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We prove well-posedness for the 3-D compressible Euler equations with movingphysical vacuum boundary, with an equation of state given by the so-calledgamma gas-law for gamma > 1. The physical vacuum singularity requires the soundspeed c to go to zero as the square-root of the distance to the movingboundary, and thus creates a degenerate and characteristic hyperbolicfree-boundary system wherein the density vanishes on the free-boundary, theuniform Kreiss--Lopatinskii condition is violated, and manifest derivative lossensues. Nevertheless, we are able to establish the existence of uniquesolutions to this system on a short time-interval, which are smooth (in Sobolevspaces) all the way to the moving boundary, and our estimates have noderivative loss with respect to initial data. Our proof is founded on anapproximation of the Euler equations by a degenerate parabolic regularizationobtained from a specific choice of a degenerate artificial viscosity term,chosen to preserve as much of the geometric structure of the Euler equations aspossible. We first construct solutions to this degenerate parabolicregularization using a new higher-order Hardy-type inequality; we thenestablish estimates for solutions to this degenerate parabolic system which areindependent of the artificial viscosity parameter. Solutions to thecompressible Euler equations are found in the limit as the artificial viscositytends to zero. Our regular solutions can be viewed as degenerate viscositysolutions. Out methodology can be applied to many other systems of degenerateand characteristic hyperbolic systems of conservation laws.

Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum