Galloping instability of viscous shock waves

Abstract

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 in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.

Introduction

Motivated by physical and numerical observations of time-oscillatory “galloping” or “pulsating” instabilities of detonation waves [76], [9], [22], [76], [3], [1], [19], [20], we study the Poincaré–Hopf bifurcation of viscous shock waves in one spatial dimension. Our main result is to obtain a rigorous (nonlinear) characterization in terms of spectral information. The complementary problem of verifying this spectral information has been studied already in [65], [67],where it was shown that the transition from stability to instability of viscous detonation waves generically involves a Poincaré–Hopf type bifurcation in the spectral configuration of the linearized operator about the wave. By essentially the same analysis we obtain also a corresponding multi-dimensional result applying to planar viscous shock fronts traveling in a cylinder of finite cross-section, with artificial periodic boundary conditions. This gives a simplified mathematical model for the time-oscillatory instabilities observed in detonation waves moving within a duct, which, besides the longitudinal galloping instabilities described above, also include transverse “cellular” or “spinning” instabilities. The method of analysis appears to be of general application, in particular, with suitable elaboration, to extend to the original motivating case of viscous detonation waves of the reactive Navier–Stokes equations with physical viscosity.

From a mathematical standpoint, the main issue in our analysis is that the linearized equations for a standing shock wave have no spectral gap between convective modes corresponding to the essential spectrum of the linearized operator $L$ of the wave and the oscillatory modes corresponding to pure imaginary point spectrum. This prohibits the usual PDE analysis by center manifold reduction to a finite-dimensional subspace; likewise, at the linearized level, decay to the center subspace is at a time-algebraic rather than a time-exponential rate, so that oscillatory and other modes are strongly coupled. We overcome this difficulty by the introduction of a nonstandard Implicit Function Theorem framework suitable for an infinite-dimensional Lyapunov–Schmidt reduction in the absence of the spectral gap (Section 2.1), augmented by a finite-dimensional “weak” Implicit Function Theorem suitable for finite-dimensional bifurcation analysis in situations of limited regularity (Section 2.3). The latter, based on the Brouwer Fixed-point Theorem rather than the standard Contraction-mapping construction, seems of interest in its own right.

The study of bifurcation from stability is a natural followup to our previous work on the stability of viscous shock and detonation waves; see, e.g., [97], [73], [74], [40], [65], [67], [66]. Interestingly, the key estimate needed to apply our bifurcation framework turns out to be a space-time stability estimate on the transverse linearized solution operator (meaning the part complementary to oscillatory modes) quite similar to estimates on the full solution operator arising in the stability analysis of viscous traveling waves. To carry out this estimate requires rather detailed pointwise information on the Green function of the linearized equations for the wave; see Sections 1.6.4 Refined analysis, 5 Proof of. Indeed, we use the full power of the pointwise semigroup techniques developed in [97], [71], [73].

The equations of compressible gas dynamics in one spatial dimension, like many equations in continuum mechanics, take the form of hyperbolic conservation laws

$$u_t+f{\left(u\right)}_x=0\text{.}$$

(Euler equations) or hyperbolic–parabolic conservation laws

$$u_t+f{\left(u\right)}_x={\left(B\left(u\right)u_x\right)}_x\text{.}$$

(Navier–Stokes equations), depending whether second-order transport effects–in this case, viscosity and heat-conduction–are neglected or included. Here, $x,t\in {\mathbb{R}}^1$ are spatial location and time, $u\in {\mathbb{R}}^n$ is a vector of densities of conserved quantities — mass, momentum, and energy in the case of gas dynamics, $f\in {\mathbb{R}}^n$ is a vector of corresponding fluxes, and $B\in {\mathbb{R}}^{n\times n}$ is a matrix of transport coefficients.

Such equations are well known to support traveling-wave solutions of the form

$$u(x,t)=\bar{u}(x-st)≔\{\begin{array}{cc}{u}_{-}& x-st\le 0\text{,}\\ u_{+}& x-st>0\end{array}$$

(discontinuous) and

$$u(x,t)=\bar{u}(x-st)\text{,}\underset{z\to \pm \infty }{lim}\bar{u}\left(z\right)=u_{\pm }$$

(smooth), respectively, known as ideal and viscous shock waves. These waves may be observed physically; indeed, the corresponding physical objects appear to be quite stable [5].

Similarly, the equations of reacting, compressible gas dynamics take the form

$$u_t+\tilde{f}{(u,z)}_x-qK\varphi \left(u\right)z=0$$$$z_t+{\left(v(u,z)z\right)}_x+K\varphi \left(u\right)z=0$$

(reactive Euler, or Zeldovich–von Neumann–Doering equations (ZND)) or

$$u_t+\tilde{f}{(u,z)}_x-qK\varphi \left(u\right)z={\left(\tilde{B}(u,z)u_x\right)}_x\text{,}$$$$z_t+{\left(v(u,z)z\right)}_x+K\varphi \left(u\right)z={\left(\tilde{D}(u,z)z_x\right)}_x$$

(reactive Navier–Stokes equations (rNS)), where $x,t\in {\mathbb{R}}^1$, $u\in {\mathbb{R}}^n$ is as before, $z\in {\mathbb{R}}^r$ is a vector of mass fractions of different reactant species, $v\in {\mathbb{R}}^1$ is fluid velocity, and $\tilde{f}$, $\tilde{B}$ and $\tilde{D}$ are flux vectors and transport matrices depending (through $z$) on the chemical makeup of the gas, with $\tilde{f}(u,0)=f\left(u\right)$, $\tilde{B}(u,0)=B\left(u\right)$. The matrix $K$ models reaction dynamics, $q$ is a constant heat release coefficient ($q>0$ corresponding to an exothermic reaction), and $\varphi \in {\mathbb{R}}^1$ is an ignition function serving to “turn on” the reaction: zero for temperatures below a certain critical temperature and positive for temperatures above.

Eqs. (1.5), (1.6) support traveling wave solutions

$$u(x,t)=\bar{u}(x-st)$$

analogous to (1.3), (1.4), known as ideal and viscous detonation waves, which may likewise be observed physically. However, in contrast to the shock wave case, detonations appear to be rather unstable. What is typically observed is not the planar, steadily progressing solution (1.7), but rather a nearby solution varying time-periodically about (1.7): that is, an apparent bifurcation with exchange of stability. In the typical experimental setting of a detonation moving along a duct, these may be longitudinal “galloping”, or “pulsating”, instabilities for which the planar structure is maintained, but the form of the profile changes time-periodically, or they may be transverse instabilities in which the front progresses steadily in the longitudinal (i.e., axial) direction, but develops time-oscillatory structure in the transverse (cross-sectional) directions of a form depending on the geometry of the cross-section: “cellular” instabilities for a polygonal (e.g. rectangular) cross-section; “spinning” instabilities for a circular cross-section, with one or more “hot spots”, or “combustion heads”, moving spirally along the duct.

