Dynamics of the Ericksen-Leslie Equations with General Leslie Stress I: The Incompressible Isotropic Case

TL;DR

This paper establishes local and global well-posedness, stability, and exponential convergence to equilibrium for the Ericksen-Leslie equations modeling nematic liquid crystals, without relying on structural assumptions or Parodi's relation.

Contribution

It provides a comprehensive dynamic theory for the Ericksen-Leslie model in the isotropic case, proving well-posedness and stability without structural restrictions.

Findings

Local well-posedness in the Lp-setting

Existence of global strong solutions near equilibria

Exponential convergence to equilibrium

Abstract

The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven {\em without} any structural assumptions on the Leslie coefficients and in particular {\em without} assuming Parodi's relation.