Global well-posedness for the 3-D density-dependent liquid crystal flows in the critical Besov spaces

TL;DR

This paper establishes local and global well-posedness results for 3-D density-dependent liquid crystal flows in critical Besov spaces, allowing for large density variations but requiring certain smallness conditions for global existence.

Contribution

It proves the local well-posedness without small density variation assumptions and identifies conditions for global solutions when initial data are close to equilibrium.

Findings

Local well-posedness in critical Besov spaces.

Global existence under smallness conditions on initial data.

No smallness assumption on density variation.

Abstract

In this paper, we prove the local well-posedness of 3-D density-dependent liquid crystal flows with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial density is close enough to a positive constant and the critical Besov norm of the liquid crystal orientation field and the horizontal components of the initial velocity field polynomially small compared with the critical Besov norm to the veritcal component of the initial velocity field, then the system has a unique global solution.