Stability of shocks and detonations may be studied within a unified mathematical framework; see, e.g., [17], [18], [56], [57], [16], [8], [21], [61], [91], [58], [68], [69], [70], [23], [77], [14] in the inviscid case (1.1), (1.5), and [86], [30], [31], [53], [54], [75], [62], [64], [32], [87], [27], [97], [11], [12], [13], [10], [59], [93], [94], [95], [96], [72], [73], [74], [33], [34], [35], [38], [4], [65], [67], [50], [41], [42], [6], [81], [24], [66] in the viscous case (1.2), (1.6). In particular, stability in each case has been shown to reduce to spectral considerations accessible by standard normal-modes analysis (see Section 1.2). However, they are studied with different motivations, and yield different results. Historically, it appears that instability motivated the physical study, which has focused on the detonation case and spectral stability criteria [5]. By contrast, the mathematical study has focused on stability and the somewhat simpler shock wave case, with the main (difficult!) issue being to establish full nonlinear stability assuming that spectral stability holds in the form of a suitable “Lopatinski” or “Evans” condition [70], [77], [93], [34]. Regarding bifurcation, there is strong evidence that the detonation instabilities described above correspond to a Poincaré–Hopf bifurcation (indeed, our description above benefits much by hindsight). In particular, Erpenbeck [18] and Bourlieux, Majda, and Roytburd [9] have carried out formal asymptotics in support of this viewpoint for the one-dimensional (galloping, or pulsating) case, in the context of the ZND equations. More recently, Kasimov and Stewart [52] have carried out a definitive study in the multidimensional case by numerical linearized normal modes analysis, also for the ZND equations, in which they demonstrate that the onset of such instabilities indeed corresponds to crossing of the imaginary axis by a conjugate pair of eigenvalues toward the unstable (positive real part) side, with, moreover, excellent correspondence between observed nonlinear oscillations and the associated normal modes. Depending whether the maximally unstable mode is in the longitudinal or transverse direction, the oscillatory behavior is seen to be of galloping or spinning (cellular) type. However, up to now, no rigorous analysis of these phenomena has been carried out. To fill this gap is the object of the present study.

The main difficulty in the analysis of shock or detonation waves is the lack of a spectral gap between stationary or oscillatory modes and the essential spectrum of the linearized operator for the wave. In the inviscid, hyperbolic case (1.1), (1.5), the linearized problem has essential spectrum filling the imaginary axis. In the viscous case (1.2), (1.6), the essential spectrum is, in contrast, tangent at the origin to the imaginary axis.

For example, without loss of generality taking a standing-wave solution $u=\bar{u}\left(x\right)$, $s=0$ (i.e., working in coordinates moving with the wave), and linearizing (1.2) about $\bar{u}$, we obtain, taking $B\equiv I$ for simplicity,

$$u_t=Lu≔u_{xx}-{\left(Au\right)}_x\text{,}A≔df\left(\bar{u}\left(x\right)\right)\text{,}$$

where the asymptotically constant-coefficient operator $L$ is what we have called the linearized operator for the wave. A standard result of Henry ([37], Theorem A.2, chapter 5) on the spectrum of operators with asymptotically constant coefficients asserts that the rightmost (i.e. largest real part) envelope of the essential spectrum with respect to any $L^p$ is the envelope of the union of the rightmost envelopes of the spectra of the limiting, constant-coefficient operators at $\pm \infty$: in this case,

$$L_{\pm }≔{\partial }_x^2-A_{\pm }{\partial }_x\text{,}{A}_{\pm }≔df\left(u_{\pm }\right)\text{.}$$

The $L^2$ spectra of $L_{\pm }$ may be computed by Fourier transform to be the curves traced out by dispersion relations

$${\lambda }_j\left(\xi \right)≔i{a}_j^{\pm }\xi -{\xi }^2\text{,}\xi \in {\mathbb{R}}^1\text{,}$$

where $a_j^{\pm }$ are the eigenvalues of $A_{\pm }$. These eigenvalues are real and nonzero under the physical assumptions that (1.1) be hyperbolic in a neighborhood of $u_{\pm }$ and the shock be noncharacteristic. Likewise [86], $L$ has always an $L^2$ zero-eigenvalue with eigenfunction ${\bar{u}}^{\prime }\left(x\right)$, associated with translation of the wave.

In the absence of a spectral gap, standard stability and bifurcation theorems do not apply, and so we must carry out a refined analysis.

The spectral configuration just described translates in the $x\text{–}t$ plane, in the stable case that there exist no other eigenvalues $\lambda$ of $L$ in the nonnegative complex half-plane $\Re \lambda \ge 0$, to the following description developed in [62], [64], [97], [96], [73] of the Green function $G(x,t;y)$ associated with ${\partial }_t-L$. A point source, or delta-function initial datum, originating at $y>0$ will propagate initially as an approximate superposition of Gaussians with constant mass (total integral), centered along hyperbolic characteristics $dx/dt=a_j^{+}$ determined by the asymptotic system at $+\infty$. Those propagating in the positive direction will continue out to $+\infty$; those propagating in the negative direction will continue until they strike the shock layer at approximately $x=0$, whereupon they will be transmitted and reflected along outgoing characteristic directions $dx/dt=a_k^{-}$, $a_k^{-}<0$ and $dx/dt=a_k^{+}$, $a_k^{+}>0$. In addition, there will be deposited at the shock layer a certain amount of mass in the stationary eigenmode ${\bar{u}}^{\prime }\left(x\right)$ (integral $\int \bar{u}\left(x\right)dx=u_{+}-u_{-}\ne 0$), corresponding to translation of the background wave.

We refer the reader to [97], [73] for further discussion and details. For the present purpose, it suffices to note that the Green function may be modeled qualitatively as the sum of terms

$$K(x,t;y)≔t^{-1/2}{e}^{-{(x-y-at)}^2/4t}$$

and

$$J(x,t;y)≔{\bar{u}}^{\prime }\left(x\right)\mathrm{errfn}((-y-at)/2t^{1/2})$$

propagating with noncharacteristic speed $a<0$, where

$$\mathrm{errfn}\left(z\right)≔\frac{1}{2\pi }{\int }_{-\infty }^z{e}^{-{\xi }^2}d\xi \text{,}$$

the first modeling moving heat kernels (Gaussian signals), and the second the excitation of the zero-eigenfunction by incoming signals from $y\ge 0$.

From this description, we see explicitly that decaying modes decay time-algebraically and not exponentially, in agreement with the lack of spectral gap. Moreover, the low-frequency/large-time–space behavior is approximately hyperbolic, propagating along characteristics associated with the endstates $u_{\pm }$.

