Suppose that a family of $k$-dimensional surfaces in $\mathbb{R}^n$ evolves by the generalized mean curvature flow with a given transport vector in the sense of Brakke's formulation of velocity. When the flow is locally close to a time-dependent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,\alpha}$ function over the plane. On the other hand, it is not known if the graph satisfies the corresponding PDE pointwise in general. For this problem, when $k=n-1$ and the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. This talk is based on a joint work [arXiv:2109.06380] with Professor Yoshihiro Tonegawa (Tokyo Tech) and Eita Tomimatsu (Tokyo Tech).

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