### Abstract

Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of sound c_{0} and the shock speed s.

Original language | English (US) |
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Pages (from-to) | 161-194 |

Number of pages | 34 |

Journal | Communications in Mathematical Physics |

Volume | 86 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1982 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*86*(2), 161-194. https://doi.org/10.1007/BF01206009

**Shock profile solutions of the Boltzmann equation.** / Caflisch, Russel; Nicolaenko, Basil.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 86, no. 2, pp. 161-194. https://doi.org/10.1007/BF01206009

}

TY - JOUR

T1 - Shock profile solutions of the Boltzmann equation

AU - Caflisch, Russel

AU - Nicolaenko, Basil

PY - 1982/6

Y1 - 1982/6

N2 - Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of sound c0 and the shock speed s.

AB - Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of sound c0 and the shock speed s.

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UR - http://www.scopus.com/inward/citedby.url?scp=0007086538&partnerID=8YFLogxK

U2 - 10.1007/BF01206009

DO - 10.1007/BF01206009

M3 - Article

VL - 86

SP - 161

EP - 194

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -