Long time existence of smooth solutions to semigeostrophic equations on a torus

TL;DR

This paper proves the long-term existence of smooth solutions to semigeostrophic equations on a torus, given bounded smooth initial dual density, by refining estimates on the Monge-Ampère equation's C2 norm.

Contribution

It introduces a more precise estimate on the C2 norm of Monge-Ampère equations, enabling proof of long-time smooth solutions for semigeostrophic equations on a torus.

Findings

Long-time existence of smooth solutions established.

Refined C2 norm estimates for Monge-Ampère equations.

Applicability to semigeostrophic equations on a torus.

Abstract

In this work,we show the long time existence of smooth solu- tions to semigeostrophic equations on a torus when the initial dual density is bounded between two positive constants and smooth.The key ingredient is a more precise estimate on C 2 norm of Monge-Amp`re equations established e in [1] .