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In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of compressible fluid mechanics will be treated in a forthcoming paper.

Having taken a search through MathSciNet I did not see anything obvious as the mentioned followup. Does anyone have a reference as to which paper that is referring to?

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I have worked for decades on compressible gas dynamics, and I am not aware of such a followup by Ebin & Marsden. But I think you will find something in the book of Arnol'd & Khesin.

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    $\begingroup$ (Just for my later reference: this is Topological Methods in Hydrodynamics.) $\endgroup$
    – Willie Wong
    Commented Dec 3, 2021 at 21:52
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The compressible case uses semidirect products of groups (group of diffeomorphisms times functions). To my knowledge, the first paper that discusses this in detail is Marsden, Ratiu, Weinstein: Semidirect Products and Reduction in Mechanics (PDF). There the authors state

By 1980 it was known that the equations for compressible flow on R3 were formally Lie-Poisson equations for the semidirect product...

So even back then it was already folklore.

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