We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge–Ampère equation \(\begin{aligned} \left\{ \begin{aligned} -u_t + \det D^2u&= \psi (x,t) \quad \quad \ \text { in } Q_T,\\ u&=\phi \quad \quad \quad \quad \, \ \text { on } \partial _pQ_T, \end{aligned}\right. \end{aligned}\) 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 \(0<\gamma \le 1\) .
