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

TL;DR

This paper proves the global existence of weak solutions for a complex model of compressible nematic liquid crystal flows with large initial energy in bounded domains, overcoming supercritical nonlinearities through advanced mathematical techniques.

Contribution

It introduces a novel approach combining maximum principles, interpolation inequalities, and approximation schemes to establish global solutions without smallness constraints.

Findings

Proved global existence of weak solutions for the system.

Handled large initial energy without density or velocity restrictions.

Overcame supercritical nonlinearities in the equations.

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.