Back to Search
Start Over
Global existence of near-affine solutions to the compressible Euler equations
- Publication Year :
- 2017
-
Abstract
- We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the pressure-density relation $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}} $. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}} $ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.<br />Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational Mech. Anal
- Subjects :
- Mathematics - Analysis of PDEs
35Q31, 76N10, 76N15, 35B35, 35L65
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1710.08368
- Document Type :
- Working Paper