Digital Object Identiﬁer (DOI) 10.1007/s00205-017-1144-x
Arch. Rational Mech. Anal. 226 (2017) 767–808
Global m-Equivariant Solutions of Nematic
Liquid Crystal Flows in Dimension Two
Yuan Chen & Yong Yu
Communicated by S. Serfaty
In this article we construct a global solution of the simpliﬁed Ericksen-Leslie
system. We show that the velocity of the solution can be decomposed into the sum of
three parts. The main ﬂow is governed by the Oseen vortex with the same circulation
Reynolds number as the initial ﬂuid. The secondary ﬂow has ﬁnite kinetic energy
and decay in the speed (1 + t)
as t →∞. The third part is a minor ﬂow whose
kinetic energy decays faster than the secondary ﬂow. As for the orientation variable,
our solution has a phase function which diverges logarithmically to ∞ as t →∞.
This indicates that the orientation variable will keep rotating around the z-axis
while t →∞. This phenomenon results from a non-trivial coupling between the
orientation variable and a ﬂuid with a non-zero circulation Reynolds number.
1.1. Background and Motivation
In this article we study the simpliﬁed Ericksen-Leslie system for nematic liquid
crystals proposed by Lin in . The spatial domain is supposed to be R
. With all
the parameters normalized to be 1, the system can be written as follows:
φ + u ·∇φ − φ =|∇φ|
φ, in R
u + u ·∇u − u =−∇ς −∇·
, in R
div u = 0, in R
In (1.1) φ is an S
−valued macroscopic orientation of a nematic liquid crystal. u :
× (0,∞) → R
represents the velocity ﬁeld of ﬂuid. ς is the pressure function.
As one can see, system (1.1) is a coupled system between the incompressible
Navier–Stokes equation and the transported heat ﬂow of harmonic maps.