Classical Solutions to Semi-geostrophic System with Variable Coriolis Parameter

We prove the short time existence and uniqueness of smooth solutions (in $${C^{k+2,\alpha}}$$ with $${k \geqq 2}$$ ) to the 2-D semi-geostrophic system and the semi-geostrophic shallow water system with variable Coriolis parameter f and periodic boundary conditions, under the natural convexity condition on the initial data. The dual space used in analysis of the semi-geostrophic system with constant f is not available for the variable Coriolis parameter case, and we develop a time-stepping procedure in Lagrangian coordinates in the physical space to overcome this difficulty.