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 Navier-Stokes equations. The primary result, following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, 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 Navier-Stokes equations are included
Confirming a conjecture of Lyng--Raoofi--Texier--Zumbrun, we show that stability of strong detonation waves in the ZND, or small-viscosity, limit is equivalent to stability of the limiting ZND detonation together with stability of the viscous profile associated with the component Neumann shock. More, on bounded frequencies the nonstable eigenvalues of the viscous detonation wave converge to those of the limiting ZND detonation, while on frequencies of order one over viscosity, they converge to one over viscosity times thos of the associated viscous Neumann shock. This yields immediately a numb…
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,…
It has long been a standard practice to neglect diffusive effects in stability analyses of detonation waves. Here, with the principal 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---particular traveling-wave solutions of the Navier-Stokes equations modeling a mixture of reacting gases. Remarkably, our results show a surprising synergy between the high-activation-energy limit typically studied in stability analyses of deto…
We describe recent analytical and numerical results on stability and behavior of viscous and inviscid detonation waves obtained by dynamical systems/Evans function techniques like those used to study shock and reaction diffusion waves. In the first part, we give a broad description of viscous and inviscid results for 1D perturbations; in the second, we focus on inviscid high-frequency stability in multi-D and associated questions in turning point theory/WKB expansion.
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 efficient 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 result…
The rigorous study of spectral stability of strong detonations was begun by Erpenbeck in the 1960s. Working with the Zeldovitch-von Neumann-Döring (ZND) model, he identified two fundamental classes of detonation profiles, referred to as those of decreasing (D) and increasing (I) type, which appeared to exhibit very different behavior with respect to high-frequency perturbations. Using a combination of rigorous and non-rigorous arguments, Erpenbeck concluded that type I detonations were unstable to some oscillatory perturbations for which the (vector) frequency was of arbitrarily large magnitud…
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 o…
For the ZND model, we show also singularity formation on the downstream side for arbitrary exponentially-growing weighted norms. For the Majda model on the other hand, we establish for appropriate such weighted norms a set of energy estimates implying not only non-formation of singularities near waves of arbitrary amplitude but also full asymptotic orbital stability for small-amplitude ones.
We establish one-dimensional spectral, or "normal modes", stability of ZND detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed, verifying numerical observations of Erpenbeck in the 1960s. The key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite, regular perturbation problem, and a careful high-frequency analysis depending uniformly on model parameters. The latter recovers the important result of high-frequency stability established by Erpenbeck by somewhat…
We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda's model for reacting flow, and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of RoqueJoffre-Vila that viscous detonations of Majda's model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.
We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also he…
Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^2$ perturbatoins, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial…
For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.
Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun--Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability cond…
Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincaré--Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction …
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…
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, which leads to undifferentiated quadratic source terms that cannot be handled using the estimates of the conservative case. The first is resolved by treating the equations in the more favorable Lagrangi…
A classical model of fluid dynamics is considered which describes the shape evolution of a viscous liquid droplet on a homogeneous substrate. All equilibria are characterized and their stability is analyzed by a geometric reduction argument.
A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic stability of the viscous shock wave is established under some smallness conditions. The proof is given by an elementary energy estimate.
This paper is concerned with the asymptotic stability of a composite wave consisting of two viscous shock waves to the Cauchy problem for a one-dimensional system of heat-conductive ideal gas without viscosity. We extend the results by Huang-Matsumura \cite{Huang-Matsumura} where they treated the equation of viscous and heat-conductive ideal gas. That is, even forthe non-viscous and heat-conductive case, we show that if the strengths of the viscous shock waves and the initial perturbation are suitably small, the unique global solution in time exists and asymptotically tends toward the correspo…
In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes $γ$-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in Matsumura-Nishihara [On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous ga…
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling-wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions.
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 $L^p$, $1\le p\le \infty$.
We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instanteous location of a viscous shock-- we show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Guès, Métivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection
In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun ["Instantaneous shock location ..."] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give b…
This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity. We show that, for the non-viscous case, if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends toward the corresponding the viscous contact wave or the composition of a viscous contact wave with rarefaction waves determined by the initial condition, which extended the results by Huang-Li-Matsumura[13], where they treated the viscous and heat-conductive ideal gas.
