A global existence result for the semigeostrophic equations in three dimensional convex domains

TL;DR

This paper proves the global existence of solutions to the semigeostrophic equations in three-dimensional convex domains, using recent regularity estimates for the Monge-Ampère equation under certain initial data assumptions.

Contribution

It establishes the first global-in-time existence result for semigeostrophic equations in 3D convex domains leveraging Monge-Ampère regularity estimates.

Findings

Proves global-in-time existence of solutions in 3D convex domains.

Utilizes recent advances in Monge-Ampère equation regularity.

Provides a framework for future research in geophysical fluid dynamics.

Abstract

Exploiting recent regularity estimates for the Monge-Amp\`ere equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 3-dimensional convex domains.