Global weak flow of three-dimensional nematic directors without a hemisphere restriction
Let $\Omega \subset \mathbb{R}^3$ be a smooth bounded domain and consider the simplified Ericksen–Leslie system for a velocity field $u$, a pressure $p$ and a director field $d$ with values in $\mathbb{S}^2$, $$u_t + u \cdot \nabla u - \Delta u + \nabla p = -\nabla \cdot (\nabla d \odot \nabla d), \qquad \nabla \cdot u = 0,$$ $$d_t + u \cdot \nabla d = \Delta d + |\nabla d|^2 d, \qquad |d| = 1,$$ where $(\nabla d \odot \nabla d)_{ij} = \partial_i d \cdot \partial_j d$, with initial data $(u_0, d_0)$ and the time-independent boundary values $u = 0$, $d = d_0$ on $\partial\Omega$. The data are smooth and compatible: $u_0 \colon \overline\Omega \to \mathbb{R}^3$ with $\nabla \cdot u_0 = 0$ and $u_0|_{\partial\Omega} = 0$, and $d_0 \colon \overline\Omega \to \mathbb{S}^2$. A global weak solution is here a distributional solution, with the elastic stress exactly as displayed and no defect measure, such that on every finite time interval $u \in L^\infty_t L^2_x \cap L^2_t H^1_{0,x}$, $d \in L^\infty_t H^1_x$, $\tau = d_t + u \cdot \nabla d \in L^2_{t,x}$, and $$\int_\Omega \bigl(|u(t)|^2 + |\nabla d(t)|^2\bigr) + 2\int_0^t \!\! \int_\Omega \bigl(|\nabla u|^2 + |\tau|^2\bigr) \le \int_\Omega \bigl(|u_0|^2 + |\nabla d_0|^2\bigr).$$ Lin, Lin and Wang proved existence of global weak solutions in two dimensions. In three dimensions, Lin and Wang proved existence of a global weak solution satisfying this energy inequality for almost every $t$ when $d_0$ takes values in the closed upper hemisphere; without that assumption, they note, defect measures may arise in their approximation scheme. Huang, Lin, Liu and Wang constructed solutions in three dimensions that develop singularities in finite time.
Problem. For all such data, with no hemisphere or small-energy restriction on $d_0$, does a global weak solution exist?
Source: Lin and Wang (2016), §1, as an outstanding open problem for data in L² × H¹(Ω, S²); see also Lin and Wang (2014), Question 2.2(i), for the general Ericksen–Leslie system. Stated for smooth data, in this form, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Ericksen–Leslie system, nematic liquid crystals, weak solutions
References
@article{lin2016,
author = {Lin, Fanghua and Wang, Changyou},
title = {Global existence of weak solutions of the nematic liquid crystal flow in dimension three},
journal = {Comm. Pure Appl. Math.},
volume = {69},
number = {8},
year = {2016},
pages = {1532--1571},
doi = {10.1002/cpa.21583},
eprint = {1408.4146},
archivePrefix = {arXiv},
}
@article{lin2014,
author = {Lin, Fanghua and Wang, Changyou},
title = {Recent developments of analysis for hydrodynamic flow of nematic liquid crystals},
journal = {Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.},
volume = {372},
number = {2029},
year = {2014},
pages = {20130361},
doi = {10.1098/rsta.2013.0361},
eprint = {1408.4138},
archivePrefix = {arXiv},
}
@article{linlinwang2010,
author = {Lin, Fanghua and Lin, Junyu and Wang, Changyou},
title = {Liquid crystal flows in two dimensions},
journal = {Arch. Ration. Mech. Anal.},
volume = {197},
number = {1},
year = {2010},
pages = {297--336},
doi = {10.1007/s00205-009-0278-x},
}
@article{huang2016,
author = {Huang, Tao and Lin, Fanghua and Liu, Chun and Wang, Changyou},
title = {Finite time singularity of the nematic liquid crystal flow in dimension three},
journal = {Arch. Ration. Mech. Anal.},
volume = {221},
number = {3},
year = {2016},
pages = {1223--1254},
doi = {10.1007/s00205-016-0983-1},
eprint = {1504.01080},
archivePrefix = {arXiv},
}
@misc{aim-liquid-crystal-three-dimensional-weak,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global weak flow of three-dimensional nematic directors without a hemisphere restriction}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/377-liquid-crystal-three-dimensional-weak.md},
note = {Entry 377 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00611, "Global weak flow of three-dimensional nematic directors without a hemisphere restriction", clopen.solutions.