Weak stability of Lagrangian solutions to the semigeostrophic equations

TL;DR

This paper extends the existence of Lagrangian solutions to the semigeostrophic equations to initial potential vorticities in $L^1$, demonstrating strong convergence and providing a counterexample for convergence in the space of bounded measures.

Contribution

It proves strong $L^q$ convergence of Lagrangian solutions for initial data in $L^1$, extending previous results from $L^p$ with $p>1$, and introduces new analytical techniques.

Findings

Strong convergence of solutions in $L^q$ for $q< inite$

Counterexample for convergence in $m$

Extension of existence results to $p=1$ case

Abstract

In [1], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in $L^p$, $p>1$. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in $L^1$ converges strongly in $L^q$, $q<\infty$, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case $p=1$. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in $\mathcal{BM}$. The analytical tools used include techniques from optimal transportation, Ambrosio's results on transport by $BV$ vector fields, and Orlicz spaces. [1] M. Cullen and M. Feldman, {\it Lagrangian solutions of semigeostrophic equations in physical space.} SIAM J. Math. Anal., {\bf 37} (2006), 1371--1395.