Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system

TL;DR

This paper proves the local well-posedness of the Ericksen-Leslie system for liquid crystals and demonstrates that solutions from the Ginzburg-Landau approximation converge smoothly to the true solutions within a certain time frame.

Contribution

It establishes the convergence of the Ginzburg-Landau approximation to the Ericksen-Leslie system and proves local well-posedness in specific Sobolev spaces.

Findings

Solutions of the Ginzburg-Landau approximation converge smoothly to the Ericksen-Leslie system solutions.

The system is well-posed locally in time for initial data in specified Sobolev spaces.

Convergence holds for any time within the maximal existence interval of the Ericksen-Leslie system.

Abstract

We establish the local well-posedness of the general Ericksen-Leslie system in liquid crystals with the initial velocity and director field in $H^1 \times H_b^2$. In particular, we prove that the solutions of the Ginzburg-Landau approximation system converge smoothly to the solution of the Ericksen-Leslie system for any $t \in (0,T^\ast)$ with a maximal existence time $T^\ast$ of the Ericksen- Leslie system.