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), we establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T ),W ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, we show the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt Lx ∩LtH x, ∇P ∈ L 4 3 t L 4 3 x , and ∇d ∈ Lt Lx ∩LtH x; or (ii) for n = 3, u ∈ Lt Lx ∩ LtH x ∩ C([0, T ), L), P ∈ L n 2 t L n 2 x , and ∇d ∈ LtL 2 x ∩C([0, T ), L). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.