On a class of fully nonlinear flow in K\"ahler geometry

In this paper, we study a class of fully nonlinear metric flow on K"ahler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the corresponding Euler equation, which is fully nonlinear of Monge-Amp`ere type. As an application, we also discuss a complex Monge-Amp`ere type equation including terms of mixed degrees, which was first posed by Chen.