Linear Dynamics of the Semi-geostrophic Equations in Eulerian Coordinates on {mathbb {R}}^{3}

Abstract

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 of the semi-geostrophic equations by looking at plane wave solutions of the associated linearised problem, and discuss differences in the case of the quasi-geostrophic equations.