  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.
