Pluripotential Monge-Amp{\`e}re flows in big cohomology class

46 pages. arXiv admin note: text overlap with arXiv:1304.2087 by other authorsWe study pluripotential complex Monge-Amp\`ere flows in big cohomology classes on compact K{\"a}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{\"a}hler-Ricci flows on compact K{\"a}hler manifolds of general type as well as on K{\"a}hler varieties with semi-log canonical singularities