# Long-time dynamics of Ericksen-Leslie system on $\mathbb S^2$

## TL;DR

This paper investigates the long-term behavior of the Ericksen-Leslie system modeling nematic liquid crystals on a spherical surface, establishing energy inequalities and conditions for convergence of solutions over time.

## Contribution

It provides new energy inequalities and convergence conditions for the full Ericksen-Leslie system on rac{S^2}{2}, extending previous results under weaker assumptions.

## Key findings

- Established key energy inequality for global weak solutions.
- Identified sufficient conditions for uniform convergence in L^2 and H^k spaces.
- Proved convergence results under small initial data.

## Abstract

In this paper, we study the long-time behavior of full Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals between two dimensional unit spheres. Under a weaker assumption for Leslie's coefficients, we give the key energy inequality for the global weak solution. At last, inspired by the conditions on the simplified system, we establish several sufficient conditions which guarantee the uniform convergence of the system in $L^2$ and $H^k$ spaces as time tends to infinity under small initial data.