Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy

TL;DR

This paper proves the global well-posedness of the 2D Beris-Edwards system for nematic liquid crystals with a general free energy, under small initial conditions, using an innovative approximation method.

Contribution

It establishes the existence and uniqueness of global weak solutions for the 2D Beris-Edwards system with a broad class of free energies, including cubic terms.

Findings

Global weak solutions exist under small initial $Q$-tensor norm.

The approximation system preserves the $L^ abla$-norm of the $Q$-tensor.

The approach handles unbounded free energy contributions.

Abstract

In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an evolution equation for the order parameter $Q$-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial $L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the $L^\infty$-norm of the $Q$-tensor along the time evolution.