Uniqueness of harmonic map heat flows and liquid crystal flows

Abstract

In this paper, we prove a limiting uniqueness criterion to harmonic map heat flows and liquid crystal flows. We firstly establish the uniqueness of harmonic map heat flows from $R^n$ to a smooth, compact Riemannian manifold $N$ in the class $C([0,T),BMO_T(R^n,N))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$ For the nematic liquid crystal flows $(v,d)$, we show that the mild solution is unique under the class $C([0,T),BMO_T^{-1}(R^n))\cap L^\infty_{loc}((0,T);L^\infty(R^n))\times C([0,T),BMO_T(R^n,S^2))\cap L^\infty_{loc}((0,T);\dot{W}^{1,\infty}(R^n))$ for $0< T ≤ +\infty.$