Global solution to the three-dimensional compressible flow of liquid crystals

TL;DR

This paper proves the existence and uniqueness of a global strong solution for the three-dimensional compressible flow of nematic liquid crystals near equilibrium, using critical Besov spaces and linearized system estimates.

Contribution

It establishes the first global well-posedness result for this flow model in critical Besov spaces with initial data close to equilibrium.

Findings

Global strong solution exists and is unique near equilibrium

Solutions are established in critical Besov spaces

Method involves linearized system uniform estimates

Abstract

The Cauchy problem for the three-dimensional compressible flow of nematic liquid crystals is considered. Existence and uniqueness of the global strong solution are established in critical Besov spaces provided that the initial datum is close to an equilibrium state $(1,{\bf 0}, \hat{\d})$ with a constant vector $\hat{\d}\in S^2$. The global existence result is proved via the local well-posedness and uniform estimates for proper linearized systems with convective terms.