# Existence and regularity of mean curvature flow with transport term

## Yoshihiro Tonegawa

### (Hokkaido University)

Date: Oct 10, 2013, time: 10:00

Abstract. We consider the following problem. Suppose on $\mathbb R^n$ ($2\leq n$) that we are given a vector field $u$ and a bounded $C^1$ hypersurface $M_0$. Prove the existence and regularity of a family of hypersurfaces $\{M_t\}_{t>0}$ starting from $M_0$ such that the normal velocity of $M_t$ is equal to $h+u^{\perp}$, where $u^{\perp}$ is the normal projection of $u$. When $u=0$, it is the usual mean curvature flow (MCF). If $u\in L^q_{loc}([0,\infty);W^{1,p}({\mathbb R}^n))$ with $2<q<\infty$ and $\frac{nq}{2(q-1)}<p<\infty$, ($\frac43\leq p$ in addition if $n=2$), we prove that there exists a time-global weak solution of this problem. $M_t$ remains $C^1$ for at least a short time, and is a.e. $C^1$ away from the higher multiplicity region. I discuss the definition of weak solution, the strategy of the existence proof, and how the partial regularity follows from our recent local regularity results on the MCF.