Blow-up suppression for the nematic liquid crystal flow via Couette flow on $\mathbb{R}^2$

TL;DR

This paper demonstrates that a sufficiently large Couette flow can prevent blow-up in the nematic liquid crystal flow on ^2, even with initial energy exceeding the critical threshold, highlighting the stabilizing effect of large-scale velocities.

Contribution

It shows that large-scale Couette flow can suppress blow-up in nematic liquid crystal flow, extending understanding of flow stabilization mechanisms.

Findings

Large Couette flow prevents blow-up with initial energy > 8\u03a0.

Constructed examples with initial energy exceeding 8\u03a0 that are stabilized.

Blow-up suppression depends on the amplitude of the Couette flow.

Abstract

As is well known, for the harmonic heat flow or liquid crystal flow in two-dimension, the solution may blow up when the initial energy is greater than $8\pi$. Motivated by Lai--Lin--Wang--Wei--Zhou (CPAM, 2022), where singular solutions were constructed in the presence of small-scale velocity fields, it is natural to ask whether large-scale velocities may play a stabilizing role, preventing the concentration of blow-up. Here we show that the blow-up phenomenon can be suppressed by a Couette flow whose amplitude is large enough under a weak assumption on the anisotropic norm of the initial data. In particular, we construct examples with initial energy exceeding $8\pi$ that satisfy our assumptions.