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

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.