  We consider the hydrodynamics for biaxial nematic phases described by a field
of orthonormal frame, which can be derived from a molecular-theory-based tensor
model. We prove the uniqueness of global weak solutions to the Cauchy problem
of the frame hydrodynamics in dimensional two. The proof is mainly based on the
suitable weaker energy estimates within the Littlewood--Paley analysis. We take
full advantage of the estimates of nonlinear terms with rotational derivatives
on $SO(3)$, together with cancellation relations and dissipative structures of
the biaxial frame system.

Analysis of PDEs