  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.
