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

Abstract

In this paper, we study the Cauchy problem of the Poiseuille flow of 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 uniformly bounded while its two components approach infinity at formations of cusp singularities.

On singularities of Ericksen-Leslie system in dimension three

“…1 ⊂ R n denote the unit ball centered at 0. Inspired by the results in [10], we first consider the domain Ω = B 2 1 × [0, 1] and an axisymmetric solution (u, P, d) to the Ericksen-Leslie system (1.1) in the following special form      u(r, θ, z, t) := v(r, t)e r + w(z, t)e 3 , d(r, θ, z, t) := sin ϕ(r, t)e r + cos ϕ(r, t)e 3 , P (r, θ, z, t) := Q(r, t) + R(z, t).…”

“…The system (6.1) can be derived from the Ericksen-Leslie system (1.1) by using the Poiseuille flows (see Appendix in [2] for details) u = (0, 0, w(x, t)), d = (sin ϕ(x, t), 0, cos ϕ(x, t)).…”

A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation

“…In a recent paper [16], the cusp singularity formation and global existence of Hölder continuous solution for the 1-d model of (1.14) has been established. By 1-d model we mean…”

“…where (x, t) ∈ R × R + . The existence framework in [11] for the variational wave equation (1.11) serves as the basis for the existence result in [16] for (1.16), because the major wave parts on u in these two equations are same. Furthermore, the recent study shows that this 1-d framework will very likely direct to some interesting results for the exterior problem of (1.14) out of a small cylinder including the center line.…”

“…16) for θ ∈ [0, 1]\{θ 1 , • • • , θ N }, here C 1 > is a constant depending only on T and the upper bound of the total energy. Indeed, according to Definition 6.1, fix > 0 and choose a relabeling of the variables X, Y , such that, at time t = 0, it holds that ˆ1 0|J θ k |W − dX + |H θ k |W + dY dθ ≤ γ 0 + .…”