Pointwise Green function bounds and stability of combustion waves

Lyng, Gregory; Raoofi, Mohammadreza; Texier, Benjamin; Zumbrun, Kevin

doi:10.1016/j.jde.2006.10.006

Cited by 25 publications

(35 citation statements)

References 45 publications

(195 reference statements)

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“…A detailed overview of these results can be found in the recent work of Humpherys et al [17]. Notably, however, outside of the Evans-function framework, namely [22,23], we know of no stability results for weak-detonation solutions of the Majda model.…”

Section: 2mentioning

confidence: 99%

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Stability of Viscous Weak Detonation Waves for Majda’s Model

et al. 2015

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Continuing the program initiated by Humpherys, Lyng, & Zumbrun [17] for strong detonation waves, we use a combination of analytical and numerical Evans-function techniques to analyze the spectral stability of weak detonation waves in a simplified model for gas-dynamical combustion. Combining these new spectral stability results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [22], we conclude that these waves are nonlinearly stable. The principal novelty of this analysis is the treatment of weak detonation waves. In contrast to the case of strong detonation waves, weak detonation waves are undercompressive and the stability of these waves is delicate and has not been treated by standard weighted-norm techniques. The present analysis thus provides a case study illustrating the flexibility and power of the Evans-functionbased approach to stability. As in the case of strong detonations, we find that all tested waves are spectrally stable, hence nonlinearly stable.

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Section: 2mentioning

confidence: 99%

“…To describe our results more precisely, we now introduce the version of the Majda model to which will be the setting for our analysis 2 . We begin with the Majda model [22]:…”

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confidence: 99%

Stability of Viscous Weak Detonation Waves for Majda’s Model

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“…Proposition 1 (Lyng-Raoofi-Texier-Zumbrun [42]). Under the Evans-function condition E(·) has precisely one zero in {Re λ ≥ 0} (necessarily at λ = 0) ,…”

Section: Stabilitymentioning

confidence: 99%

“…Basic Assumptions. Following Lyng, Raoofi, Texier & Zumbrun [42], we begin with the following version of the Majda model:…”

Section: Preliminariesmentioning

confidence: 99%

Stability of viscous detonations for Majda’s model

Humpherys

Lyng

Zumbrun

2013

Physica D: Nonlinear Phenomena

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Abstract. Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of Majda's scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J. Differential Equations 233 (2007), no. 2, 654-698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonationwave solutions of the Majda model without a smallness assumption. Notably, our results indicate that, for the simplified Majda model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to "galloping" or "pulsating" solutions as is observed in the full reactive Navier-Stokes equations. This answers in the negative a question posed by Majda as to whether the scalar detonation model captures this aspect of detonation behavior.

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“…New physical applications beyond those of [10] are to undercompressive waves in MHD and, with slight modification following [16], to weak detonation waves in reactive compressible Navier-Stokes equations. The latter we intend to treat in a future work.…”

Section: Definition 12 An Ideal Shockmentioning

confidence: 99%

Stability of undercompressive viscous shock profiles of hyperbolic–parabolic systems

Raoofi

Zumbrun

2009

Journal of Differential Equations

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Extending to systems of hyperbolic-parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in L p , 1 p ∞.

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Pointwise Green function bounds and stability of combustion waves

Cited by 25 publications

References 45 publications

Stability of Viscous Weak Detonation Waves for Majda’s Model

Stability of Viscous Weak Detonation Waves for Majda’s Model

Stability of viscous detonations for Majda’s model

Stability of undercompressive viscous shock profiles of hyperbolic–parabolic systems

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