On the convexity condition for the Semi-Geostrophic system

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are some well-defined sense critical points of time-shifted energy functional involving measure-preserving rearrangements absolute density and momentum, which arise as one-parameter flow maps continuously differentiable, compactly supported divergence free vector fields. also directly, with no recourse Monge-Kantorovich theory, convexity requirement on modified pressure potentials arises naturally if these local minimizers said for any admissible field. The obligatory connection theory is addressed briefly.