*(English)*Zbl 1032.53058

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 $t\to \infty $ 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.

##### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) |

35K55 | Nonlinear parabolic equations |

35K60 | Nonlinear initial value problems for linear parabolic equations |

53C42 | Immersions (differential geometry) |