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We derive a priori estimates for the compressible free-boundary Eulerequations with surface tension in three spatial dimensions in the case of aliquid. These are estimates for local existence in Lagrangian coordinates whenthe initial velocity and initial density belong to $H^3$, with an extraregularity condition on the moving boundary, thus lowering the regularity ofthe initial data. Our methods are direct and involve two key elements: theboundary regularity provided by the mean curvature, and a new compressibleCauchy invariance.

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid