Well-Posedness of Nematic Liquid Crystal Flow in $L^3_{\hbox{uloc}}(\R^3)$

TL;DR

This paper proves the local well-posedness of the simplified nematic liquid crystal flow in three dimensions for initial data with small uniformly locally $L^3$-norm, ensuring existence, uniqueness, and smoothness of solutions.

Contribution

It establishes the first well-posedness result for the nematic liquid crystal flow in $L^3_{uloc}$ spaces, extending understanding of solutions with small initial data.

Findings

Existence of a unique, global, smooth solution for small initial data.

Solutions have monotone decreasing $L^3$-energy over time.

The result applies to initial data with small $L^3_{uloc}$-norm.

Abstract

In this paper, we establish the local well-posedness for the Cauchy problem of the simplified version of hydrodynamic flow of nematic liquid crystals (\ref{LLF}) in $\mathbb R^3$ for any initial data $(u_0,d_0)$ having small $L^3_{\hbox{uloc}}$-norm of $(u_0,\nabla d_0)$. Here $L^3_{\hbox{uloc}}(\mathbb R^3)$ is the space of uniformly locally $L^3$-integrable functions. For any initial data $(u_0, d_0)$ with small $\displaystyle |(u_0,\nabla d_0)|_{L^3(\mathbb R^3)}$, we show that there exists a unique, global solution to (\ref{LLF}) which is smooth for $t>0$ and has monotone deceasing $L^3$-energy for $t\ge 0$.