Further information may be obtained at the spectral level by the comparison of Evans and Lopatinski determinants $D$ and $\Delta$ for the viscous and inviscid problems (1.2), (1.1). The Evans function $D\left(\lambda \right)$, defined as a Wronskian of functions spanning the decaying manifolds of solutions of the eigenvalue equation

$$(L-\lambda )u=0$$

associated with $L$ at $x\to +\infty$ and $x\to -\infty$ is an analytic function with domain containing $\{\Re \lambda \ge 0\}$, whose zeros away from the essential spectrum correspond in location and multiplicity with eigenvalues of $L$ with respect to any $L^p$. Its behavior is also closely linked with that of the resolvent kernel of $L$,i.e., the Laplace transform with respect to time of the Green function $G$; see [2], [27], [97], [98], [93], [94] for history and further details. The corresponding object for the inviscid linearized problem is the Lopatinski determinant $\Delta \left(\lambda \right)$ defined in [58], [68], [69], [77], a homogeneous function of degree one: in the one-dimensional case, just linear.

An important relation between these two objects, established in [27], [98], [78], is the low-frequency expansion

$$D\left(\lambda \right)=\gamma \Delta \left(\lambda \right)+o\left(\left|\lambda \right|\right)$$

quantifying the above observation that low-frequency behavior is essentially hyperbolic, where $\gamma$ is a constant measuring transversality of the profile $\bar{u}$ as a connecting orbit of equilibria $u_{\pm }$ in the associated traveling-wave ODE, nonvanishing for transversal connections.

A corresponding expansion

$$D_{rNS}\left(\lambda \right)=\gamma {\Delta }_{\mathrm{CJ}}\left(\lambda \right)+o\left(\left|\lambda \right|\right)$$

holds for the Evans function $D_{rNS}$ associated with detonation wave solutions of (1.6), where ${\Delta }_{\mathrm{CJ}}$ denotes the Lopatinski condition, not for (1.5), but for the still simpler inviscid–instantaneous-reaction-rate Chapman–Jouguet model in which detonations are modeled by piecewise constant solutions (1.3) across which combustion proceeds instantaneously; see [93], [65], [67], [50], [14] for further details. That is, the low-frequency behavior is not only hyperbolic but “instantaneous”, with small-scale details of the profile structure lost. Both (1.13), (1.14) extend to the corresponding multidimensional problem of a planar shock moving in ${\mathbb{R}}^d$ (different from the finite-cross-sectional case considered here).

As calculated in [68], [65], respectively, neither $\Delta$ nor ${\Delta }_{\mathrm{CJ}}$ vanishes for $\lambda \ne 0$ for an ideal-gas equation of state.¹ Likewise, $\gamma$ does not vanish for any choice of parameters for ideal gas dynamics by a well-known result of Gilbarg [29], or, by results of [28] for reactive gas dynamics with an ideal gas equation of state in the ZND-limit $\left|B\right|\to 0$. From these observations, combined with (1.13), (1.14), we may deduce that $D$ and $D_{rNS}$ have for all choices of physical parameters precisely one zero at $\lambda =0$, corresponding to translation-invariance of the background equations.

In particular, as physical parameters are varied, starting from a stable viscous traveling-wave, transition to instability, signalled by passage from the stable complex half-plane $\{\Re \lambda <0\}$ to the unstable complex half-plane $\{\Re \lambda >0\}$ of one or more eigenvalues of the linearized operator about the wave, cannot occur through passage of a real eigenvalue through the origin, but rather must occur through the passage of one or more nonzero complex conjugate pairs through the imaginary axis: that is, a Poincaré–Hopf-type configuration. This observation, made in [65], [67], gives a rigorous corroboration at the spectral level of the numerical observations of Kasimov and Stewart [52]. What remains is to convert this spectral information into a rigorous nonlinear existence result.

From physical/numerical observations, one expects for an ideal gas equation of state that such a transition seldom or never occurs for shocks but frequently occurs for detonations. However, from a mathematical point of view, the situation for shocks and detonations appears to be entirely parallel. We may thus phrase a common mathematical problem:

(P)Let ${\bar{u}}^{\epsilon }$ denote a family of traveling-wave (either shock or detonation) solutions indexed by bifurcation parameter $\epsilon \in {\mathbb{R}}^1$, with associated linearized operators $L\left(\epsilon \right)$ transitioning at $\epsilon =0$ from stability to instability as follows.

Assuming the “generic” spectral situation that

• the $L^2$ essential spectrum of each $L\left(\epsilon \right)$ is contained in $\{\Re \lambda <0\}\cup \left\{0\right\}$,

• the translational zero-eigenvalue of each $L\left(\epsilon \right)$ is simple in the sense that the associated Evans function $D_{\epsilon }$ vanishes at $\lambda =0$ with multiplicity one, and

• a single complex conjugate pair of $L^2$ eigenvalues of $L\left(\epsilon \right)$ (zeros of $D_{\epsilon }$)

  $${\lambda }_{\pm }\left(\epsilon \right)=\gamma \left(\epsilon \right)\pm i\tau \left(\epsilon \right)\text{,}\gamma \left(0\right)=0\text{,}\tau \left(0\right)\ne 0\text{,}(d\gamma /d\epsilon )\left(0\right)>0$$

  crosses the imaginary axis with positive speed, and the rest of the $L^2$ point spectrum of $L\left(\epsilon \right)$ is located in $\{\Re \lambda <0\}\cup \left\{0\right\}$,

show that there occurs a Poincaré–Hopf bifurcation from the family of traveling wave solutions ${\bar{u}}^{\epsilon }$ to nearby time-oscillatory solutions.

We examine this problem in a series of increasingly realistic contexts.

Let us first recall the situation considered in [88], of a one-parameter family of standing-wave solutions ${\bar{u}}^{\epsilon }\left(x\right)$ of a smoothly-varying family of equations

$$u_t=\mathcal{F}(\epsilon ,u)≔u_{xx}-F(\epsilon ,u,u_x)$$

(possibly shifts $F(\epsilon ,u,u_x)≔f(u,u_x)-s\left(\epsilon \right)u_x$ of a single equation written in coordinates $x\to x-s\left(\epsilon \right)t$ moving with traveling-wave solutions of varying speeds $s\left(\epsilon \right)$), with linearized operators $L\left(\epsilon \right)≔\partial \mathcal{F}/\partial u{|}_{u={\bar{u}}^{\epsilon }}$ for which a spectral gap may be recovered in an appropriate exponentially-weighted norm

$${\Vert f\Vert }_{H_{\eta }^2}^2≔\sum _{j=0}^2{\Vert {(d/dx)}^{j}f\left(x\right)\Vert }_{L_{\eta }^2}^2\text{,}{\Vert f\Vert }_{L_{\eta }^2}≔{\Vert e^{\eta {(1+{\left|x\right|}^2)}^{1/2}}f\left(x\right)\Vert }_{L^2}\text{,}\eta >0\text{.}$$

We call this the weighted norm condition.

