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

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.

60 pages

References in corpus (3)

• Evolution of curves and surfaces by mean curvature
• A general regularity theory for weak mean curvature flow
• A second derivative Hölder estimate for weak mean curvature flow

Cited by in corpus (8)

• A general regularity theory for weak mean curvature flow
• Convergence of the Allen-Cahn equation with a zero Neumann boundary condition on non-convex domains
• On the existence of canonical multi-phase Brakke flows
• Existence of weak solution to volume preserving mean curvature flow in higher dimensions
• A diffused interface with the advection term in a Sobolev space
• An example of a mean-convex mean curvature flow developing infinitely many singular epochs
• End-time regularity theorem for Brakke flows
• Existence of BV flow via elliptic regularization