Global regularity to the 2D inhomogeneous liquid crystal flows with large initial data and vacuum. (English) Zbl 1505.35074
Summary: We study the 2D incompressible nematic liquid crystal equations in a smooth bounded domain, where the velocity \(u\) and macroscopic molecular orientation \(d\) admit the Dirichlet and Neumann boundary condition, respectively. Under a geometric condition for the initial orientation field, we establish the global existence of strong solutions with large initial data. In particular, the initial density can be allowed to vanish. Furthermore, this result extends the corresponding results of J. Li [Methods Appl. Anal. 22, No. 2, 201–220 (2015; Zbl 1322.35117)] and X. Li [Discrete Contin. Dyn. Syst. 37, No. 9, 4907–4922 (2017; Zbl 1364.35242)] to the Neumann boundary condition and removes the smallness conditions on the initial data.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A15 | Liquid crystals |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
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