### 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}
^{2}H
_{x}
^{1}, ▽ P ∈ L
_{t}
^{4/3}L
_{x}
^{4/3}, and ▽d ∈ L
_{t}
^{∞}L
_{x}
^{2} ∩ L
_{t}
^{2}H
_{x}
^{2}; or (ii) for n = 3, u ∈ L
_{t}
^{∞}L
_{x}
^{2} ∩ L
_{t}
^{2}H
_{x}
^{1} ∩ C ([0, T), L
^{n}), P ∈ L
_{t}
^{n/2}L
_{x}
^{n/2}, and ▽d ∈ L
_{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.

Original language | English (US) |
---|---|

Pages (from-to) | 921-938 |

Number of pages | 18 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 31 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2010 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Chinese Annals of Mathematics. Series B*, vol. 31, no. 6, pp. 921-938. https://doi.org/10.1007/s11401-010-0612-5

}

TY - JOUR

T1 - On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

AU - Lin, Fang-Hua

AU - Wang, Changyou

PY - 2010/11

Y1 - 2010/11

N2 - 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.

AB - 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.

KW - Harmonic maps

KW - Hydrodynamic flow

KW - Nematic liquid crystals

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=78650186120&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650186120&partnerID=8YFLogxK

U2 - 10.1007/s11401-010-0612-5

DO - 10.1007/s11401-010-0612-5

M3 - Article

VL - 31

SP - 921

EP - 938

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

SN - 0252-9599

IS - 6

ER -