Methods and Applications of Analysis

Volume 27 (2020)

Number 4

Special Issue for the 60th Birthday of John Urbas: Part I

Guest Editors: Neil Trudinger and Xu-Jia Wang

From Euler to the semi-geostrophic system: convergence under uniform convexity

Pages: 375 – 386



Michael Cullen (Met Office, Exeter, Devon, United Kingdom)

Mikhail Feldman (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Adrian Tudorascu (Department of Mathematics, West Virginia University, Morgantown, W.V., U.S.A.)


We prove that if the initial data is well prepared, then certain solutions to the Euler system converge to a solution of the Semi-Geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre–Fenchel transform of the modified pressure.


SG system, Euler system, flows of maps, optimal mass transport

2010 Mathematics Subject Classification

35Dxx, 35G25

The work of Mikhail Feldman was supported in part by the National Science Foundation under Grant DMS-1764278, and the Van Vleck Professorship Research Award by the University of Wisconsin, Madison.

Adrian Tudorascu was partially supported by the National Science Foundation under Grant DMS-1600272.

Received 12 February 2020

Accepted 17 June 2020

Published 24 September 2021