Mathematical Insights into Hydrostatic Modeling of Stratified Fluids

We review recent mathematical results concerning the analysis of hydrostatic equations in the context of stably stratified fluids. Beginning with the simpler and better understood setting of homogeneous fluids, we emphasize the additional mathematical challenges posed by non-homogeneous framework. We present both positive and negative results, including well-posedness and proof of the hydrostatic limit with a suitable regularization, alongside ill-posedness in the fully inviscid setting and the breakdown of the hydrostatic limit in specific scenarios.