  We prove the existence of local-in-time smooth solutions of the
incompressible semi-geostrophic equations expressed in Eulerian co-ordinates in
3-dimensional smooth bounded simply-connected domains. Our solutions adhere to
Cullen's Stability Principle in that the geopotential is guaranteed to be a
convex map for all times of its existence. We achieve our results by appealing
to the theory of so-called div-curl systems (or Hodge systems), making use of
recent results of Wang, which yield useful estimates on the ageostrophic
velocity field. To our knowledge, this work constitutes the first time that any
notion of bounded solution of the semi-geostrophic equations in Eulerian
co-ordinates has been constructed on a bounded domain. Indeed, our work solves
an open problem as highlighted by, among others, A. Figalli in his CIME
lectures on the semi-geostrophic equations. Our methods are largely elementary.
We discuss the application of the novel ideas in this work to the case of
variable Coriolis force in the final section of the article.
