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

Yubo Chen, Wendong Wang, Juncheng Wei, Guoxu Yang

TL;DR

This work shows that a strong Couette shear $U(y)=(Ay,0)$ can suppress blow-up in the 2D nematic liquid crystal flow around the Couette background, even when the initial energy exceeds $8\pi$. By introducing an anisotropic $Y_{m,\epsilon}$ norm and a space–time $X_{a,m,\epsilon}$ framework, the authors establish a bootstrap argument enabled by time-dependent Fourier multipliers that yield enhanced dissipation on the $|D_x|^{1/3}$-scale and inviscid damping. A frequency-based two-layer decomposition controls nonlinear interactions, ensuring energy transfer to high frequencies dominates nonlinear focusing for sufficiently large $A$. The main result provides global-in-time solutions with weighted $L^{\infty}$ bounds for the velocity and the director field, demonstrating blow-up suppression in a higher-dimensional liquid crystal model and highlighting a potentially broad scope for anisotropic, shear-driven stabilization in complex fluids. The analysis integrates advanced harmonic-analytic tools (multiplier method, region-wise convolution estimates) with an intricate bootstrap that couples the fluid and director dynamics under a robust, large-shear regime.

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 $8π$. 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 $8π$ that satisfy our assumptions.

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

TL;DR

This work shows that a strong Couette shear can suppress blow-up in the 2D nematic liquid crystal flow around the Couette background, even when the initial energy exceeds . By introducing an anisotropic norm and a space–time framework, the authors establish a bootstrap argument enabled by time-dependent Fourier multipliers that yield enhanced dissipation on the -scale and inviscid damping. A frequency-based two-layer decomposition controls nonlinear interactions, ensuring energy transfer to high frequencies dominates nonlinear focusing for sufficiently large . The main result provides global-in-time solutions with weighted bounds for the velocity and the director field, demonstrating blow-up suppression in a higher-dimensional liquid crystal model and highlighting a potentially broad scope for anisotropic, shear-driven stabilization in complex fluids. The analysis integrates advanced harmonic-analytic tools (multiplier method, region-wise convolution estimates) with an intricate bootstrap that couples the fluid and director dynamics under a robust, large-shear regime.

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 . 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 that satisfy our assumptions.
Paper Structure (21 sections, 19 theorems, 158 equations)

This paper contains 21 sections, 19 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Stability mechanisms
  4. Enhanced dissipation
  5. Inviscid damping
  6. The key idea in the proof
  7. Construction of the multipliers
  8. Frequency-based two-layer decomposition
  9. Energy transfer mechanism and blow-up suppression
  10. Notations and Outlines
  11. Preliminaries and proof of the main theorem
  12. Space–time estimates
  13. $L^\infty$ embedding
  14. Inequalities in different frequency regions
  15. Proof of the main theorem under Proposition \ref{['main prop2']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\frac{1}{3}<\epsilon<\frac{1}{2}<m$, $0<a<\frac{1}{16(1+2 \pi)}$ and $\delta>1$. Assume that the initial data $u_{\mathrm{in}}\in H^1(\mathbb{R}^2)$ and $d_{\mathrm{in}}$ satisfy There exists a positive constant $\bar{A}_1$ satisfying such that if $A>\bar{A}_1$ and the solutions to eq:main are global in time satisfying for all $t\geq0$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4: Physical mechanism of blow-up suppression
  • Remark 1.5: Motivation for the $|D_x|^{1/3}$ index
  • Remark 1.6: The anisotropic norm
  • Remark 1.7: Examples of initial data with arbitrarily large energy
  • Proposition 2.1: Proposition 2.3 in CWY2025b
  • Lemma 2.2: Lemma A.3 in CWY2025b
  • Lemma 2.3
  • ...and 28 more