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

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 ∈ L ∞ L 2 ∩ L 2 H 1 , ▿ P ∈ L 4/3 L 4/3 , and ▿d ∈ L ∞ L 2 ∩ L 2 H 2 ; or (ii) for n = 3, u ∈ L ∞ L 2 ∩ L 2 H 1 ∩ C ([0, T), L n ), P ∈ L /2 L /2 , and ▿d ∈ L 2 L 2 ∩ C ([0, T), L n ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.