Weak stability of Lagrangian solutions to the semigeostrophic equations
| Autor: | Faria, Josiane C. O., Filho, Milton C. Lopes, Lopes, Helena J. Nussenzveig |
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| Rok vydání: | 2009 |
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| Zdroj: | Nonlinearity 22 (2009), 2521-2539 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/0951-7715/22/10/011 |
| Popis: | 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. Comment: 19 pages |
| Databáze: | arXiv |
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