This approach, introduced by Sattinger [86], applies to the case (see Section 1.2.1) that signals in the far field are convected under the linearized evolution equation $u_t=L\left(\epsilon \right)u$ inward toward the background profile, hence time-exponentially decaying in the weighted norm ${\Vert \cdot \Vert }_{H_{\eta }^2}$ penalizing distance from the origin. The method encompasses both (1.2), (1.6) in the scalar case $u$, $z\in {\mathbb{R}}^1$, with artificial viscosity $B=D=I$. It has also interesting applications to certain reaction–diffusion systems and the related Poincaré–Hopf phenomenon of “breathers” [79], [44], [43]. However, it does not apply to either shock or detonation waves in the system case $u\in {\mathbb{R}}^n$, $n\ge 2$.

In this case, we have the following rather complete result, including stability along with bifurcation description.

Proposition 1.1 [88]

Let ${\bar{u}}^{\epsilon }$ ,(1.15)be a family of traveling-waves and systems satisfying the weighted norm condition and assumptions (P) , with $F\in C^4$ . Then, for $a\ge 0$ sufficiently small and $C>0$ sufficiently large, there are $C^1$ functions $\epsilon \left(a\right)$ , $\epsilon \left(0\right)=0$ , and $T^{\ast }\left(a\right)$ , $T^{\ast }\left(0\right)=2\pi /\tau \left(0\right)$ , and a $C^1$ family of solutions

$$u^a(x,t)=u^a(x-{\sigma }^{a}t,t)$$

of(1.15)with $\epsilon =\epsilon \left(a\right)$ , where $u^a(\cdot ,t)$ is time-periodic with period $T^{\ast }\left(a\right)$ and ${\sigma }^a$ is a constant drift, such that

$$C^{-1}a\le {\Vert u^a(\cdot ,t)-{\bar{u}}^{\epsilon \left(a\right)}\Vert }_{H_{\eta }^2}\le Ca$$

for all $t\ge 0$ . Up to fixed translations in $x$ , $t$ , for $\epsilon$ sufficiently small, these are the only nearby solutions of this form, as measured in $H_{\eta }^2$ . Moreover, solutions $u^a$ are time-exponentially phase-asymptotically orbitally stable with respect to $H_{\eta }^2$ if $d\epsilon /da>0$ , in the sense that perturbed solutions converge time-exponentially to a specific shift in $x$ and $t$ of the original solution, and unstable if $d\epsilon /da<0$ .²

Proposition 1.1 was established in [88] by center-manifold reduction, with the main issue being to accommodate the underlying group invariance of translation. The basic idea is to coordinatize $u$ as $(v,\alpha )$, where $\alpha$ parameterizes the group invariance, then work on the quotient space $v$, reducing the problem from relative to a standard Poincaré–Hopf bifurcation, afterward recovering the location $\alpha$ (a function of $a$ and $t$) driven by the solution on the quotient space, by quadrature. The integral of the periodic driving term yields a periodic part that may be subsumed in the profile and a drift ${\sigma }^{a}t$. See [88] for further discussion and details.

Remark 1.2

A consequence of (1.17) (by Sobolev embedding) is

$$|u^a(x,t)-{\bar{u}}^{\epsilon }|\le C{e}^{-\eta \left|x\right|}\text{.}$$

That is, the existence result in weighted norm space includes quite strong information on the structure of the wave. On the other hand, the stability result is somewhat weakened by the appearance of a spatial weight, being restricted to exponentially decaying perturbations.

The purpose of the present paper is to extend the analysis initiated in [88] for scalar models to the more physically realistic system case. For simplicity, we restrict to the somewhat simpler case of viscous shock solutions of systems with artificial viscosity $B\equiv I$; however, the method of analysis in principle applies also in the general case; see discussion, Section 1.7.

Specifically, consider a one-parameter family of standing viscous shock solutions

$$u(x,t)={\bar{u}}^{\epsilon }\left(x\right)\text{,}\underset{z\to \pm \infty }{lim}{\bar{u}}^{\epsilon }\left(z\right)=\underset{\pm }{\overset{\epsilon }{u}}\text{(constant for fixed }\epsilon \text{) ,}$$

of a smoothly-varying family of conservation laws

$$u_t=\mathcal{F}(\epsilon ,u)≔u_{xx}-F{(\epsilon ,u)}_x\text{,}u\in {\mathbb{R}}^n\text{,}$$

with associated linearized operators

$$L\left(\epsilon \right)≔\frac{\partial \mathcal{F}}{\partial u}{|}_{u={\bar{u}}^{\epsilon }}=-{\partial }_x{A}^{\epsilon }\left(x\right)+{\partial }_x^2\text{,}$$

denoting

$$A^{\epsilon }\left(x\right)≔F_u(\epsilon ,{\bar{u}}^{\epsilon }\left(x\right))\text{,}{A}_{\pm }^{\epsilon }≔\underset{z\to \pm \infty }{lim}{A}^{\epsilon }\left(z\right)=F_u(\epsilon ,\underset{\pm }{\overset{\epsilon }{u}})\text{.}$$

We take ${\bar{u}}^{\epsilon }$ to be of standard Lax type, meaning that the hyperbolic convection matrices $A_{+}^{\epsilon }$ and $A_{-}^{\epsilon }$ at plus and minus spatial infinity have, respectively, $n-p$ positive and $p-1$ negative real eigenvalues for $1\le p\le n$, where $p$ is the characteristic family associated with the shock: in other words, there are precisely $n-1$ outgoing hyperbolic characteristics in the far field.

