Gregory Lyng

Stability of Large-Amplitude Shock Waves of Compressible Navier–Stokes Equations

Abstract We summarize recent progress on one-dimensional and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the large-amplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multidimensions by a co-dimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently small-amplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently large-amplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory. 1

A stability index for detonation waves in Majda's model for reacting flow

Abstract Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [SIAM J. Math. Anal. 32 (2001) 929; Commun. Pure Appl. Math. 51 (7) (1998) 797], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier–Stokes equations of reacting flow in [G. Lyng, One dimensional stability of detonation waves, Doctoral Thesis, Indiana University, 2002; G. Lyng, K. Zumbrun, Stability of detonation waves, Preprint, 2003]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [SIAM J. Sci. Statist. Comput. 7u (1986) 1059] and analytical results of [Commun. Math. Phys. 204 (3) (1999) 551; Commun. Math. Phys. 202 (3) (1999) 547] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, “bump-type” ignition function) deflagration profiles, as discussed in [SIAM J. Math. Anal. 24 (1993) 968; SIAM J. Appl. Math. 55 (1995) 175] for the full equations.

One-Dimensional Stability of Viscous Strong Detonation Waves

Abstract.Building on Evans-function techniques developed to study the stability of viscous shocks, we examine the stability of strong-detonation-wave solutions of the Navier-Stokes equations for reacting gas. The primary result, following [1, 17], is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the Zeldovich-von Neumann-Döring limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided that the underlying shock is stable. Finally, for completeness, we include the calculation of the stability index for a viscous shock solution of the Navier-Stokes equations for a nonreacting gas.

VISCOUS HYPERSTABILIZATION OF DETONATION WAVES IN ONE SPACE DIMENSION

It is a standard practice to neglect diffusive effects in stability analyses of detonation waves. Here, with the aim of quantifying the impact of these oft-neglected effects on the stability characteristics of such waves, we use numerical Evans-function techniques to study the (spectral) stability of viscous strong detonation waves---particularly traveling-wave solutions of the Navier--Stokes equations modeling a mixture of reacting gases. Our results show a surprising synergy between the high-activation-energy limit typically studied in stability analyses of detonation waves and the presence of small but nonzero diffusive effects. While our calculations do show a modest delay in the onset of instability in agreement with recently reported calculations by direct numerical simulation of the physical equations, our Evans-function-based approach provides additional spectral information. In particular, for each of the families of detonation waves in our computational domain, we find an unexpected kind of hyst...

Multidimensional stability of large-amplitude Navier-Stokes shocks

Extending results of Humpherys–Lyng–Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier–Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients

Spectral Stability of Ideal-Gas Shock Layers

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POINTWISE GREEN FUNCTION BOUNDS AND STABILITY OF COMBUSTION WAVES

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