| Abstract: |
| The semi-geostrophic equation is a reduced model for the approximate description of large-scale atmospheric or oceanic flows. The corresponding Lagrangian flow is measure preserving and the density of this image measure is evolved by a transport equation with a velocity field that is determined through the solution of a Monge-Ampere equation where the right hand side is the transported density. This coupled system is the so called dual semi-geostrophic equation for which the global existence of smooth solutions is still open.
On the other hand, by a suitable linearization of the Monge-Ampere equation one obtains a Poisson equation and hence, in two dimensions, the dual semi-geostrophic equation is formally approximated by the Euler vorticity equation which has globally smooth solutions. G. Loeper made this approximation result rigorous to leading order on time scales of order $O(1/\varepsilon) $ with an error of order $O (\varepsilon) $. We present a generalization of this result for higher order approximations with the higher order correction terms being determined by nonlocal linear inhomogeneous transport equations. This allows us to reduce the order of the error to $O(\varepsilon^k)$ for arbitrary $k \in \mathbb{N}$. This is work supported by the Hellenic Foundation for Research and Innovation (HFRI). |
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