This is the only type occurring for gas dynamics with a standard (e.g., ideal gas) equation of state; for reacting gas dynamics, the corresponding object is a strong detonation, which is the only (nondegenerate) type occurring in the ZND limit $B,D\to 0$ [28]. The special features of Lax-type shocks (resp. strong detonations), as compared to more general undercompressive shocks (resp. weak detonations) that can occur in other settings, turn out to be important for the analysis (see Remark 3.10), in sharp contrast with the generality of [88]. Note, for the system case $n\ge 2$, that there is at least one outgoing characteristic, so that the weighted norm methods of the previous section do not apply. In particular, we see no way to construct a center manifold for this problem, and suspect that one may not exist; the slow (time-algebraic) decay rate of outgoing modes evident in our description of the Green function in Section 1.2.1 suggests an essential obstacle to such construction. On the other hand, the conservative form of system (1.20) implies that relative mass $\int u\left(x\right)dx$ is conserved for all time for perturbations $u=\tilde{u}-{\bar{u}}^{\epsilon }$, $\tilde{u}$ satisfying (1.20). Thus, at least at a formal level, there is a convenient invariant subspace of (1.20) consisting of perturbations with zero excess mass, on which the zero eigenvalue associated with translational invariance is removed. For, recall that $\int {\bar{u}}^{\prime }\left(x\right)dx=u_{+}-u_{-}\ne 0$, so that the associated zero-eigenfunction is not in the subspace. On the other hand, nonzero eigenfunctions always have zero mass [97], since $\lambda \varphi =L\left(\epsilon \right)\varphi$ implies $\lambda \int \varphi \left(x\right)dx=0$ by divergence form of $L\left(\epsilon \right)$, hence the crossing nonzero imaginary eigenvalues ${\lambda }_{\pm }\left(\epsilon \right)$ in (P) persist. Thus, the spectral scenario (P) translates in the zero-mass subspace to a standard Poincaré–Hopf scenario, with no additional zero-eigenvalue, for which the translational group-invariance need not be taken into account. Recall, that this was the main issue in the analysis of the scalar case.

In short, the two problems (Model analysis I vs. Model analysis II) have essentially complementary mathematical difficulties, hence little technical contact. Accordingly, the analysis has a quite different flavor in the system case, depending on both the full, pointwise Green function bounds of [73] and the special, conservative structure of the equations.

Our result in this case, and the main result of the paper, is as follows.

Theorem 1.3

Let ${\bar{u}}^{\epsilon }$ ,(1.20)be a family of traveling-waves and systems satisfying assumptions (P) , with $F\in C^2$ . Assume further that ${\bar{u}}^{\epsilon }$ is of Lax type, and that the eigenvalues of $A_{\pm }^{\epsilon }$ , as defined in(1.22), are real, nonzero and simple. Then, for $a\ge 0$ sufficiently small and $C>0$ sufficiently large, there are $C^1$ functions $\epsilon \left(a\right)$ , $\epsilon \left(0\right)=0$ , and $T^{\ast }\left(a\right)$ , $T^{\ast }\left(0\right)=2\pi /\tau \left(0\right)$ , and a $C^1$ family of solutions $u^a(x,t)$ of(1.20)with $\epsilon =\epsilon \left(a\right)$ , time-periodic of period $T^{\ast }\left(a\right)$ , such that

Up to fixed translations in $x$ , $t$ , for $\epsilon$ sufficiently small, these are the only nearby solutions as measured in norm ${\Vert f\Vert }_{X_1}≔{\Vert (1+\left|x\right|)f\left(x\right)\Vert }_{L^{\infty }\left(x\right)}$that are time-periodic with period $T\in [T_0,T_1]$ , for any fixed $0<T_0<T_1<+\infty$ . Indeed, they are the only nearby solutions of the more general form(1.16).

Note that the statement of Theorem 1.3 is considerably weaker than that of Proposition 1.1, asserting no stability information, and only the relatively weak structural information of algebraic decay (1.23) at $\pm \infty$, as compared to the exponential bound (1.18). As suggested by our formal “zero-mass” discussion, the periodic solutions constructed have zero mean drift $\sigma$, in contrast to (1.16). However, somewhat surprisingly, they do not appear to have zero excess mass. See Remark 1.8, Remark 2.10 for further discussion and explanation.

One-dimensional traveling-wave solutions (1.4) of (1.2) with $B\equiv I$ may alternatively be viewed as the restriction to one dimension of a planar viscous shock solution

$$u(x,t)=\bar{u}(x_1-st)$$

of a multidimensional system of viscous conservation laws

$$u_t+\sum f^j{\left(u\right)}_{x_j}={\Delta }_{x}u\text{,}u\in {\mathbb{R}}^n,x\in {\mathbb{R}}^d,t\in {\mathbb{R}}^{+}$$

on the whole space. Likewise, traveling-wave solutions (1.24) may be viewed as planar traveling-wave solutions of (1.25) on an infinite cylinder

$$\mathcal{C}≔\{x:(x_1,\tilde{x})\in {\mathbb{R}}^1\times \Omega \}\text{,}\tilde{x}=(x_2,\dots ,x_d)$$

with bounded, cross-section $\Omega \in {\mathbb{R}}^{d-1}$, under artificial Neumann boundary conditions

$$\partial u/{\partial }_{\tilde{x}}\cdot {\nu }_{\Omega }=0\text{ for}\tilde{x}\in \partial \Omega \text{,}$$

or, in the case that $\Omega$ is reflection symmetric, periodic boundary conditions

$$u(x_1,\tilde{x})=u(x_1,-\tilde{x})\text{ for}\tilde{x}\in \partial \Omega \text{.}$$

We take this as a simplified mathematical model for flow in a duct, in which we have neglected boundary-layer phenomena along the wall $\partial \Omega$ in order to isolate the oscillatory phenomena of our main interest.

Consider a one-parameter family of standing planar viscous shock solutions ${\bar{u}}^{\epsilon }\left(x_1\right)$ of a smoothly-varying family of conservation laws

$$u_t=\mathcal{F}(\epsilon ,u)≔{\Delta }_{x}u-\sum _{j=1}^d{F}^j{(\epsilon ,u)}_{x_j}\text{,}u\in {\mathbb{R}}^n$$

in a fixed cylinder $\mathcal{C}$, with periodic boundary conditions (typically, shifts $\sum F^j{(\epsilon ,u)}_{x_j}≔\sum f^j{\left(u\right)}_{x_j}-s\left(\epsilon \right)u_{x_1}$ of a single Eq. (1.25) written in coordinates $x_1\to x_1-s\left(\epsilon \right)t$ moving with traveling-wave solutions of varying speeds $s\left(\epsilon \right)$), with linearized operators $L\left(\epsilon \right)≔\partial \mathcal{F}/\partial u{|}_{u={\bar{u}}^{\epsilon }}$. Let

$$A_{\pm }^1\left(\epsilon \right)≔\underset{z\to \pm \infty }{lim}\underset{u}{\overset{1}{F}}(\epsilon ,{\bar{u}}^{\epsilon })\text{.}$$

We take ${\bar{u}}^{\epsilon }$, considered as a shock wave in one dimension, to be of standard Lax type (an assumption on $A_{\pm }^1\left(\epsilon \right)$, as in the previous subsection). For simplicity, we take $\Omega ={\mathbb{T}}^{d-1}$, i.e., $\Omega ={[0,1]}^{d-1}$ with periodic boundary conditions (1.26). Then, we have the following result generalizing Theorem 1.3.

