TY - JOUR

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

AU - Lin, Fanghua

AU - Wang, Changyou

N1 - Funding Information:
Manuscript received May 27, 2010. Published online October 22, 2010. ∗Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: linf@courant.nyu.edu ∗∗Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA. E-mail: cywang@ms.uky.edu ∗∗∗Project supported by the National Science Foundations (Nos. 0700517, 1001115).

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),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 ∈ Lt∞Lx2 ∩ Lt2Hx1, ▽ P ∈ Lt4/3Lx4/3, and ▽d ∈ Lt∞Lx2 ∩ Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞Lx2 ∩ 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.

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),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 ∈ Lt∞Lx2 ∩ Lt2Hx1, ▽ P ∈ Lt4/3Lx4/3, and ▽d ∈ Lt∞Lx2 ∩ Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞Lx2 ∩ 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.

KW - Harmonic maps

KW - Hydrodynamic flow

KW - Nematic liquid crystals

KW - Uniqueness

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U2 - 10.1007/s11401-010-0612-5

DO - 10.1007/s11401-010-0612-5

M3 - Article

AN - SCOPUS:78650186120

SN - 0252-9599

VL - 31

SP - 921

EP - 938

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

IS - 6

ER -