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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2505.07506 |
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| _version_ | 1866917205346615296 |
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| author | Canevari, Giacomo Dipasquale, Federico Luigi Stroffolini, Bianca |
| author_facet | Canevari, Giacomo Dipasquale, Federico Luigi Stroffolini, Bianca |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics Canevari, Giacomo Dipasquale, Federico Luigi Stroffolini, Bianca Analysis of PDEs 35Q56, 76A15, 49Q15, 26B30 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. |
| title | The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics |
| topic | Analysis of PDEs 35Q56, 76A15, 49Q15, 26B30 |
| url | https://arxiv.org/abs/2505.07506 |