Local Well-Posedness of the Motion of Inviscid Liquid Crystals with a Free Surface Boundary

TL;DR

This paper establishes the local well-posedness of free-boundary inviscid liquid crystal flow equations with surface tension, overcoming challenges posed by symmetry loss in linearized systems.

Contribution

It develops an effective approximate system that captures fluid motion, harmonic heat flows, and boundary regularity, enabling well-posedness proof.

Findings

Proved local existence of solutions in Lagrangian coordinates.

Constructed an asymptotically consistent approximate system.

Addressed the loss of symmetry in linearized equations.

Abstract

In this article, we prove the local well-posedness of the free-boundary Lin-Liu equations describing the motion of inviscid nematic liquid crystals in the presence of surface tension in Lagrangian coordinates. It is well known that a priori energy estimates alone are insufficient for establishing local existence in free-boundary problems involving inviscid fluid equations, primarily due to the loss of symmetry in the linearized equations. The main challenge is to develop an effective approximate system of equations that is asymptotically consistent with the free-boundary Lin-Liu model expressed in the Lagrangian coordinates. This system must accurately capture the coupling between the fluid motion and the harmonic heat flows within the interior, as well as the regularity of the moving boundary.