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 of the parabolic Monge-Amp\`ere equation &uₜ= D²u in QT, \\ &u=φ on ∂ₚQT, eqnarray. where φ is a smooth function, QT=Ω×(0,T], ∂ₚ QT is the parabolic boundary of QT, and Ω is a uniformly convex domain in Rⁿ with smooth boundary. Our approach can also be used to prove similar results for γ-Gauss curvature flow with any 0<γ≤ 1.