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

Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$