Sharp Pointwise Bounds for Perturbed Viscous Shock Waves

Howard, Peter; Raoofi, Mohammadreza; Zumbrun, Kevin

doi:10.1142/s021989160600080x

Cited by 17 publications

(14 citation statements)

References 37 publications

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“…The full reactive Navier-Stokes equations with real, or physical viscosity may be treated by essentially the same techniques, using the more complicated arguments (and more detailed Green function bounds) developed in [18,19,36,37,41,48] for the treatment of viscous shocks with real viscosity. However, these arguments so far are limited to the strong detonation case.…”

Section: Extension To the Navier-stokes Equations With Physical Viscomentioning

confidence: 99%

Pointwise Green function bounds and stability of combustion waves

Lyng

Raoofi

Texier

et al. 2007

Journal of Differential Equations

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Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier-Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman-Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu-Ying and Tesei-Tan for Majda's model and the reactive Navier-Stokes equations, respectively.

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Section: Extension To the Navier-stokes Equations With Physical Viscomentioning

confidence: 99%

Pointwise Green function bounds and stability of combustion waves

Lyng

Raoofi

Texier

et al. 2007

Journal of Differential Equations

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“…2 The study of stability of viscous shock layers, was initiated at the one-dimensional scalar level by Hopf [34] and Il'in-Oleȋnik [42]. For one-dimensional systems, it was begun in the 1980's by Kawashima-Matsumura, Kawashima-Matsumua-Nishihara, Liu, and Goodman [46,47,50,26,27], and essentially concluded in [73,51,24,82,61,62,63,39,41,38,37,67]. We note in particular the proof by Mascia-Zumbrun and Humpherys-Zumbrun [62,39] for the first time of small-amplitude (one-dimensional) stability of ordinary gas-dynamical and Laxtype magnetohydrodynamic Navier-Stokes shocks with general equation of state, and the proof by Mascia-Zumbrun and Raoofi-Zumbrun [63,67] of nonlinear (one-dimensional) stability of large-amplitude shock solutions of arbitrary type for a class of systems generalizing the Kawashima class [44,45], including gas dynamics, viscoelasticity, and magnetohydrodynamics (MHD), assuming a numerically verifiable Evans-function condition encoding spectral stability in an appropriate sense; that is, the Evans-function condition accounts for the lack of spectral gap/accumulating essential spectrum at the origin that is an fundamental feature of the shock stability problem.…”

Section: Background and Description Of Main Resultsmentioning

confidence: 99%

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

Humpherys

Lyng

Zumbrun

2017

Arch Rational Mech Anal

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Abstract. 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 µ, µ + η, and ν = κ/c v in the physical ratios predicted by statistical mechanics, with Mach number M > 1.035. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any γ-law gas, γ > 1, and arbitrary µ > |η| ≥ 0, ν > 0 or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).

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“…Finally, we note that the second order term in [5] Theorem 1.1 is the analogue here of the diffusion waves of Liu [41], and could be recovered in the current setting by an analysis similar to that of [29,30,49].…”

Section: Introductionmentioning

confidence: 89%

Asymptotic Behavior Near Transition Fronts for Equations of Generalized Cahn–Hilliard Form

Howard

2006

Commun. Math. Phys.

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We consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary PDE of generalized Cahn-Hilliard form in one space dimension. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into its two components with their concentrations separated by sharp transition layers. Motivated by work of Bricmont, Kupiainen, and Taskinen [5], we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions. A critical feature of the Cahn-Hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from standing wave solutions, establishing phase-asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate.

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Sharp Pointwise Bounds for Perturbed Viscous Shock Waves

Cited by 17 publications

References 37 publications

Pointwise Green function bounds and stability of combustion waves

Pointwise Green function bounds and stability of combustion waves

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

Asymptotic Behavior Near Transition Fronts for Equations of Generalized Cahn–Hilliard Form

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