Analysis and numerical approximation of energy-variational solutions to the Ericksen--Leslie equations

TL;DR

This paper introduces energy-variational solutions for the Ericksen--Leslie equations, establishing their existence, uniqueness properties, and computational approximation methods, with implications for modeling liquid crystal flows.

Contribution

It defines a new solution concept for the Ericksen--Leslie equations, proves existence and uniqueness properties, and develops a structure-preserving numerical scheme.

Findings

Energy-variational solutions are finer than dissipative solutions.

Existence of measure-valued solutions is implied under certain conditions.

Numerical experiments demonstrate the algorithm's applicability.

Abstract

We define the concept of energy-variational solutions for the Ericksen--Leslie equations in three spatial dimensions. This solution concept is finer than dissipative solutions and satisfies the weak-strong uniqueness property. For a certain choice of the regularity weight, the existence of energy-variational solutions implies the existence of measure-valued solutions and for a different choice, we construct an energy-variational solution with the help of an implementable, structure-inheriting space-time discretization. Computational studies are performed in order to provide some evidence of the applicability of the proposed algorithm.