Weak limits of the J-flow and the deformed Hermitian-Yang-Mills flow on K\"ahler surfaces: boundary cases

We prove that if a pair of K\"ahler classes is J-nef, the J-flow on a compact K\"ahler surface converges to a weak solution of the Monge-Amp\`ere equation in the sense of currents. We also establish the same convergence behavior for the deformed Hermitian-Yang-Mills flow. The method is based on a property of a limit of viscosity subsolutions.