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20
Stability of largeamplitude shock waves of compressible NavierStokes equations
, 2003
"... We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be ..."
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Cited by 38 (24 self)
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We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude 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 codimension 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 smallamplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently largeamplitude 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.
An Evans function approach to spectral stability of smallamplitude viscous shock profiles, preprint
 34 HOWARD and K. ZUMBRUN, Stability of Undercompressive Shock Profiles
, 2002
"... Abstract. In recent work, the second author and various collaborators have shown using Evans function/refined semigroup techniques that, under very general circumstances, the problems of determining one or multidimensional nonlinear stability of a smooth shock profile may be reduced to that of det ..."
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Cited by 35 (31 self)
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Abstract. In recent work, the second author and various collaborators have shown using Evans function/refined semigroup techniques that, under very general circumstances, the problems of determining one or multidimensional nonlinear stability of a smooth shock profile may be reduced to that of determining spectral stability of the corresponding linearized operator about the wave. It is expected that this condition should in general be analytically verifiable in the case of small amplitude profiles, but this has so far been shown only on a casebycase basis using clever (and difficult to generalize) energy estimates. Here, we describe how the same set of Evans function tools that were used to accomplish the original reduction can be used to show also smallamplitude spectral stability by a direct and readily generalizable procedure. This approach both recovers the results obtained by energy methods, and yields new results not previously obtainable. In particular, we establish onedimensional stability of small amplitude relaxation profiles, completing the Evans function program set out in Mascia&Zumbrun [MZ.1]. Multidimensional stability of small amplitude viscous profiles will be addressed in a companion paper [PZ], completing the program of Zumbrun [Z.3]. Section
Viscous And Inviscid Stability Of Multidimensional Planar Shock Fronts
 Indiana Univ. Math. J
, 1999
"... . We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock profile. Our main result, generalizing earlier onedimensional calculations, is that the Evans function reduces in the longwave limit t ..."
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Cited by 35 (18 self)
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. We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock profile. Our main result, generalizing earlier onedimensional calculations, is that the Evans function reduces in the longwave limit to the KreissSakamoto Lopatinski determinant obtained by Majda in the inviscid case, multiplied by a constant measuring transversality of the shock connection in the underlying (viscous) traveling wave ODE. Remarkably, this result holds independently of the nature of the viscous regularization, or the type of the shock connection. Indeed, the analysis is more general still: in the overcompressive case, we obtain a simple longwave stability criterion even in the absence of a sensible inviscid problem. An immediate consequence is that inviscid stability is necessary (but not sufficient) for viscous stability; this yields a number of interesting results on viscous instability through the in...
Alternate Evans functions and viscous shock waves
, 1999
"... We present various alternative definitions of the classical Evans function, and show how they may be used to compute quantities related to stability of viscous shock waves. This allows the extension of Evans function methods to settings in which the standard definition does not apply. At the same ti ..."
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Cited by 31 (19 self)
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We present various alternative definitions of the classical Evans function, and show how they may be used to compute quantities related to stability of viscous shock waves. This allows the extension of Evans function methods to settings in which the standard definition does not apply. At the same time, we show how to treat systems of dimension n > 2 and how to evaluate cases of neutral stability. The former resolves a problem left open by Gardner and Zumbrun in their treatment of viscous shock waves.
Pointwise Green’s function bounds and stability of relaxation shocks
 Indiana Univ. Math. J
"... Abstract. We establish sharp pointwise Green’s function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stab ..."
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Cited by 31 (26 self)
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Abstract. We establish sharp pointwise Green’s function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, with no additional assumptions on the structure or strength of the shock. Restricting to Lax type shocks, we establish the further result of nonlinear stability with respect to small L 1 ∩ H 2 perturbations, with sharp rates of decay in L p, 2 ≤ p ≤ ∞, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in W 3,1 ∩ W 3, ∞, we obtain sharp rates of decay in L p, 1 ≤ p ≤ ∞, for (Lax type) shocks of arbitrary strength. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin–Xin and Broadwell models, for which spectral stability has been established in [HL,JH] and [KM], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [ZH] for the study of traveling waves of parabolic systems; however, significant extensions are required to deal with the nonsectorial generator and more singular shorttime behavior of the associated (hyperbolic) linearized equations. Our main technical innovation is a systematic method for refining largefrequency (shorttime) estimates on the resolvent kernel, suitable in the absence of parabolic smoothing. This seems particularly interesting from the viewpoint of general linear theory, replacing the zeroorder estimates of existing theory with a series expansion to arbitrary order. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD. Section
Planar stability criteria for viscous shock waves of systems with real viscosity
, 2004
"... We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with “real,” or partial viscosity. The main result is the establishment of necessary, or “weak”, and sufficient, or “strong”, conditions ..."
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Cited by 17 (14 self)
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We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with “real,” or partial viscosity. The main result is the establishment of necessary, or “weak”, and sufficient, or “strong”, conditions for nonlinear stability analogous to those established by Majda [Ma.1–3] in the inviscid case but (generically) separated by a codimensionone set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stability and instability is thereby determined, lying on the boundary set between the open regions of strong stability and strong instability (the latter defined as failure of weak stability). Strong stability holds always for smallamplitude shocks of classical “Lax” type [PZ.1–2, FreS]; for largeamplitude shocks, however, strong instability may occur [ZS, Z.3].
Existence and Stability of Standing Pulses in Neural Networks
 I. Existence. SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse soluti ..."
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Cited by 15 (1 self)
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Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse solutions coexist, the small pulse is unstable, and the large pulse is stable. The large single pulse is bistable with the “alloff ” state. This bistable localized activity may have strong implications for the mechanism underlying working memory. We show that dimple pulses have similar stability properties to large pulses but double pulses are unstable.
A stability index for detonation waves in Majda’s model for reacting flow
 Phys. D
"... Abstract. Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified mo ..."
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Cited by 11 (9 self)
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Abstract. Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], 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 [Ly,LyZ]. 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 [CMR] and analytical results of [Sz,LY] 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, “bumptype ” ignition function) deflagration profiles, as discussed in [GS.1–2] for the full equations. Section
Stability of viscous shocks in isentropic gas dynamics
, 2007
"... Abstract. In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or psystem with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which smallamplitude viscous s ..."
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Cited by 10 (9 self)
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Abstract. In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or psystem with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which smallamplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire righthalf plane, thus demonstrating numerically that largeamplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finitedifference simulations of perturbed largeamplitude shocks converge to a translate of the original shock wave, as expected. 1.
Onedimensional 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 viscous strong detonation wave solutions of the reacting NavierStokes equations. The primary result, following [1, 17], is the calculation of a stability index whose sign ..."
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Cited by 10 (9 self)
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Abstract. Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting NavierStokes equations. 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 ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting NavierStokes equations are included.