Poiseuille Flow of Nematic Liquid Crystals via the Full Ericksen–Leslie Model

In this paper, we study the Cauchy problem of the Poiseuille flow of the full Ericksen–Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parame...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2020-05, Vol.236 (2), p.839-891
Main Authors: Chen, Geng, Huang, Tao, Liu, Weishi
Format: Article
Language:English
Subjects:
Cauchy problems
Classical Mechanics
Complex Systems
Cusps
Fluid- and Aerodynamics
Flux density
Laminar flow
Liquid crystals
Mathematical and Computational Physics
Nematic crystals
Physical properties
Physics
Physics and Astronomy
Singularity (mathematics)
Theoretical
Wave equations
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Summary:In this paper, we study the Cauchy problem of the Poiseuille flow of the full Ericksen–Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities—blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen–Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are Hölder continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation—the flux density remains bounded while its two components approach infinity at formations of cusp singularities.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-019-01484-4