A priori estimates for parabolic Monge-Amp\`ere type equations

We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Amp\`ere equationn \left\{\begin{eqnarray} &&-u_t+\det D^2u= \psi(x,t) \quad\quad\ \text{ in } Q_T,\newline &&u=\phi\quad\text{ on }\partial_pQ_T, \end{eqnarray}\right. where $\psi,\phi$ are given functions, $Q_T=\Omega\times(0,T]$, $\partial_p Q_T$ is the parabolic boundary of $Q_T$, and $\Omega\subset\mathbb{R}^n$ is a uniformly convex domain. Our approach can also be used to prove similar results for the $\gamma$-Gauss curvature flow with any $0<\gamma\le 1$.