Semigeostrphic Equations in Physical Space with Free Upper Boundary

We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the the work of [5] and [11] to the case of free upper boundary. As a byproduct of our proof, we obtain the existence of measure-valued dual space solutions when the initial measure ν0 ∈ P2(R) and is supported on {− 1δ ≤ y3 ≤ −δ}