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The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

Giacomo Canevari, Federico Luigi Dipasquale, Bianca Stroffolini

TL;DR

We study a 2D ferronematic model coupling a $\mathbf{Q}$-tensor and a magnetisation $\mathbf{M}$ via a Ginzburg–Landau type energy with a singular coupling. As $\varepsilon\to0$, the $\mathbf{Q}$-energy concentrates at a finite set of points, while the $\mathbf{M}$-energy concentrates along a rectifiable 1D set, linked by a coupling that induces a balance between a limiting Ginzburg–Landau varifold and a vectorial Allen–Cahn structure. A key innovation is a change of variables that decouples the system into a Ginzburg–Landau component for $\mathbf{Q}$ and a perturbed vectorial Allen–Cahn component for $\mathbf{M}$, enabling compactness and detailed structure results via varifold and Jacobian analysis. The work establishes a comprehensive asymptotic picture, including regularity and a hierarchy of singular sets, and extends the analysis to minimisers under mixed boundary conditions, with a rigorous balance law between the first variations of the limiting 1D varifold and the Ginzburg–Landau renormalised energy. These results advance understanding of gradient-driven singular structures in coupled vectorial systems and provide a robust framework for ferronematic energetics in the small-$\varepsilon$ regime.

Abstract

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\mathbf{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\mathbf{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\mathbf{Q}$ and~$\mathbf{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\varepsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~$\mathbf{Q}$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~$\mathbf{M}$-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component.

The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

TL;DR

We study a 2D ferronematic model coupling a -tensor and a magnetisation via a Ginzburg–Landau type energy with a singular coupling. As , the -energy concentrates at a finite set of points, while the -energy concentrates along a rectifiable 1D set, linked by a coupling that induces a balance between a limiting Ginzburg–Landau varifold and a vectorial Allen–Cahn structure. A key innovation is a change of variables that decouples the system into a Ginzburg–Landau component for and a perturbed vectorial Allen–Cahn component for , enabling compactness and detailed structure results via varifold and Jacobian analysis. The work establishes a comprehensive asymptotic picture, including regularity and a hierarchy of singular sets, and extends the analysis to minimisers under mixed boundary conditions, with a rigorous balance law between the first variations of the limiting 1D varifold and the Ginzburg–Landau renormalised energy. These results advance understanding of gradient-driven singular structures in coupled vectorial systems and provide a robust framework for ferronematic energetics in the small- regime.

Abstract

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~-tensor for the liquid crystal component and a magnetisation vector field~, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~ and~. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the -component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~-component.
Paper Structure (41 sections, 96 theorems, 720 equations, 1 figure)

This paper contains 41 sections, 96 theorems, 720 equations, 1 figure.

Table of Contents

  1. Preliminary estimates
  2. The space of $\mathbf{Q}$-tensors
  3. Properties of the potential $f_\varepsilon$.
  4. First consequences of the Euler-Lagrange equations
  5. Estimates for $\mathbf{Q}_\varepsilon$ and $\mathbf{M}_\varepsilon$
  6. Bounds in $W^{1,p}(\Omega)$ for $\mathbf{Q}_\varepsilon$
  7. $\eta$-ellipticity for $\mathbf{Q}_\varepsilon$
  8. An estimate for the energy of $\mathbf{M}_\varepsilon$
  9. Summary of the energy estimates
  10. Compactness
  11. Changes of variables: from $\mathbf{Q}$ to $\mathbf{q}$ and from $\mathbf{M}$ to $\mathbf{u}$
  12. Compactness for $\mathbf{Q}_\varepsilon$
  13. Compactness for $\mathbf{u}_\varepsilon$
  14. Compactness for $\mathbf{M}_\varepsilon$
  15. Refined energy estimates for $\mathbf{M}_\varepsilon$
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\Omega\subseteq\mathbb{R}^2$ be a bounded, simply connected domain of class $C^2$. Let $\{(\mathbf{Q}_\varepsilon, \,\mathbf{M}_\varepsilon)\}\subset W^{1,2}(\Omega, \, \mathscr{S}_0^{2\times 2})\times W^{1,2}(\Omega, \, \mathbb{R}^2)$ be a sequence of critical points of $\mathscr{F}_\varepsilo

Figures (1)

  • Figure 1: Example of a connection for $\{a_1 \, \ldots, a_6\}$ relative to $\Omega$.

Theorems & Definitions (237)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.4
  • Lemma 1.1: CanevariMajumdarStroffoliniWang
  • Lemma 1.2
  • ...and 227 more