Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy

TL;DR

This paper proves that measure-valued solutions to the Ericksen-Leslie model with Oseen-Frank energy are unique when a strong solution exists, using a relative energy inequality in three dimensions.

Contribution

It establishes the weak-strong uniqueness property for measure-valued solutions of the Ericksen-Leslie system with Oseen-Frank energy in 3D.

Findings

Measure-valued solutions satisfy a weak-strong uniqueness property.

The uniqueness is shown via a relative energy inequality.

The results apply to solutions fulfilling an energy inequality.

Abstract

We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i. e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional.