Well-posedness of frame hydrodynamics for biaxial nematic liquid crystals

Abstract: We consider the hydrodynamics for the biaxial nematic phase characterized by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. In dimension two and three, we establish the local well-posedness and the blow-up criterion for smooth solutions to the frame hydrodynamic model. Furthermore, we prove the global existence of weak solutions in $\mathbb{R}^2$ which are nonsmooth at finitely many singular times.