Short Time Existence for the Curve Diffusion Flow with a Contact Angle

TL;DR

This paper proves short-time existence for curve diffusion flow with a prescribed contact angle, handling low regularity initial data through advanced PDE techniques and weighted Sobolev space estimates.

Contribution

It establishes local well-posedness for a fourth-order PDE modeling curve evolution with contact angle boundary conditions, even with low regularity initial curves.

Findings

Short-time existence of solutions is proven.

The approach handles initial data with low regularity.

A contraction mapping argument is used in weighted Sobolev spaces.

Abstract

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $\alpha \in (0, \pi)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^{\gamma}$ with $\gamma \in (\tfrac{3}{2}, 2]$. For the proof the evolving curve is represented by a height function over a reference curve: The local well-posedness of the resulting quasilinear, parabolic, fourth-order PDE for the height function is proven with the help of contraction mapping principle. Difficulties arise due to the low regularity of the initial curve. To this end, we have to establish suitable product estimates in time weighted anisotropic $L_2$-Sobolev spaces of low regularity for proving that the non-linearities are well-defined and contractive for small times.