### Abstract

An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x^{1/2} close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local-in-time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.

Original language | English (US) |
---|---|

Pages (from-to) | 1327-1385 |

Number of pages | 59 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 62 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2009 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Well-posedness for compressible Euler equations with physical vacuum singularity.** / Jang, Juhi; Masmoudi, Nader.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 62, no. 10, pp. 1327-1385. https://doi.org/10.1002/cpa.20285

}

TY - JOUR

T1 - Well-posedness for compressible Euler equations with physical vacuum singularity

AU - Jang, Juhi

AU - Masmoudi, Nader

PY - 2009/10

Y1 - 2009/10

N2 - An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x1/2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local-in-time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.

AB - An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x1/2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local-in-time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity.

UR - http://www.scopus.com/inward/record.url?scp=68049131315&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68049131315&partnerID=8YFLogxK

U2 - 10.1002/cpa.20285

DO - 10.1002/cpa.20285

M3 - Article

VL - 62

SP - 1327

EP - 1385

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 10

ER -