Semi-geostrophic System with Variable Coriolis parameter

TL;DR

This paper establishes short-term existence and uniqueness of smooth solutions for 2-D semi-geostrophic systems with variable Coriolis parameter, overcoming analytical challenges posed by the variable $f$ through a novel time-stepping approach.

Contribution

It extends the mathematical theory of semi-geostrophic systems to variable Coriolis parameters by developing a new analytical method that replaces the dual space approach.

Findings

Proves short-time existence and uniqueness of solutions.

Develops a new time-stepping procedure for variable $f$ case.

Addresses analytical difficulties due to the absence of dual space.

Abstract

We prove short time existence and uniqueness of smooth solutions ( in $C^{k+2,\alpha}$ with $k\geq 2$) to the 2-D semi-geostrophic system and 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 $f$ does not exist for the variable Coriolis parameter case, and we develop a time-stepping procedure to overcome this difficulty.