  In this paper, we prove the long time regularity of the interface in the
$p$-Gauss curvature flow with flat side in all dimensions for $p>\frac1n$. Here
the interface is the boundary of the flat part in the flow. In dimension $2$,
this problem was solved in \cite{DL2004} for $p=1$ and in \cite{KimLeeRhee2013}
for $p\in(1/2,1)$. We utilize the duality method to transform the Gauss
curvature flow to a singular parabolic Monge-Amp\`ere equation, and prove the
regularity of the interface by studying the asymptotic cone of the parabolic
Monge-Amp\`ere equation in the polar coordinates.
