Well-posedness in smooth function spaces for moving-boundary 1-D compressible euler equations in physical vacuum
Article first published online: 16 SEP 2010
Copyright © 2010 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 64, Issue 3, pages 328–366, March 2011
How to Cite
Coutand, D. and Shkoller, S. (2011), Well-posedness in smooth function spaces for moving-boundary 1-D compressible euler equations in physical vacuum. Comm. Pure Appl. Math., 64: 328–366. doi: 10.1002/cpa.20344
- Issue published online: 17 DEC 2010
- Article first published online: 16 SEP 2010
- Manuscript Revised: MAY 2009
- Manuscript Received: MAR 2009
- National Science Foundation. Grant Number: DMS-0701056
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 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.