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

2009/10/16 by Coutand, Daniel, Shkoller, Steve · 1 citation
#35L65 #35L70 #35L80 #35Q35 #35R35 #76B03 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.3136

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= γρ^γ-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.

Cited by

• Global Existence and Asymptotic Equivalence to Barenblatt-type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D

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