Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two

Abstract

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)−2 as $${t \rightarrow \infty}$$ . 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 $${\infty}$$ as $${t \rightarrow \infty}$$ . This indicates that the orientation variable will keep rotating around the z-axis while $${t \rightarrow \infty}$$ . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number.