In [1], a computational fluid dynamic style reformulation is introduced to solve the classical Monge-Kantorovich problem. Though the described augmented Lagrangian method involves an Hilbertian framework, the discussion was purely formal. Taking advantage of the recent progress in the optimal transport theory [4],[4],[6][10]and despite the lack of coercivity of the Hilbertian problem, we establish an existence result. Then under a reasonable assumption of positivity for the density, we prove the existence of saddle-points for both Lagrangian defined in [1], and finally prove the convergence of the numerical method.