An elementary proof of eigenvalue preservation for the co-rotational Beris-Edwards system

TL;DR

This paper provides a straightforward proof demonstrating that the eigenvalues of the Q-tensor in the co-rotational Beris-Edwards system are preserved over time, applicable in both bounded and unbounded domains.

Contribution

It offers an alternative, direct proof of eigenvalue preservation for the Q-tensor in the co-rotational Beris-Edwards system, extending validity to bounded domains.

Findings

Eigenvalues of the Q-tensor are preserved over time.

The proof applies to both bounded and unbounded spatial domains.

The approach simplifies understanding of eigenvalue preservation in liquid crystal models.

Abstract

We study the co-rotational Beris-Edwards system modeling nematic liquid crystals and revisit the eigenvalue preservation property discussed in \cite{XZ16}. We give an alternative but direct proof to the eigenvalue preservation of the initial data for the $Q$-tensor. It is noted that our proof is not only valid in the whole space case, but in the bounded domain case as well.