimensional
Class

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. / Lin, Fang-Hua; Wang, Changyou.

In: Chinese Annals of Mathematics. Series B, Vol. 31, No. 6, 11.2010, p. 921-938.

Research output: Contribution to journalArticle

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