| Contents |
| In this talk we consider a family of hypersurfaces $ \{ \Gamma (t) \}_{t\in
[0,\infty)}$ in $\mathbb{R}^n$ whose velocity is
\begin{equation}
V_{\Gamma } =H+(u\cdot \nu)\nu \qquad \text{on} \qquad \Gamma (t), \qquad t \geq 0.
\end{equation}
Here $H$ and $\nu$ are the mean curvature vector and the unit normal vector of
$\Gamma (t)$ respectively and $u: \mathbb{R}^n \times [0,\infty) \to \mathbb{R}^n$
is a given vector valued function.
In 1978, Brakke proved the existence of a weak solution for the mean curvature flow
by using the geometric measure theory, and the weak solution is called Brakke's
mean curvature flow. In 2010, Liu, Sato and Tonegawa proved that there exists
Brakke's mean curvature flow with transport term when $n=2,3$ by using the
phase
field method and the monotonicity formula. Regarding the results, we consider
the existence of mean curvature flow with transport term for $n \geq 2$. This is a
joint work with Yoshihiro Tonegawa (Hokkaido University). |
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