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

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.