Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain $\Omega \subset R^2$, where the velocity $u$ and the orthogonal unit vector fields $(m,n)$ admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.