### Abstract

The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in [9]. He used a normal mode analysis to define a stability function V(λ, η), whose zeros in ℜλ > 0 correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [11] he was able to prove that for large classes of steady ZND profiles, unstable zeros of V always exist in the high frequency regime, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well. In this paper we begin a rigorous study of the implications for nonlinear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of V(λ, η), one can prove the finite (but arbitrarily long) time existence of slightly curved, nonsteady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, △_{ZND} (λ̂ η̂), which turns out to coincide with the high frequency limit of V(λ η) | λ η | in ℜ λ̂ > 0. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where △_{ZND}(λ̂ η̂ ) is bounded away from zero inℜ λ̂ > 0. We also revisit the argument of [11] in order to simplify and complete some of the analysis in the proof of the main result there. Indiana University Mathematics Journal

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

Pages (from-to) | 1405-1461 |

Number of pages | 57 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Aug 3 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*56*(3), 1405-1461. https://doi.org/10.1512/iumj.2007.56.2972

}

*Indiana University Mathematics Journal*, vol. 56, no. 3, pp. 1405-1461. https://doi.org/10.1512/iumj.2007.56.2972

**Existence and stability of curved multidimensional detonation fronts.** / Costanzino, N.; Jenssen, Helge Kristian; Lyng, G.; Williams, Mark.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence and stability of curved multidimensional detonation fronts

AU - Costanzino, N.

AU - Jenssen, Helge Kristian

AU - Lyng, G.

AU - Williams, Mark

PY - 2007/8/3

Y1 - 2007/8/3

N2 - The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in [9]. He used a normal mode analysis to define a stability function V(λ, η), whose zeros in ℜλ > 0 correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [11] he was able to prove that for large classes of steady ZND profiles, unstable zeros of V always exist in the high frequency regime, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well. In this paper we begin a rigorous study of the implications for nonlinear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of V(λ, η), one can prove the finite (but arbitrarily long) time existence of slightly curved, nonsteady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, △ZND (λ̂ η̂), which turns out to coincide with the high frequency limit of V(λ η) | λ η | in ℜ λ̂ > 0. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where △ZND(λ̂ η̂ ) is bounded away from zero inℜ λ̂ > 0. We also revisit the argument of [11] in order to simplify and complete some of the analysis in the proof of the main result there. Indiana University Mathematics Journal

AB - The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in [9]. He used a normal mode analysis to define a stability function V(λ, η), whose zeros in ℜλ > 0 correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [11] he was able to prove that for large classes of steady ZND profiles, unstable zeros of V always exist in the high frequency regime, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well. In this paper we begin a rigorous study of the implications for nonlinear stability of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of V(λ, η), one can prove the finite (but arbitrarily long) time existence of slightly curved, nonsteady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, △ZND (λ̂ η̂), which turns out to coincide with the high frequency limit of V(λ η) | λ η | in ℜ λ̂ > 0. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where △ZND(λ̂ η̂ ) is bounded away from zero inℜ λ̂ > 0. We also revisit the argument of [11] in order to simplify and complete some of the analysis in the proof of the main result there. Indiana University Mathematics Journal

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U2 - 10.1512/iumj.2007.56.2972

DO - 10.1512/iumj.2007.56.2972

M3 - Article

VL - 56

SP - 1405

EP - 1461

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -