Uniqueness for the dispersive Hunter-Saxton equation with low regularity solutions

It is well-known that solutions for the dispersive Hunter-Saxton equation in C ([0,T];H-S (S))boolean AND c(1) ([0,T];HS-1 (S)) with s > 3/2 are unique, see M. Li, Z. Yin [Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete Contin Dyn Syst Ser. 2017;37:6471-6485 and Z. Yin [On the structure of solutions to the periodic Hunter-Saxton equation. SIAM J Math Anal. 2004;36:272-283]. In this paper, we show that c ([0,T];HS(S))boolean AND c(1) ([0,T]; HS-1 (S)) with s > 3/2 is not a critical space for uniqueness. We firstly establish the energy conservation for weak solutions to the dispersive Hunter-Saxton equation in c(w) ([0,T];H-1 (S)boolean AND B-3,2(1) (S)), and then prove that every weak solution in C-w ([0,T];H-7/6 (S)) is unique. This weakens the traditional regularity condition required for the uniqueness.