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

Matthias Hieber, Jinkai Li, Mathias Wilke

TL;DR

This work establishes the first strong well-posedness result for the general Ericksen-Leslie system with general Leslie stress and anisotropic elasticity in bounded domains, incorporating a fully nonlinear boundary condition for the director field derived from thermodynamic principles. By deriving the Ericksen operator’s explicit principal part and boundary operator, the authors prove normal ellipticity and satisfy the Lopatinskii-Shapiro condition, enabling maximal $L_p$-regularity for the linearized problem. They then develop a nonlinear fixed-point framework in time-weighted spaces to obtain local-in-time, strong solutions with $|d|=1$ preserved, and they prove continuous dependence on initial data. The results hold without Parodi relations and under general Frank coefficient conditions, providing a rigorous foundation for the thermodynamically consistent Ericksen-Leslie model with anisotropic elasticity in bounded domains.

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.

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

TL;DR

This work establishes the first strong well-posedness result for the general Ericksen-Leslie system with general Leslie stress and anisotropic elasticity in bounded domains, incorporating a fully nonlinear boundary condition for the director field derived from thermodynamic principles. By deriving the Ericksen operator’s explicit principal part and boundary operator, the authors prove normal ellipticity and satisfy the Lopatinskii-Shapiro condition, enabling maximal -regularity for the linearized problem. They then develop a nonlinear fixed-point framework in time-weighted spaces to obtain local-in-time, strong solutions with preserved, and they prove continuous dependence on initial data. The results hold without Parodi relations and under general Frank coefficient conditions, providing a rigorous foundation for the thermodynamically consistent Ericksen-Leslie model with anisotropic elasticity in bounded domains.

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 -setting. More precisely, the existence and uniqueness of a local, strong -solution to the general system is proved and it is shown that the director satisfies provided this holds for its initial data . 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.
Paper Structure (11 sections, 11 theorems, 256 equations)

This paper contains 11 sections, 11 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. The general Ericksen-Leslie model and Main Results
  3. The Ericksen operator
  4. Computation of the Ericksen-operator
  5. Boundary condition for the Ericksen operator
  6. Functional analytic setting
  7. Properties of the linearized Ericksen operator
  8. Strong Ellipticity
  9. The Lopatinskii-Shapiro condition
  10. Linearized problems and maximal regularity
  11. Proofs of the Main Results

Key Result

Theorem 2.1

Let $1<p<\infty$, $\mu \in (\frac{1}{2} + \frac{5}{2p},1]$, $\Omega\subset{\mathbb R}^3$ be a bounded domain with boundary $\partial\Omega\in C^3$, and assume that the assumptions (R), (P) and (F) are satisfied. Then, given satisfying $|d_0(x)|_2= 1$ for all $x\in\Omega$, the compatibility conditions (B) as well as $u_0=0$ on $\partial\Omega$, there exists $T=T(u_0,d_0)>0$ such that problem eq:el

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.3
  • Proposition 5.1: Strong ellipticity
  • proof
  • Remark 5.2
  • Proposition 5.3
  • proof
  • Proposition 5.4: Lopatinskii-Shapiro condition
  • proof
  • Remark 5.5
  • ...and 13 more