  This paper studies singularities of mean curvature flows with integral mean
curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For
such flows, any tangent flow is given by the flow of a stationary cone
$\mathbf{C}$. When $p = \infty$ and $\mathbf{C}$ is a regular cone, we prove
that the tangent flow is unique. These results hold for general integral Brakke
flows of arbitrary codimension in an open subset $U \subseteq \mathbb{R}^N$
with $H \in L^\infty L^p_{loc}$. For smooth, codimension one mean curvature
flows with $H \in L^\infty L^\infty_{loc}$, we also show that, at points where
a tangent flow is given by an area-minimizing Simons cone, there is an
accompanying limit flow given by a smooth Hardt-Simon minimal surface.

Differential Geometry;Analysis of PDEs