Here we consider a reduced version of the model presented in the introduction, which was first stated in  and focuses on the forces and Our aim is to establish local existence (of smooth solutions) to the correponding coupled system assuming that the curve can be parametrised as a graph
The corresponding system becomes
subject to initial conditions
and periodic boundary conditions
Here denotes the pulled back surface quantity.
We will assume as well as smooth periodic initial values .
Derivation of system in graph case
for a suitable scalar . Here are the curvature, unit normal and unit tangent in this parametrisation, i.e.
(To obtain the expression for note that by Frenet-Serret.)
Taking the normal component of this equation and multiplying the equation with , we find
Function spaces and a priori estimates
The spaces where a solution to the coupled problem is constructed are
for geometric equation and surface PDE respectively.
Note that we have the embedding and there exists such that for all
The main tool in the local existence proof are the following a priori estimates for smooth solutions to the system. We have a local-in-time estimate for the non-linear geometric equation (note in particular the dependence of the bound on )
The function and all constants can be chosen to depend continuously on their arguments.
Construction of an approximate sequence
Given smooth periodic initial data we constructed an approximate sequence by iteratively solving the problems
to initial data respectively, and subject to periodic boundary conditions.
Compactness and convergence
Then for some subsequence in suitable topologies, where solves the coupled problem.
-  P. Pozzi and B. Stinner. Curve shortening flow coupled to lateral diffusion. arXiv preprint arXiv:1510.06173, 2015
-  A. Cesaroni, M. Novaga, and E. Valdinoci. Curve shortening flow in heterogeneous media. Interfaces Free Bound, 13(4):485–505, 2011.