# Existence of PDE system (mean curvature flow coupled with surface PDE)

Hi all,

What should I look for if I want to study existence/uniqueness of the system of PDEs: $$u_t -\Delta u + u\nabla \cdot v = f(u) \quad\text{on \Gamma(t)}$$ $$X_t = \kappa N(X) + u \quad \text{for \Gamma(t)} \tag{MCF with forcing}$$

where $N$ is the normal vector, $\Gamma(t)$ is the surface parametrised by $X$, $\kappa$ is the curvature and $v$ is the velocity. Take $f \equiv 0$ if it makes things easier.

So we have a geometric PDE flowing by mean curvature flow with forcing that gives a hypersurface $\Gamma(t)$ and a surface PDE that resides on this hypersurface $\Gamma(t)$. But the geometric PDE depends on the solution to the surface PDE.

Can anyone point me to some references/literature and what I should be looking for?

Thanks.

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