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| On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals |
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Citation： |
Fanghua LIN,Changyou WANG.On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals[J].Chinese Annals of Mathematics B,2010,31(6):921~938 |
| Page view： 1993
Net amount： 1274 |
Authors： |
Fanghua LIN; Changyou WANG; |
Foundation： |
the National Science Foundations (Nos. 0700517, 1001115). |
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| 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 $n=2$,
$u\in L^\infty_t L^2_x\cap L^2_tH^1_x$,\ $\nabla P\in
L^{\frac43}_tL^{\frac43}_x$, and $\nabla d\in L^\infty_t L^2_x\cap L^2_t H^2_x$;
or (ii) for $n=3$, $u\in L^\infty_tL^2_x\cap L^2_tH^1_x\cap C([0,T), L^n)$, $P\in L^{\frac{n}2}_t L^{\frac{n}2}_x$,
and $\nabla d\in L^2_tL^2_x\cap C([0,T), L^{n})$.
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 |
Classification： |
35K55 |
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