Global Existence and Stability for a Hydrodynamic System in the Nematic Liquid Crystal Flows

TL;DR

This paper proves the global existence and stability of solutions for a coupled hydrodynamic system modeling nematic liquid crystal flows, using advanced mathematical techniques to establish uniqueness and self-similarity.

Contribution

It introduces a novel analysis framework for proving global solutions and weak-strong uniqueness in nematic liquid crystal flow models with initial data in critical Besov spaces.

Findings

Existence of unique local solutions for initial data in critical Besov spaces.

Global solutions exist when initial data is sufficiently small.

Weak-strong uniqueness ensures solutions coincide with weak solutions for the same initial data.

Abstract

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solutions. In order to figure out the relation between the solution obtained here and weak solution of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.