### Abstract

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].

Original language | English (US) |
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Pages (from-to) | 103-130 |

Number of pages | 28 |

Journal | Communications in Mathematical Physics |

Volume | 114 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1988 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*114*(1), 103-130. https://doi.org/10.1007/BF01218291

**Stability of shock waves for the Broadwell equations.** / Caflisch, Russel; Liu, Tai Ping.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 114, no. 1, pp. 103-130. https://doi.org/10.1007/BF01218291

}

TY - JOUR

T1 - Stability of shock waves for the Broadwell equations

AU - Caflisch, Russel

AU - Liu, Tai Ping

PY - 1988/3

Y1 - 1988/3

N2 - For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].

AB - For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].

UR - http://www.scopus.com/inward/record.url?scp=0011609562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011609562&partnerID=8YFLogxK

U2 - 10.1007/BF01218291

DO - 10.1007/BF01218291

M3 - Article

VL - 114

SP - 103

EP - 130

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -