Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy

TL;DR

This paper establishes the existence of measure-valued solutions for the Ericksen-Leslie model with Oseen-Frank energy, introducing generalized gradient Young measures and demonstrating energy inequalities.

Contribution

It introduces generalized gradient Young measures and proves existence of measure-valued solutions for the Ericksen-Leslie system with energy inequalities.

Findings

Existence of measure-valued solutions proven.

Introduction of generalized gradient Young measures.

Solutions satisfy an energy inequality.

Abstract

In this article, we prove the existence of measure-valued solutions to the Ericksen-Leslie system equipped with the Oseen-Frank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.