Linear dynamics of the semi-geostrophic equations in Eulerian coordinates on $\mathbb{R}^{3}$

We consider a class of steady solutions of the semi-geostrophic equations on $\mathbb{R}^3$ and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in $L^2(\mathbb{R}^3;\mathbb{R}^3)$ introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on $\mathbb{R}^{3}$. We investigate stability of the steady solutions by looking at plane wave solutions of the linearised problem, and discuss differences in the case of the quasi-geostrophic equations.