Liouville theorems for a stationary and non-stationary coupled system of liquid crystal flows in local Morrey spaces

TL;DR

This paper establishes Liouville-type theorems for a coupled liquid crystal flow system in local Morrey spaces, showing under certain conditions that solutions must be trivial, thereby extending classical results for Navier-Stokes equations.

Contribution

It introduces new Liouville theorems for the Ericksen-Leslie system in Morrey spaces, enhancing understanding of solution triviality for coupled liquid crystal flows.

Findings

Solutions vanish under certain a priori conditions

Results apply to both stationary and non-stationary cases

Improves classical Liouville theorems for Navier-Stokes equations

Abstract

We consider here the simplified Ericksen-Leslie system on the whole three-dimensional space. This system deals with the incompressible Navier-Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both the stationary (time independing) case and the non-stationary (time depending) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi-product, our theorems also improve some well-known results on Liouville-type theorems for the particular case of classical Navier-Stokes equations