Existence and regularity of mean curvature flow with transport term in higher dimensions

TL;DR

This paper proves the global existence and regularity of mean curvature flow with an added transport term in higher dimensions, showing hypersurfaces remain mostly smooth despite potential singularities.

Contribution

It establishes the existence and regularity of mean curvature flow with a transport term in higher dimensions, including behavior after singularities.

Findings

Hypersurfaces remain $C^1$ for a short time.

Flow exists globally in time.

Singularities occur but hypersurfaces are $C^1$ almost everywhere.

Abstract

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $C^1$ for a short time and, even after some singularities occur, almost everywhere $C^1$ away from higher multiplicity region.