Theorem 1.4

Let ${\bar{u}}^{\epsilon }$ ,(1.27)be a family of traveling-waves and systems satisfying assumptions (P) , with $F\in C^2$ , $\Omega ={\mathbb{T}}^{d-1}$ . Assume further that ${\bar{u}}^{\epsilon }$ is of Lax type, and that the eigenvalues of $A_{\pm }^1\left(\epsilon \right)$ , as defined in(1.28), are real, nonzero and simple. Then, for $a\ge 0$ sufficiently small and $C>0$ sufficiently large, there are $C^1$ functions $\epsilon \left(a\right)$ , $\epsilon \left(0\right)=0$ , and $T^{\ast }\left(a\right)$ , $T^{\ast }\left(0\right)=2\pi /\tau \left(0\right)$ , and a $C^1$ family of solutions $u^a(x_1,t)$ of(1.27)with $\epsilon =\epsilon \left(a\right)$ , time-periodic of period $T^{\ast }\left(a\right)$ , such that

$$C^{-1}a\le \underset{x_1\in \mathbb{R}}{sup}(1+\left|x_1\right|)|u^a(x_1,t)-{\bar{u}}^{\epsilon \left(a\right)}\left(x_1\right)|\le Ca\text{,}\text{<mi mathvariant="italic" is="true">for all</mi>}t\ge 0\text{.}$$

Up to fixed translations in $x$ , $t$ , for $\epsilon$ sufficiently small, these are the only nearby solutions as measured in norm ${\Vert f\Vert }_{X_1}≔{\Vert (1+\left|x_1\right|)f\left(x\right)\Vert }_{L^{\infty }\left(x\right)}$that are time-periodic with period $T\in [T_0,T_1]$ , for any fixed $0<T_0<T_1<+\infty$ . Indeed, they are the only nearby solutions of the more general form(1.16).

Remark 1.5

Extension to a circular cross-section $\Omega ≔\{\left|\tilde{x}\right|\le R\}$ follows by Bessel function expansion as in [52]. Extensions to general cross-sectional geometries and or Neumann boundary conditions are interesting directions for further investigation.

We now briefly discuss the ideas behind the proofs of Theorem 1.3, Theorem 1.4. Under the spectral assumptions (P) (stated in Section 1.2.3), there exist smooth (in $\epsilon$) $L\left(\epsilon \right)$-invariant projections onto the two-dimensional eigenspace ${\Sigma }^{\epsilon }$ of $L\left(\epsilon \right)$ associated with the pair of crossing eigenvalues ${\lambda }_{\pm }\left(\epsilon \right)=\gamma \left(\epsilon \right)\pm i\tau \left(\epsilon \right)$ and its complement ${\tilde{\Sigma }}^{\epsilon }$. Projecting onto these subspaces, and rewriting in polar coordinates the flow on ${\Sigma }^{\epsilon }$, we may thus express (1.20) in standard fashion as

$$\dot{r}=\gamma \left(\epsilon \right)r+N_r(\epsilon ,r,\theta ,v)\text{,}$$$$\dot{\theta }=\tau \left(\epsilon \right)+N_{\theta }(\epsilon ,r,\theta ,v)\text{,}$$$$\dot{v}=\tilde{L}\left(\epsilon \right)v+N_v(\epsilon ,r,\theta ,v)\text{,}$$

where the “transverse linearized operator” $\tilde{L}\left(\epsilon \right)$ is the restriction of $L\left(\epsilon \right)$ to ${\tilde{\Sigma }}^{\epsilon }$, and $N_j$ are higher-order terms coming from the nonlinear part of (1.20): $N_r$ and $N_v$ quadratic order in $r,v$ and their derivatives, and $N_{\theta }$ linear order; in particular $N_j(\epsilon ,0,\theta ,0)\equiv 0$ for $j=r,\theta ,v$. We are precisely interested in the case that $\tilde{L}$ has no spectral gap, i.e., $\Re \sigma \left(L\right)⁄\le -\theta <0$ for any $\theta >0$. One may think for example of the finite-dimensional case that $\tilde{L}\left(0\right)$ has additional pure-imaginary spectra besides ${\lambda }_{\pm }\left(0\right)=\pm i\tau \left(0\right)$. Thus, the center manifold may be of higher dimension, and one cannot follow the usual course of reduction to a center manifold involving only $r$, $\theta$, $\epsilon$ and applying the standard, two-dimensional Poincaré–Hopf Theorem. Instead, we proceed by a direct analysis, combining the two-dimensional Poincaré return map construction with Lyapunov–Schmidt reduction.

Specifically, truncating $\left|v\right|\le C_{0}r$, $C_0\gg 1$, in the arguments of $N_r$, $N_{\theta }$, we obtain

$$\left|N_{\theta }\right|\le Cr\text{,}$$

and therefore $\dot{\theta }\ge \tau \left(0\right)/2>0$ for $\epsilon$, $r$ sufficiently small. Thus, in seeking periodic solutions

$$(r,\theta ,v)\left(T\right)=(r,\theta ,v)\left(0\right)\text{,}$$

we may eliminate $\theta$, solving for $T(\epsilon ,a,b)$ as a function of initial data

$$(a,b)≔(r,v)\left(0\right)$$

and the bifurcation parameter $\epsilon$, with $T(0,0,0)=2\pi /\tau \left(0\right)$, and seek solutions (1.30) as fixed points

$$(a,b)=(r,v)\left(T(\epsilon ,a,b)\right)$$

of the Poincaré return map $(r,v)\left(T(\epsilon ,\cdot ,\cdot )\right)$.

Using Duhamel’s formula/variation of constants, we may express (1.31) in a standard way (see, e.g., [36]) as

$$0=f(\epsilon ,a,b)=(e^{\gamma \left(\epsilon \right)T(\epsilon ,a,b)}-1)a+N_1\text{,}$$$$0=g(\epsilon ,a,b)=(e^{\tilde{L}\left(\epsilon \right)T(\epsilon ,a,b)}-\mathrm{Id})b+N_2\text{,}$$

where

$$N_1≔{\int }_0^{T(\epsilon ,a,b)}{e}^{\gamma \left(\epsilon \right)(T(\epsilon ,a,b)-s)}{N}_r(\epsilon ,r,\theta ,v)\left(s\right)ds\text{,}$$$$N_2≔{\int }_0^{T(\epsilon ,a,b)}{e}^{\tilde{L}\left(\epsilon \right)(T(\epsilon ,a,b)-s)}{N}_v(\epsilon ,r,\theta ,v)\left(s\right)ds$$

are, formally, quadratic order terms in $a$, $b$. Note that $(r,\theta ,v)$ are functions of $(\epsilon ,a,b)$ as well as $s$, through the flow of (1.29), with

$$\Vert (r,v)\Vert \le C\Vert (a,b)\Vert$$

for any “reasonable” norm in the sense that (1.29) are locally well-posed (in practice, no restriction). By quadratic dependence of $N_j$, we have, evidently, that $(\epsilon ,a,b)=(\epsilon ,0,0)$ is a solution of (1.32) for all $\epsilon$.

