We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density.
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Linfeng Li (2024) studied this question.
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