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),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_x² ∩ L_t²H_x¹, ▽ P ∈ L_t^4/3L_x^4/3, and ▽d ∈ L_t^∞L_x² ∩ L_t²H_x²; or (ii) for n = 3, u ∈ L_t^∞L_x² ∩ L_t²H_x¹ ∩ C ([0, T), Lⁿ), P ∈ L_t^n/2L_x^n/2, and ▽d ∈ L_t²L_x² ∩ C ([0, T), Lⁿ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.