Global Well-posedness and Long-time Behavior of the Two-dimensional General Ericksen--Leslie System in the Isotropic Case under a Magnetic Field
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917065159344128 |
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| author | Wu, Qingtong |
| author_facet | Wu, Qingtong |
| contents | This paper establishes the global well-posedness and long-time dynamics of the general Ericksen--Leslie system for isotropic nematic liquid crystals under a constant magnetic field. On the two-dimensional torus $\mathbb{T}^2$, a liquid crystal molecule coincides with itself under rotations by integer multiples of $π$, which results in special boundary conditions. We prove the existence of global-in-time strong solutions by developing novel high-order energy estimates and employing compactness techniques. A key challenge lies in controlling the orientation of the liquid crystal molecules. After achieving a uniform bound for the molecular orientation angle in $\mathbb{S}^1$, we further characterize the long-time behavior of the solutions. This is accomplished by applying the Lojasiewicz--Simon inequality, which reveals the convergence of the solutions as time approaches infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Well-posedness and Long-time Behavior of the Two-dimensional General Ericksen--Leslie System in the Isotropic Case under a Magnetic Field Wu, Qingtong Analysis of PDEs Mathematical Physics This paper establishes the global well-posedness and long-time dynamics of the general Ericksen--Leslie system for isotropic nematic liquid crystals under a constant magnetic field. On the two-dimensional torus $\mathbb{T}^2$, a liquid crystal molecule coincides with itself under rotations by integer multiples of $π$, which results in special boundary conditions. We prove the existence of global-in-time strong solutions by developing novel high-order energy estimates and employing compactness techniques. A key challenge lies in controlling the orientation of the liquid crystal molecules. After achieving a uniform bound for the molecular orientation angle in $\mathbb{S}^1$, we further characterize the long-time behavior of the solutions. This is accomplished by applying the Lojasiewicz--Simon inequality, which reveals the convergence of the solutions as time approaches infinity. |
| title | Global Well-posedness and Long-time Behavior of the Two-dimensional General Ericksen--Leslie System in the Isotropic Case under a Magnetic Field |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2411.06748 |