A diffuse-interface Landau-de Gennes model for free-boundary problems in the theory of nematic liquid crystals
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866918144543555584 |
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| author | Wu, Dawei Shi, Baoming Han, Yucen Zhang, Pingwen Majumdar, Apala Zhang, Lei |
| author_facet | Wu, Dawei Shi, Baoming Han, Yucen Zhang, Pingwen Majumdar, Apala Zhang, Lei |
| contents | We introduce a diffuse-interface Landau-de Gennes free energy for nematic liquid crystals (NLC) systems, with free boundaries, in three dimensions submerged in isotropic liquid, and a phase field is introduced to model the deformable interface. The energy consists of the original Landau-de Gennes free energy, three penalty terms and a volume constraint. We prove the existence and regularity of minimizers for the diffuse-interface energy functional. We also prove a uniform maximum principle of the minimizer under appropriate assumptions, together with a uniqueness result for small domains. Then, we establish a sharp-interface limit where minimizers of the diffuse-interface energy converge to a minimizer of a sharp-interface energy using methods from $Γ$-convergence. Finally, we conduct numerical experiments with the diffuse-interface model and the findings are compared with existing works. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_21437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A diffuse-interface Landau-de Gennes model for free-boundary problems in the theory of nematic liquid crystals Wu, Dawei Shi, Baoming Han, Yucen Zhang, Pingwen Majumdar, Apala Zhang, Lei Analysis of PDEs 76A15, 49J27, 49K20, 35B50, 49J45, 35F30 We introduce a diffuse-interface Landau-de Gennes free energy for nematic liquid crystals (NLC) systems, with free boundaries, in three dimensions submerged in isotropic liquid, and a phase field is introduced to model the deformable interface. The energy consists of the original Landau-de Gennes free energy, three penalty terms and a volume constraint. We prove the existence and regularity of minimizers for the diffuse-interface energy functional. We also prove a uniform maximum principle of the minimizer under appropriate assumptions, together with a uniqueness result for small domains. Then, we establish a sharp-interface limit where minimizers of the diffuse-interface energy converge to a minimizer of a sharp-interface energy using methods from $Γ$-convergence. Finally, we conduct numerical experiments with the diffuse-interface model and the findings are compared with existing works. |
| title | A diffuse-interface Landau-de Gennes model for free-boundary problems in the theory of nematic liquid crystals |
| topic | Analysis of PDEs 76A15, 49J27, 49K20, 35B50, 49J45, 35F30 |
| url | https://arxiv.org/abs/2407.21437 |