Advances in Differential Equations

Singularities and nonuniqueness in cylindrical flow of nematic liquid crystals

E. Vilucchi and R. van der Hout

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The subject of this paper is the behavior of the director field of a nematic liquid crystal in flow through a tube with circular cross-section. Both the flow and the director field are assumed to have cylindrical symmetry. The requirement of finite Frank-Oseen energy forces ``admissible" director fields to be axially directed at the location of the symmetry axis. Thus, the angle between the axis and the director field at the location of the axis amounts to $k\pi$, $k$ being an integer. In the steady case, it is shown that $k$ is largely (but not uniquely) determined by the (Dirichlet) boundary conditions. In particular, this may give rise to line singularities with finite energy density. Moreover, the associated Dirichlet problem may have several distinct solutions with identical value of $k$. Analogously to the heat flow of harmonic mappings, finite-time blow-up phenomena for the associated parabolic problem are established.

Article information

Adv. Differential Equations, Volume 6, Number 7 (2001), 799-820.

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35K55: Nonlinear parabolic equations 58E50: Applications 76A15: Liquid crystals [See also 82D30]


van der Hout, R.; Vilucchi, E. Singularities and nonuniqueness in cylindrical flow of nematic liquid crystals. Adv. Differential Equations 6 (2001), no. 7, 799--820.

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