Continuing in this standard fashion, we should next like to perform a Lyapunov–Schmidt reduction, using the Implicit Function Theorem to solve the $g$ equation for $b$ in terms of $(\epsilon ,a)$, i.e., to find a function

$$b=B(\epsilon ,a)\text{,}B(0,0)=0\text{,}$$

satisfying $g(\epsilon ,a,B(\epsilon ,a))\equiv 0$: equivalently, to reduce to the nullcline of $g$.

If this were possible, from $\left|N_2\right|\le C{\left|(a,b)\right|}^2$ in (1.32)(ii), we would find, further, that $\left|B(\epsilon ,a)\right|\le C{\left|a\right|}^2$, from which straightforward Duhamel/Gronwall estimates on (1.29)(i),(ii) would yield

$$\left|v\left(s\right)\right|\le C_1{r}^2$$

for $s\in [0,T]$, justifying a posteriori the truncation $\left|v\right|\le C_{0}r$, of $N_{\theta }$ performed in the first step, provided that $(\epsilon ,a,b)$ are taken sufficiently small: specifically, small enough that $r$ remains less than or equal to $C_0/C_1$ for $s\in [0,T]$.

Substituting into the $f$ equation, we would then obtain a reduced, scalar bifurcation problem

$$0=f^{\ast }(\epsilon ,a)≔f(\epsilon ,a,B(\epsilon ,a))\text{,}$$

$f^{\ast }(\epsilon ,0)=0$ for all $\epsilon$, that would be solvable in the usual way (i.e., by dividing out $a$ and applying the Implicit Function Theorem a second time, using

$${\partial }_{\epsilon }\left(a^{-1}{f}^{\ast }\right)(0,0)={\partial }_{\epsilon }(\gamma /\tau )\left(0\right)={\partial }_{\epsilon }\gamma \left(0\right)/\tau \left(0\right)\ne 0$$

to solve for $\epsilon$ as a function of $a$; see, e.g., [36], or Section 2.3.1). The Poincaré return map system (1.32) would then be solved, and a solution of (1.32) would generate a periodic solution of (1.29).

Let us ask ourselves now under what circumstances this basic reduction procedure may actually be carried out. Formal differentiation of (1.32) yields

$${\partial }_{b}g(0,0,0)=e^{2\pi (\tilde{L}/\tau )\left(0\right)}-\mathrm{Id}\text{.}$$

In the finite-dimensional ODE case, the usual condition of application of the (standard) Implicit Function Theorem is thus that $(e^{2\pi (\tilde{L}/\tau )\left(0\right)}-\mathrm{Id})$ be invertible, i.e., that $\tilde{L}\left(0\right)$ have no “resonant” oscillatory modes, i.e., pure imaginary eigenvalues that are integer multiples of the crossing modes ${\lambda }_{\pm }\left(0\right)=\pm i\tau \left(0\right)$. Note that this includes interesting cases for which $\tilde{L}$ has no spectral gap, and thus that do not yield to the standard center-manifold reduction. The analogous criterion in the infinite-dimensional (e.g., PDE) setting is that all possible resonant modes $ni\tau \left(0\right)$, $n\in \mathbb{Z}$, lie in the resolvent set of $\tilde{L}$. This is encouraging, and shows that the requirements of Lyapunov–Schmidt reduction are much less than those for center-manifold reduction: in particular, only spectral separation (from both $\pm i\tau \left(0\right)$ and their integer multiples $ni\tau \left(0\right)$) rather than a spectral gap is needed. Unfortunately, in the setting (1.20), (1.27) of our interest we do not even have spectral separation, since the essential spectra of $\tilde{L}\left(\epsilon \right)$ accumulates at $\lambda =0$ for every $\epsilon$. In this case, or others for which separation fails, we must follow a different approach.

Alternatively, we may rewrite (1.32)(ii) formally as a fixed-point equation

$$b={(\mathrm{Id}-e^{\tilde{L}\left(\epsilon \right)T(\epsilon ,a,b)})}^{-1}{N}_2(\epsilon ,a,b)\text{,}$$

then apply the Contraction-mapping Principle to carry out an Implicit Function construction “by hand” (reminiscent of, but not exactly the standard proof of the Implicit Function Theorem). In the standard case that $\left|e^{\tilde{L}t}\right|$ is exponentially decaying ($\sim$spectral gap), we may rewrite (1.34) using Neumann expansion as

$$b=\sum _{j=0}^{\infty }{e}^{j\tilde{L}\left(\epsilon \right)T(\epsilon ,a,b)}{N}_2(\epsilon ,a,b)\text{.}$$

The basis for our analysis is the simple observation that, more generally, provided the series on the righthand side converges (conditionally) for $\Vert (a,b)\Vert$ bounded, then (i) (by scaling argument) the righthand side is contractive in $b$ for $\Vert (\epsilon ,a,b)\Vert$ sufficiently small, and (ii) (by standard telescoping sum argument, applying $(\mathrm{Id}-e^{\tilde{L}\left(\epsilon \right)T(\epsilon ,a,b)})$ to (1.34)) the resulting solution

$$b=B(\epsilon ,a)$$

guaranteed by the Contraction-mapping Principle is in fact a solution of the original (1.32)(ii). See Section 2 for further details.

In the context of (1.20), by divergence form of the equation, $N_2={\left(n_2\right)}_x$, with $n_2=O\left({\left|(r,v)\right|}^2\right)$. For simplicity of discussion, model $n_2$ as a bilinear form in $(r,v)$, so that (by quadratic scaling) $n_2\in L^1$ for $v\in L^2$, from which we obtain $N_2=n_x$, $n\in L^1$ for $b\in L^2$. Thus, replacing the righthand side of (1.35) by its continuous approximant

$$N_2+T^{-1}{\int }_T^{\infty }{e}^{\tilde{L}\left(\epsilon \right)t}{N}_{2}dt=N_2+T^{-1}{\int }_T^{\infty }\left(e^{\tilde{L}\left(\epsilon \right)t}{\partial }_x\right)ndt\text{,}n\in L^1\text{,}$$

we see that convergence (as a map from $L^2$ to $L^2$) reduces, roughly, to a space-time stability estimate

$${\Vert {\int }_{-\infty }^{+\infty }\left({\int }_T^{\infty }{\tilde{G}}_y(x,t;y)dt\right)n\left(y\right)dy\Vert }_{L^2\left(x\right)}\le C{\Vert ,n\Vert }_{L^1}\text{,}$$

where $\tilde{G}$ is the Green kernel associated with transverse solution operator $e^{\tilde{L}t}$. Estimate (1.36) is quite similar to those arising for the full solution operator (again, through Duhamel’s principle; see, for example, [96], [74]) in the study of nonlinear stability of spectrally stable waves. That (1.36) holds for all $n\in L^1$, is equivalent to

$$\underset{y}{sup}{\Vert {\int }_T^{\infty }{\tilde{G}}_y(\cdot ,t;y)dt\Vert }_{L^2}\le C\text{.}$$

(One implication is a consequence of the triangle inequality; for the other take a sequence converging to a Dirac function in $L^1$.) This would hold, for example, if

$$\underset{y}{sup}{\int }_T^{\infty }{\Vert {\tilde{G}}_y(\cdot ,t;y)dt\Vert }_{L^2}\le C\text{.}$$

Note that the neglected first term $N_2$, corresponding to $j=0$ in (1.35), is $C^2$ from $L^2$ to $L^2$ by the smoothing action of (1.33)(ii); see Section 4.

