Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals

TL;DR

This paper proves the existence and uniqueness of global strong solutions for the three-dimensional incompressible flow of liquid crystals with small initial data, ensuring consistency among weak solutions in a bounded domain.

Contribution

It establishes the global existence and uniqueness of strong solutions for liquid crystal flow equations in 3D, and shows weak solutions coincide with strong solutions when they exist.

Findings

Global strong solutions exist for small initial data.

Weak solutions are unique and coincide with strong solutions.

The results apply to smooth bounded domains.

Abstract

The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution.