On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂RNΩ⊂RN, where N=2 or 3N=2 or 3. By exploiting a maximum principle, Nirenbergʼs interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field d0d0 to overcome the difficulties induced by the supercritical nonlinearity |∇d|2d|∇d|2d in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.