Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

TL;DR

This paper proves the global existence of weak solutions for a two-dimensional compressible nematic liquid crystal flow system, addressing challenges from supercritical nonlinearities and initial data restrictions.

Contribution

It introduces new techniques and estimates to handle supercritical nonlinearities and extends the existence results to large initial data with geometric conditions.

Findings

Established global weak solutions for small initial energy.

Proved existence of large weak solutions under geometric angle conditions.

Developed new analytical methods for supercritical nonlinearities.

Abstract

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.