Dynamics of the general $Q$-tensor model interacting with a rigid body

TL;DR

This paper investigates the interaction between nematic liquid crystals and a rigid body using the $Q$-tensor model, proving energy decay, establishing well-posedness, and analyzing the mathematical properties of the system.

Contribution

It introduces a rigorous analysis of the fluid-rigid body interaction with the $Q$-tensor model, including energy decay and well-posedness results for large and small data.

Findings

Total energy decreases over time.

Established maximal $L^p$-regularity of the linearized problem.

Proved local and global well-posedness for the interaction system.

Abstract

In this article, the fluid-rigid body interaction problem of nematic liquid crystals described by the general Beris-Edwards $Q$-tensor model is studied. It is proved first that the total energy of this problem decreases in time. The associated mathematical problem is a quasilinear mixed-order system with moving boundary. After the transformation to a fixed domain, a monolithic approach based on the added mass operator and lifting arguments is employed to establish the maximal $L^p$-regularity of the linearized problem in an anisotropic ground space. This paves the way for the local strong well-posedness for large data and global strong well-posedness for small data of the interaction problem.