# 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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## 1 Answer

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?

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Hi Mike, thanks for replying. Unfortunately I'm just a MSc student and am trying to understand this area. Is any of your work available for me to read online? –  blackcat Jul 20 '12 at 11:42
Ahh ok. Unfortunately I haven't put anything online at the moment. I've proven that when you have isotropic evolution of the metric and couple this to the soln of the SPDE then you have existence and uniqueness. Your problem is a lot harder and as it stands ill posed. Since your parameterisation X is a vector, the equation where you add u to the expression is incorrect, as u is a scalar. Where did this PDE system come from? I know it has applications in modelling biomembranes... Which university are you at? –  Mike Jul 25 '12 at 19:31
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