Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum

Journal article

The free-boundary compressible 1-D Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws which are both characteristic and degenerate. The physical vacuum singularity (or rate-of-degeneracy) requires the sound speed $c= \gamma \rho^{\gamma -1}$ to scale as the square-root of the distance to the vacuum boundary, and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions.

Authors: Coutand, D; Shkoller, S

• Journal: Communications on Pure and Applied Mathematics
• Volume: 64
• Issue: 3
• Pages: 328-366
• Publication date: 2009-10-16
• DOI: 10.1002/cpa.20344
• EISSN: 0010-3640
• ISSN: 0010-3640
• Language: English
• Keywords: math.AP; 35L65, 35L70, 35L80, 35Q35, 35R35, 76B03