Global well-posedness of radially symmetric strong solutions to two-dimensional compressible liquid crystal flows with large data and vacuum

Abstract: We study the initial boundary value problem of the two-dimensional compressible nematic liquid crystal flow with the shear viscosity $μ$ being a positive constant and bulk viscosity $λ$ being a power function of density with the power exponent $β$. Under the condition $β>1$, we establish the global existence and large time behavior of the radially symmetric strong solutions to this Vaigant--Kazhikhov type model of simplified compressible Ericksen-Leslie system of Dirichlet boundary conditions for the velocity and Neumann boundary ones for the director with arbitrary large data and vacuum. This work improves the results of Zhong and Zhou (\textit{Math. Ann.} \textbf{390}, 2024; \textit{J. Math. Pures Appl.}, \textbf{212}, 2026) for general 2D domains by removing the geometric angel condition on the director and relaxing the constrain on $β$ from $β>4/3$ to $β>1$. The key ingredient is that the rigidity mechanism arising from the radial symmetry of director prevents concentration phenomena in the transported harmonic heat flow.