  We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$
flow in the viscosity sense for compact convex hypersurfaces $\Sigma_t$
embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex
hypersurfaces with flat sides we show that, under a certain non-degeneracy
initial condition, the interface separating the flat from the strictly convex
side, becomes smooth, and it moves by the $Q_{k-1}$ flow at least for a short
time.
