The zero inertia limit from hyperbolic to parabolic Ericksen-Leslie system of liquid crystal flow

Abstract

The original Ericksen-Leslie consists the conservation laws of the nematic liquid crystals, which includes the inertial effect. Thus the full Ericksen-Leslie system has hyperbolic feature. In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the solutions of the limiting system, we derive a global in time uniform energy bound to the remainder system under the small size of the initial data. This work justifies the validity of the well-known parabolic Ericksen-Leslie system without inertial term.