# 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 $\mathcal{F}_p,$ $\mathcal{F}_{visc}$ and $\mathcal{F}_s.$ 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

$\gamma(x,t)=(x,u(x,t)),\;\;x\in[0,1].$

The corresponding system becomes

$u_t=\frac{u_{xx}}{1+u_x^2}+f(\tilde{c}(x,t))\sqrt{1+u^2_x},$

$\tilde{c}_t+\tilde{c}\frac{\left(\sqrt{1+u^2_x}\right)_t}{\sqrt{1+u^2_x}}=\frac{1}{\sqrt{1+u^2_x}}\left(\frac{\tilde{c}_x}{\sqrt{1+u^2_x}}\right)_x,\\ \label{init}$

subject to initial conditions

$u(0)=u_0, \tilde{c}(0)=\tilde{c}_0$

and periodic boundary conditions
$u(0,t)=u(1,t),\\ u_x(0,t)=u_x(1,t)$
and
$\tilde{c}(0,t)=\tilde{c}(1,t),\\ \tilde{c}_x(0,t)=\tilde{c}_x(1,t).$

Here $\tilde{c}=c(\gammma(x,t),t)$ denotes the pulled back surface quantity.
We will assume $f\in C^\infty_b(\mathbb{R})$ as well as smooth periodic initial values $u_0, \tilde{c}_0$.

##### Derivation of system in graph case
Let us first formally derive the above system from the geometric problem
$v=(\kappa+f(c))\nu,$
$\partial^\bullet c+c\nabla_\Gamma\cdot v=\partial_{s}^2c\;\;\text{ on }\Gamma.$
Equation for height function $u$. Given the graph parametrisation in order for $\Gamma$ to evolve by the forced curvature flow it is necessary and sufficient that
$\partial_t\gamma=(\tilde\kappa+f(\tilde{c}))\,\tilde\nu+S\,\tilde\tau$

for a suitable scalar $S(x,t)$. Here $\tilde\kappa, \tilde\nu,\tilde\tau$ are the curvature, unit normal and unit tangent in this parametrisation, i.e.

$\tilde\kappa=\frac{u_{xx}}{\sqrt{1+u_x^2}^3}, \hspace{1cm} \tilde\nu=\frac{(-u_x,1)}{\sqrt{1+u_x^2}}, \hspace{1cm} \tilde\tau=\frac{(1,u_x)}{\sqrt{1+u_x^2}}.$

(To obtain the expression for $\tilde\kappa$ note that $\kappa=-\nu_s\cdot\tau$ by Frenet-Serret.)
Taking the normal component of this equation and multiplying the equation with $\sqrt{1+u_x^2}$, we find

$u_t=\frac{u_{xx}}{1+u_x^2}+\sqrt{1+u_x^2}\,f(\tilde{c}).$
Equation for pulled back chemical $\tilde{c} .$ First note that $\partial^\bullet{c}$ transforms into $\partial_t\tilde{c}$ and $\partial_s^2c$ into

$\frac{1}{\sqrt{1+u_x^2}}\partial_x\left(\frac{\tilde{c}_x}{\sqrt{1+u_x^2}}\right).$
The transport term $\nabla_\Gamma\cdot v$ transforms into
$\frac{1}{1+u_x^2}\partial_x\gamma_t\cdot\partial_x\gamma =\frac{u_xu_{xt}}{1+u_x^2}=\frac{\partial_t\sqrt{1+u_x^2}}{\sqrt{1+u_x^2}}.$
Hence the surface PDE transforms into
$\tilde{c}_t+\tilde{c}\frac{\left(\sqrt{1+u^2_x}\right)_t}{\sqrt{1+u^2_x}} =\frac{1}{\sqrt{1+u^2_x}}\left( \frac{\tilde{c}_x}{\sqrt{1+u^2_x}}\right)_x.$
Combining the equations of height function and pulled back chemical, the above stated system follows.

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.