Existence and regularity of mean curvature flow with transport term

Yoshihiro Tonegawa

(Hokkaido University)

Event: ERC Workshop on Geometric Measure Theory, Analysis in Metric Spaces and Real Analysis

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.