In this paper, we are concerned with the global existence and stability of a 3-D perturbed viscous circulatory flow around an infinite long cylinder. This flow is described by 3-D compressible Navier-Stokes equations. By introducing some suitably weighted energy spaces and establishing a priori estimates, we show that the 3-D cylindrical symmetric circulatory flow is globally stable in time when the corresponding initial states are perturbed suitably small.
We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $\approx 1.86 $, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $\approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verificati…
This paper is concerned with the large-time behavior of solutions for the one-dimensional compressible Navier-Stokes system. We show that the combination of viscous contact wave with rarefaction waves for the non-isentropic polytropic gas is stable under \emph{large} initial perturbation without the condition that the adiabatic exponent $γ$ is close to 1, provided the strength of the combination waves is suitably small.
We investigate the Navier-Stokes-Fourier system for incompressible heat conducting inhomogeneous fluid. The main result concerns existence of global in time regular large solutions, provided the initial temperature is sufficiently large. The system may be viewed as a model of pudding, as we assume the viscosity grows with the temperature.
In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. Moreover the strengths of two waves could be arbitrarily large.
The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \mathbb R \times \mathbb T^{n-1} (n\geq2) $, where $\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.
Stability of Superposition of Viscous Contact Wave and Rarefaction Waves for Compressible Navier-Stokes System with Degenerate Heat-Conductivity and Large-Data
This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time …
This paper is concerned with nonlinear stability of viscous contact discontinuity to a free boundary problem for the one-dimensional full compressible Navier-Stokes equations in half space $[0,\infty)$. For the case when the local stability of the contact discontinuities was first studied by [1],later generalized by [2], local stability of weak viscous contact discontinuity is well-established by [4-8], but for the global stability of the impermeable gas with big oscillation ends $(|θ_+-θ_-|>1) $, fewer results have been obtained excluding zero dissipation [9] or $γ\to 1$ gas see [10]. Our mai…
We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with $γ\ge 2$, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.
The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier-Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N-S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearl…
Extending results of Oh and Zumbrun in dimensions $d\ge 3$, we establish nonlinear stability and asymptotic behavior of spatially-periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions $d=1,2$, under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions $d\ge 3$ are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the no…
We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$, for a class of large perturbations and for any bounded time interval. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method developed in [22], [23] to the study of an inviscid limit to a shock.
We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier-Stokes equations under the generic perturbations, in particular, without zero mass conditions. Moreover, the time-dependent shift function keeps the shock profile shape time-asymptotically. Our stability result is unconditional for the weak planar Navier-Stokes shock. Our proof is motivated by the $a$-contraction method (a kind of weighted $L^2$-relative entropy method) with time-dependent shift introduced in [10,11,13] for the stability of viscous shock …
Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real f…
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 codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stabil…
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and dif…
We study the large-time behavior of solutions to the compressible Navier-Stokes equations for a viscous and heat-conducting ideal polytropic gas in the one-dimensional half-space. A rarefaction wave and its superposition with a non-degenerate stationary solution are shown to be asymptotically stable for the outflow problem with large initial perturbation and general adiabatic exponent.
We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point…
In this paper, we study the nonlinear stability of the composite wave consisting of planar rarefaction and planar contact waves for viscous conservation laws with degenerate flux under multi-dimensional periodic perturbations. To the level of our knowledge, it is the first stability result of the composite wave for conservation laws in several dimensions. Moreover, the perturbations studied in the present paper are periodic, which keep constantly oscillating at infinity. Suitable ansatz is constructed to overcome the difficulty caused by this kind of perturbation and delicate estimates are don…
By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier--Stokes equations with $γ$-law pressure (i) in the limit as Mach number $M$ goes to infinity, for any $γ\ge 1$ (proved analytically), and (ii) for $M\ge 2,500$, $γ\in [1,2.5]$ (demonstrated numerically). This builds on and completes earlier studies by Matsumura--Nishihara and Barker--Humpherys--Rudd--Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, in…
Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$.…
