Short time existence for the curve diffusion flow with a contact angle
Abstract
We show shorttime existence for curves driven by curve diffusion flow with a prescribed contact angle α ∈ (0 , π): The evolving curve has free boundary points, which are supported on a line and it satisfies a noflux condition. The initial data are suitable curves of class W_{2}^{γ} with γ ∈ (3/2 , 2 ]. For the proof the evolving curve is represented by a height function over a reference curve: The local wellposedness of the resulting quasilinear, parabolic, fourthorder 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 nonlinearities are welldefined and contractive for small times.
 Publication:

Journal of Differential Equations
 Pub Date:
 December 2019
 DOI:
 10.1016/j.jde.2019.08.018
 arXiv:
 arXiv:1810.01502
 Bibcode:
 2019JDE...268..318A
 Keywords:

 53C44;
 35K35;
 35K55;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 53C44;
 35K35;
 35K55
 EPrint:
 38 pages