Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Fan Yang

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.

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

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 for small initial data . This is the first well-posedness result for the system with initial data in critical space.

Paper Structure

This paper contains 6 sections, 5 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Known results for active nematic liquid crystals
  3. Motivation at the critical regularity
  4. Main result
  5. Preliminaries
  6. Proof of Theorem \ref{['thm1.1']}

Key Result

Theorem 1.1

There exists a small positive constant $\epsilon_0$ such that if the initial data $(Q_0, u_0)\in L^\infty\times {\rm BMO}^{-1}$ satisfying then the system eq1.1 admits a unique local solution $(Q,u)\in \mathbb{X}_{T}\times\mathbb{Y}_{T}$ in $[0,T]$ satisfying where $C$ is a constant depending only on $|a|$, $b$, $c$, $\mu$, $\kappa$, $\lambda$ and $\Gamma$. $\blacktriangleleft$$\blacktrianglelef

Theorems & Definitions (11)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 1 more