Convergence of the weak K\"ahler-Ricci Flow on manifolds of general type

We study the K"ahler-Ricci flow on compact K"ahler manifolds whose canonical bundle is big. We show that the normalized K"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular K"ahler-Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge-Amp`ere flows in big classes that we develop, extending and refining the approach of Eyssidieux-Guedj-Zeriahi.