On Multi-dimensional Compressible Flows of Nematic Liquid Crystals with Large Initial Energy in a Bounded Domain

Abstract

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 $\Omega\subset \mathbb{R}^N$, where N=2 or 3. By exploiting a maximum principle, Nirenberg's interpolation inequality and a smallness condition imposed on the $N$-th component of initial direction field $\mf{d}_0$ to overcome the difficulties induced by the supercritical nonlinearity $|\nabla{\mathbf d}|^2{\mathbf d}$ 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.