On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals

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 ∈L∞tL2x∩L2tH1x，▽P∈L3/4tL4/3t，and ▽d∈L∞tL2x∩L2tH2x; or(ⅱ)for n=3，u ∈L∞tL2x∩L2tH1x∩C([0，T)，Ln)，P∈Ln/2tLn/2x，and▽d∈L2tL2x∩C([0，T)，Ln)．This answers affirmatively the uniqueness question posed by Lin-Lin-Wang．The proofs are very elementary.