Nematic liquid crystals: Ericksen-Leslie theory with general stress tensors

TL;DR

This paper proves the strong well-posedness of the Ericksen-Leslie model for nematic liquid crystals with general stress tensors, including nonlinear boundary conditions, without structural assumptions on Leslie coefficients.

Contribution

It establishes the first strong well-posedness results for the general Ericksen-Leslie system with minimal assumptions, handling nonlinear boundary conditions and anisotropic elasticity.

Findings

Existence and uniqueness of local strong solutions in $L_p$-spaces.

The director field maintains unit length if initially unit length.

Solutions depend continuously on initial data.

Abstract

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the $L_p$-setting. More precisely, the existence and uniqueness of a local, strong $L_p$-solution to the general system is proved and it is shown that the director $d$ satisfies $|d|_2\equiv 1$ provided this holds for its initial data $d_0$. In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi's relation.