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

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.Comment: 60 page

A second derivative H\"{o}lder estimate for weak mean curvature flow

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given \alpha-H\"{o}lder continuous vector field, then we show C^{2,\alpha} regularity almost everywhere.Comment: 32 page

Diffused interface with the chemical potential in the Sobolev space

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms $W^{1,p}$ of the associated chemical potential fields are bounded uniformly, where $p>\frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as \e tending to zero is an integral varifold with the sharp integrability condition on the mean curvatur

A singular perturbation problem with integral curvature bound

We consider a singular perturbation problem of Modica-Mortola functional as the thickness of diffused interface approaches to zero. We assume that sequence of functions have uniform energy and square-integral curvature bounds in two dimension. We show that the limit measure concentrate on one rectifiable set and has square integrable curvature

Interior gradient estimate for 1-D anisotropic curvature flow

We establish the interior gradient estimate for general 1-D anisotropic curvature flow. The estimate depends only on the height of the graph and not on the gradient at initial time