Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

TL;DR

This paper analyzes the bifurcation and stability of stationary shear flows in nematic liquid crystals modeled by the Ericksen-Leslie system, revealing multiple solutions and their stability changes at critical shear speeds.

Contribution

It establishes a one-to-one correspondence between stationary solutions and algebraic solutions, identifying bifurcation points and stability properties in the Ericksen-Leslie model.

Findings

Multiple stationary solutions arise through saddle-node bifurcations.

Stability changes occur at critical shear speeds with eigenvalue bifurcations.

Unique solutions exist at critical shear speeds, with bifurcations for larger shear speeds.

Abstract

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}] for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.