A priori estimates for parabolic Monge-Ampère type equations

Abstract

We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Ampère equationn $$ \left\{\begin{eqnarray} &&-u_t+\det D^2u= ψ(x,t) \quad\quad\ \text{ in } Q_T,\newline &&u=ϕ\quad\text{ on }\partial_pQ_T, \end{eqnarray}\right. $$ where $\psi ,\varphi$ 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 $0<\gamma \le 1$.