半地转方程拉格朗日解的弱稳定性
Weak stability of Lagrangian solutions to the semigeostrophic equations
摘要：
在文献[1]中，Cullen和Feldman证明了在$L^p$，$p>1$中具有初始位涡的半地转系统的Lagrange解的存在性.在这里，我们表明，一个子序列的拉格朗日解对应于一个强收敛序列的初始位涡在$L^1$强收敛$L^q$，$q<\infty$，拉格朗日解，特别是扩展的存在性结果的情况下，$p=1$的卡伦和费尔德曼。我们还提出了一个反例拉格朗日解对应的初始位涡序列收敛在$\mathcal{BM}$。所使用的分析工具包括技术从最佳运输，安布罗西奥的结果运输的$BV$向量场，和Orlicz空间。[1]M.卡伦和M. Feldman，{物理空间中半地转方程的\it拉格朗日解。} SIAM J. Math. Anal.，[\bf 37]（2006），1371- 1395。
英文摘要：
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.
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