The conical complex Monge-Amp\`ere equations on K\"ahler manifolds

In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Amp\`ere equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-solution of the conical complex Monge-Amp\`ere equation admits the $C^{2,\alpha,\beta}$-regularity.