Weak limits of the $J$-flow and the deformed Hermitian-Yang-Mills flow on Kähler surfaces: boundary cases

We prove that if a pair of Kähler classes is $J$-nef, the $J$-flow on a compact Kähler surface converges to a weak solution of the Monge-Ampère 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.