# Small data global regularity for 3-D Ericksen-Leslie's hyperbolic liquid crystal model without kinematic transport

## TL;DR

This paper proves global regularity for small initial data in a 3D hyperbolic liquid crystal model, combining Navier-Stokes and wave map equations, using space-time resonance techniques.

## Contribution

It establishes the first global regularity result for this complex coupled hyperbolic liquid crystal system without kinematic transport.

## Key findings

- Global regularity for small initial data is achieved.
- The space-time resonance method is effectively applied.
- The model couples Navier-Stokes with wave maps in 3D.

## Abstract

In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Global regularity for small and smooth initial data near the equilibrium is proved. The proof relies on the idea of space-time resonance.