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

Daniel Coutand, Steve Shkoller

Research output: Contribution to journalArticle

68 Citations (Scopus)

Abstract

The free-boundary compressible one-dimensional Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws that are both characteristic and degenerate. The physical vacuum singularity (or rate of degeneracy) requires the sound speed$c^2= \gamma \rho^{ \gamma -1}$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. © 2010 Wiley Periodicals, Inc.

Original languageEnglish
Pages (from-to)328-366
Number of pages39
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number3
DOIs
Publication statusPublished - Mar 2011

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