  The fully compressible semi-geostrophic system is widely used in the
modelling of large-scale atmospheric flows. In this paper, we prove rigorously
the existence of weak Lagrangian solutions of this system, formulated in the
original physical coordinates. In addition, we provide an alternative proof of
the earlier result on the existence of weak solutions of this system expressed
in the so-called geostrophic, or dual, coordinates. The proofs are based on the
optimal transport formulation of the problem and on recent general results
concerning transport problems posed in the Wasserstein space of probability
measures.
