Rearrangements of vector valued functions, with application to atmospheric and oceanic flows

This paper establishes the equivalence of four definitions of two vector valued functions being rearrangements. Properties of the monotone rearrangement of a vector valued function are used to show existence and uniqueness of the minimizer of an energy functional arising from the semigeostrophic equations, a model for atmospheric and oceanic flow. At each fixed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury, and Purser [SIAM J. Appl. Math., 51 (1991). pp. 20-31].