Weak stability of Lagrangian solutions to the semigeostrophic equations

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In [1], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in Lᵖ, p>1. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in L¹ converges strongly in Lq, q<∞, 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 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, Lagrangian solutions of semigeostrophic equations in physical space. SIAM J. Math. Anal., 37 (2006), 1371--1395.