Stationary hydrodynamic systems in General Relativity constitute a serious challenge. They are described by nonlinear  integro-differential equations with a free boundary, investigated by numericists, but not well studied in the  mathematical literature. We recapitulate recent analytic and numerical results on  new effects, that  appear for weak gravitational fields, in the so-called first post-Newtonian approximation. One of the exciting  implications is that one can estimate the  mass of a gaseous disk rotating around a spinless black hole.

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