On the General Ericksen-Leslie System: Parodi's Relation, Well-posedness and Stability

TL;DR

This paper explores how Parodi's relation influences the well-posedness and stability of the Ericksen-Leslie system modeling nematic liquid crystals, providing theoretical insights and stability results.

Contribution

It offers a formal derivation of the system under Parodi's relation and proves global well-posedness and stability results in this context.

Findings

Global well-posedness for large viscosity our

Lyapunov stability near local energy minimizers

Connection between Parodi's relation and linear stability

Abstract

In this paper we investigate the role of Parodi's relation in the well-posedness and stability of the general Ericksen-Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the Ericksen-Leslie system through an appropriate energy variational approach under Parodi's relation, in which we can distinguish the conservative/dissipative parts of the induced elastic stress. Next, we prove global well-posedness and long-time behavior of the Ericksen-Leslie system under the assumption that the viscosity $\mu_4$ is sufficiently large. Finally, under Parodi's relation, we show the global well-posedness and Lyapunov stability for the Ericksen-Leslie system near local energy minimizers. The connection between Parodi's relation and linear stability of the Ericksen-Leslie system is also discussed.