Koch-Tataru theorem for 3D incompressible active nematic liquid crystals

TL;DR

This paper proves the existence and uniqueness of solutions for a 3D active nematic liquid crystal system with small initial data in critical spaces, using a Koch-Tataru type approach.

Contribution

It provides the first well-posedness result for the active nematic system in critical function spaces, employing Kato's method and Banach contraction.

Findings

Established existence and uniqueness of solutions in critical spaces for small initial data.

First well-posedness result for the system with initial data in critical space.

Demonstrated the potential for computational and information transmission in active soft materials.

Abstract

We investigate the incompressible hydrodynamic system of the active nematic liquid crystals in the Beris-Edwards framework. Although we focus on constant activity in this paper, the simplified system derived from it exhibits the potential to perform computations and transmit information in active soft materials \cite{defect-active}. More precisely, by employing Kato's strategy for constructing mild solutions combined with the Banach contraction principle, we show the existence and uniqueness of the Koch-Tataru type solution in $\mathbb{R}^3$ for small initial data $(Q_0,u_0)\in L^\infty\times {\rm BMO}^{-1}$. This is the first well-posedness result for the system with initial data in critical space.