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MASDOC - Mathematics and Statistics Doctoral Training Centre

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.