# Short-time Existence

Here we consider a reduced version of the model presented in the introduction, which was first stated in [1] 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

**Equation for height function .**Given the graph parametrisation in order for to evolve by the forced curvature flow it is necessary and sufficient that

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

**Equation for pulled back chemical**First note that transforms into and into

**References:**

- [1] P. Pozzi and B. Stinner. Curve shortening flow coupled to lateral diffusion.
*arXiv preprint arXiv:1510.06173*, 2015 - [2] A. Cesaroni, M. Novaga, and E. Valdinoci. Curve shortening flow in heterogeneous media.
*Interfaces Free Bound*, 13(4):485–505, 2011.