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Liquid crystals and topological vorticity: smoothness of mild solutions

Fanghua Lin, Yannick Sire, Yantao Wu, Yifu Zhou

TL;DR

This work develops a class of 2D PDE models that couple a dissipative SQG-type scalar $\theta$ with a nematic director field $d$ through a nonlocal velocity $u$ and a stress-driven forcing, preserving an energy-dissipation structure. The vorticity formulation reduces the fluid system to a scalar drift-diffusion equation for $\theta$ linked to the harmonic-map heat flow for $d$, with theBiot–Savart relation $u = \nabla^{\perp}(-\Delta)^{-1+\alpha}\theta$ and fractional diffusion $(-\Delta)^a$, allowing a tunable criticality via $(a,\alpha)$. The main result proves global regularity of mild solutions in 2D under near-optimal initial data, via a three-stage bootstrap: higher integrability, Hölder regularity, and full smoothness, using parabolic Morrey–Campanato spaces and Riesz-potential techniques. These findings connect topological vorticity and conserved geometric motions with dissipative fluid-like dynamics, offering a tractable framework that mirrors Navier–Stokes/Euler features while preserving gradient-flow structure, and may inform models of ferromagnetic topological dynamics.

Abstract

We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the crystal, moving by the gradient flow of the energy of maps. The main idea of such models is to have a better insight on the vorticity formulation of the Liquid Crystal Flow and to tackle some regularity issues in the associated conserved geometric motions. One of the advantage of the present PDEs is to capture features of the Navier-Stokes equations (or Euler) through a {\sl scalar} unknown, keeping the advection-diffusion structure of the orientation field. We obtain regularity for mild solutions under natural assumptions for the initial data, which are actually near-optimal. Along the way, we also draw some links with natural models of (anti-)ferromagnets previously investigated.

Liquid crystals and topological vorticity: smoothness of mild solutions

TL;DR

This work develops a class of 2D PDE models that couple a dissipative SQG-type scalar with a nematic director field through a nonlocal velocity and a stress-driven forcing, preserving an energy-dissipation structure. The vorticity formulation reduces the fluid system to a scalar drift-diffusion equation for linked to the harmonic-map heat flow for , with theBiot–Savart relation and fractional diffusion , allowing a tunable criticality via . The main result proves global regularity of mild solutions in 2D under near-optimal initial data, via a three-stage bootstrap: higher integrability, Hölder regularity, and full smoothness, using parabolic Morrey–Campanato spaces and Riesz-potential techniques. These findings connect topological vorticity and conserved geometric motions with dissipative fluid-like dynamics, offering a tractable framework that mirrors Navier–Stokes/Euler features while preserving gradient-flow structure, and may inform models of ferromagnetic topological dynamics.

Abstract

We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the crystal, moving by the gradient flow of the energy of maps. The main idea of such models is to have a better insight on the vorticity formulation of the Liquid Crystal Flow and to tackle some regularity issues in the associated conserved geometric motions. One of the advantage of the present PDEs is to capture features of the Navier-Stokes equations (or Euler) through a {\sl scalar} unknown, keeping the advection-diffusion structure of the orientation field. We obtain regularity for mild solutions under natural assumptions for the initial data, which are actually near-optimal. Along the way, we also draw some links with natural models of (anti-)ferromagnets previously investigated.
Paper Structure (9 sections, 9 theorems, 117 equations)

This paper contains 9 sections, 9 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Derivation of the system
  3. The forcing term
  4. Ad hoc singular forcing
  5. Topological vorticity formulation
  6. Regularity of mild solutions
  7. Higher integrability: Proof of Theorem \ref{['Prop: Reg1-intro']}
  8. Hölder regularity: proof of Theorem \ref{['Prop: Reg2-intro']}
  9. Proof of the main Theorem \ref{['Thm: Reg-intro']}

Key Result

Theorem 1.1

Assume that a mild solution of eqn: SQGHMF satisfies $\theta_0\in L^{\frac{2}{2a-1}}(\mathbb{R}^2), ~\nabla d_0\in L^2(\mathbb{R}^2),~ \theta\in L^2\cap L^{p_0}((0,T)\times \mathbb{R}^2)$ for some $p_0>\frac{2a+2}{2a-1}$, and $|\nabla d| \in L^2\cap L^{q_0}((0,T)\times \mathbb{R}^2)$ for some $q_0>4

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4: Fractional Parabolic Poincare Inequality
  • proof
  • Definition 3.5: Parabolic Morrey space, Parabolic Campanato space, Maximal operator
  • ...and 9 more