Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions

TL;DR

This paper proves weak-strong uniqueness for the three-dimensional Ericksen-Leslie system with a nonconvex energy functional, using a relative energy approach adapted to the system's quadratic free energy and director norm constraints.

Contribution

It introduces a novel relative energy inequality for the Ericksen-Leslie system with nonconvex energy, establishing weak-strong uniqueness in three dimensions.

Findings

Weak-strong uniqueness holds for the system.

Relative energy inequality is established for nonconvex energy.

Method adapts techniques from fluid dynamics to liquid crystal models.

Abstract

We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty is that the relative energy inequality is proved for a system with a nonconvex energy.