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We prove the existence of local-in-time smooth solutions of theincompressible semi-geostrophic equations expressed in Eulerian co-ordinates in3-dimensional smooth bounded simply-connected domains. Our solutions adhere toCullen's Stability Principle in that the geopotential is guaranteed to be aconvex map for all times of its existence. We achieve our results by appealingto the theory of so-called div-curl systems (or Hodge systems), making use ofrecent results of Wang, which yield useful estimates on the ageostrophicvelocity field. To our knowledge, this work constitutes the first time that anynotion of bounded solution of the semi-geostrophic equations in Eulerianco-ordinates has been constructed on a bounded domain. Indeed, our work solvesan open problem as highlighted by, among others, A. Figalli in his CIMElectures on the semi-geostrophic equations. Our methods are largely elementary.We discuss the application of the novel ideas in this work to the case ofvariable Coriolis force in the final section of the article.

Local-in-time Physical Solutions of the Incompressible Semi-Geostrophic Equations in Eulerian Coordinates