This paper has two main parts. In the first part the authors consider the flow of a strictly convex graphical hypersurface by its Gauss curvature. They show that for the Neumann boundary condition and for the second boundary condition the flow has a smooth solution for all time and as it converges to a solution of the prescribed Gauss curvature equation. More general Monge-Ampère equations are also considered.
In the second part they consider Hessian flows in conjunction with the second boundary condition and prove long time existence and convergence to a stationary solution.
The results can be viewed as parabolic versions of results proved by P.-L. Lions, N. S. Trudinger and J. I. E. Urbas [Commun. Pure Appl. Math. 39, 539–563 (1986; Zbl 0604.35027)] and by the reviewer J. Urbas [Commun. Partial Differ. Equ. 26, 859–882 (2001; Zbl 1194.35158)] for the corresponding elliptic equations.