Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

Abstract

Mathematical studies of nematic liquid crystals address in general two rather different perspectives: that of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

Smectic layering: Landau theory for a complex-tensor order parameter

“…For these mesophases, the extent of ordering is measured by a real scalar order parameter S(r, t) and the local direction is measured by the director field n(r, t) of unit length. In the Ericksen-Leslie formalism [51][52][53][54][55][56][57], the free energy density is constructed from a bulk contribution in terms of S alone and a Frank-Oseen deformation contribution in terms of only gradients of n. However, de Gennes [58,59] proposed that the order parameter and director can be written together as a tensor order parameter…”

Thermal response of nematicons in a parabolic potential

“…The nonlinear response of the medium is a spatially nonlocal function of the wave intensity in many nonlinear optical systems, such as nematic liquid crystal (NLC) [23]. This indicates that the refractive index change at a particular point is affected by the light intensity in the area around that point [24]. Nonlocal responses in optics include the thermo-optic response of materials in which a point-wise heat source, such as absorbed light, affects the optical properties even at distant locations, and the reorientational nonlinearity of liquid crystals in which elastic interactions tend to spread the effect of a local electromagnetic perturbation [25,26].…”

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in bounded domains with boundaries

“…The rigorous mathematical analysis of the Ericksen-Leslie model was first made by Lin [20] and Lin-Liu [22][23][24], in which they introduced a considerably simplified version and proved the existence of global weak solutions and their partial regularities. Regarding modeling and analysis of the Ericksen-Leslie equations describing nematic liquid crystals, please refer to the works [9,16,21,26] and the survey papers [8,25,39] as well as the references therein for more discussions on the physics and mathematical results.…”