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In this paper, we prove a new type of energy estimates for the compressibleEuler's equation with free boundary, with a boundary part and an interior part.These can be thought of as a generalization of the energies in Christodoulouand Lindblad [CL] to the compressible case and do not require the fluid to beirrotational. In addition, we show that our estimates are in fact uniform inthe sound speed. As a consequence, we obtain convergence of solutions ofcompressible Euler equations with a free boundary to solutions of theincompressible equations, generalizing the result of Ebin [Eb] to when you havea free boundary. In the incompressible case our energies reduces to those in[CL] and our proof in particular gives a simplified proof of the estimates in[CL] with improved error estimates. Since for an incompressible irrotationalliquid with free surface there are small data global existence results ourresult leaves open the possibility of long time existence also for slightlycompressible liquids with a free surface.

A priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit