Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866910570528112640 |
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| author | Liu, Yuning Wu, Hao Xu, Xiang |
| author_facet | Liu, Yuning Wu, Hao Xu, Xiang |
| contents | In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an evolution equation for the order parameter $Q$-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial $L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the $L^\infty$-norm of the $Q$-tensor along the time evolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_09961 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy Liu, Yuning Wu, Hao Xu, Xiang Analysis of PDEs 35Q35, 35Q30, 76D03, 76D05 In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity $\mathbf{u}$ coupled with an evolution equation for the order parameter $Q$-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial $L^\infty$-norm of the $Q$-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the $L^\infty$-norm of the $Q$-tensor along the time evolution. |
| title | Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy |
| topic | Analysis of PDEs 35Q35, 35Q30, 76D03, 76D05 |
| url | https://arxiv.org/abs/1810.09961 |