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.



Existence of PDE system (mean curvature flow coupled with surface PDE)
To the best of my knowledge, no theory has been done for this kind of problem. 
One of the major problems is that the Laplace-Beltrami operator is no longer linear or monotone.


I am interested in similar problems though. Are you a PhD student?


