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

  • Author(s): Coutand, Daniel
  • Shkoller, Steve
  • et al.

Published Web Location

https://arxiv.org/pdf/0910.3136.pdf
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Abstract

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.

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