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In this paper we use the new regularity and stability estimates forAlexandrov solutions to Monge-Ampere equations estabilished by G.De Philippisand A.Figalli to provide a global in time existence of distributional solutionsto a semigeostrophic equation on the 2-dimensional torus, under very mildassumptions on the initial data. A link with Lagrangian solutions is alsodiscussed.

Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case