Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$

TL;DR

This paper investigates the regularity of nematic liquid crystal Q-tensor models in three dimensions, demonstrating that certain classical conditions prevent solution blow-up, thus advancing understanding of solution behavior in these complex systems.

Contribution

It establishes that classical Prodi-Serrin and Beale-Kato-Majda conditions ensure regularity for the nematic LCD system with Q-tensor, resolving an open question.

Findings

Proves solutions do not blow-up under specific conditions.

Shows classical regularity criteria prevent solution singularities.

Extends understanding of liquid crystal models in mathematical physics.

Abstract

We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.