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

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.

💡 Research Summary

This paper establishes the local well-posedness (existence, uniqueness, and continuous dependence on initial data) for a free boundary problem modeling the motion of an inviscid, incompressible nematic liquid crystal with surface tension. The system is governed by the so-called Lin-Liu equations, a simplified model for nematic liquid crystals that couples the incompressible Euler equations for the fluid velocity with a harmonic heat flow equation for the unit-length orientation vector field of the liquid crystal molecules.

The central challenge lies in the presence of a moving boundary whose shape is unknown and evolves with the fluid. The boundary conditions involve a pressure jump proportional to the mean curvature (surface tension) and a Neumann condition for the orientation field. A priori energy estimates alone are insufficient for proving local existence due to the loss of symmetry in the linearized equations, a common issue in free-boundary inviscid fluid problems.

To overcome this, the authors reformulate the problem in Lagrangian coordinates, fixing the fluid domain. This transforms the problem into studying a system on a fixed domain Ω, where the moving boundary’s geometry is encoded in the deformation tensor and its inverse. The main theorem (Theorem 1.1) states that for sufficiently regular initial data (velocity in H^5.5, orientation field and its time derivatives up to order 4 in appropriate spaces) and a fixed surface tension coefficient σ > 0, there exists a unique solution on a short time interval