In the arguments above, we have used strongly the specific structure of the nonlinearity, both quadratic dependence and divergence form, since the $L^2$-operator norm ${\left|e^{\tilde{L}t}\right|}_{L^2\to L^2}$ does not decay in $t$. However, this is still not enough. For, recall the one-dimensional discussion of Section 1.2.1, modeling $\tilde{G}$ qualitatively as the sum of a convected heat kernel $K$ as in (1.10) and an error-function term $J$ as in (1.11). From the standard bounds ${\Vert K_y(\cdot ,t;y)\Vert }_{L^2}=C{t}^{-3/4}$, ${\Vert J_y(\cdot ,t;y)\Vert }_{L^2}\le C{t}^{-1/2}$, hence (the support of $K$ and $J$ being essentially disjoint) ${\int }_T^{\infty }{\tilde{G}}_y(x,t;y)dt$ is not absolutely convergent in $L^2\left(x\right)$, and the bound (1.38) does not hold.

Instead, we must show conditional convergence, using detailed knowledge of the propagator $\tilde{G}$ to identify cancellation in ${\int }_T^{\infty }{\tilde{G}}_y(x,t;y)dt$. For example, using $K_y=a^{-1}(K_t-K_{yy})$, we find that

$${\int }_T^t{K}_y(x,s;y)ds=a^{-1}({\int }_T^t{K}_t(x,s;y)ds-{\int }_T^t{K}_{yy}(x,s;y)ds)=a^{-1}(K(x,s;y){|}_T^t-{\int }_T^t{K}_{yy}(x,t;y)ds)$$

is the sum of $-a^{-1}K(x,T;y)\in L^2\left(x\right)$ with terms $a^{-1}K(x,t;y)\sim t^{-1/4}$ and $a^{-1}{\int }_T^t{K}_{yy}(x,t;y)ds\sim {\int }_T^t{s}^{-5/4}ds$ that are respectively decaying and absolutely convergent in $L^2\left(x\right)$, hence converges uniformly with respect to $y$ in $L^2\left(x\right)$.

A similar cancellation argument yields convergence of the error-function term, along with uniform boundedness with respect to $y$. Convergence of the error-function term is not uniform with respect to $y$, a detail which necessitates further technicalities: in particular, the weighted estimate

$$\left|v\left(x\right)\right|\le C{(1+\left|x\right|)}^{-1}$$

leading eventually to (1.23). Likewise, there are further issues associated with $\epsilon$-regularity of the solution, needed for bifurcation analysis of the reduced equation. However, the main idea is contained in calculation (1.39).

The multi-dimensional case goes similarly, with the computation of the critical neutral, zero transverse wave number reducing to the one-dimensional case, and all others to the case of a spectral gap.

Remark 1.6

It is readily checked that the above arguments go through in the general case that $n_2$ is quadratic order for ${\left|v\right|}_{L^{\infty }}\le C$ and not bilinear in $(v,r)$, substituting $v\in L^2\cap L^{\infty }$ ($\Rightarrow n_2\in L^1$) for $v\in L^2$, working in the $L^2\cap L^{\infty }$ norm in place of $L^2$, and substituting for (1.37) the estimate

$$\underset{y}{sup}{\Vert {\int }_T^{\infty }{\tilde{G}}_y(x,t;y)dt\Vert }_{L^2\cap L^{\infty }\left(x\right)}\le C\text{.}$$

Similar cancellation estimates are important in the study of asymptotic behavior of stable viscous shock waves [63], [87], [64], [82], [40].

Remark 1.7

As in the result of Proposition 1.1 obtained by center manifold reduction, the family of periodic solutions obtained, up to translation, lies tangent to the $(r,\theta )$-plane corresponding to linearized oscillatory behavior.

Remark 1.8

As noted earlier, operator $S=e^{\tilde{L}T}$ by the divergence form of the equations preserves the property of zero mass, $\int udx=0$, hence each finite approximant ${\sum }_{j=0}^N{S}^j$ to ${(I-S)}^{-1}$ also preserves zero mass. However, $\int udx=0$ is not closed with respect to ${\Vert \cdot \Vert }_{L^2}$, as may be seen by the example

$$K(x,s;y){|}_T^t\to -K(x,T;y)$$

in $L^2$ as $t\to \infty$; that is, mass may “escape at infinity” in the limiting process. Comparing with (1.39), we see that the principal term in ${lim}_{N\to \infty }{\sum }_{j=0}^N{S}^j{\partial }_x$ is exactly of this form, hence the $L^2$-limit ${(I-S)}^{-1}$ does not preserve zero mass, even though each finite approximant does. Accordingly, there is no reason that the solution $b=b(a,\epsilon )={(I-S)}^{-1}{\mathcal{N}}_2(b,a,\epsilon )$ obtained by reduction should have zero mass: in the model case $n=u^2$, $G=K$, it can be seen that it does not. Thus, the idea of carrying out a simplified bifurcation analysis in the invariant subspace of zero-mass perturbations, though appealing at a formal level, in general cannot succeed. See also Remark 2.10.

Theorem 1.3, Theorem 1.4 together with the spectral observations of [65], [67] give rigorous validation in a simplified context of the formal and numerical observations of [9], [52]. An interesting problem for future work is to extend these results to the originally-motivating case of detonation waves of the full, reacting compressible Navier–Stokes equations.

We expect that our one-dimensional analysis will extend in a straightforward fashion, combining tools developed in [73], [74], [93], [66] to treat, respectively, nonreacting gas dynamics with physical, partial viscosity and reacting gas dynamics with artificial viscosity. Likewise, we expect that we can readily treat flow in a cylinder for the physical equations with artificial Neumann or periodic boundary conditions. However, the treatment of physical, no-slip boundary conditions (presumably associated with characteristic viscous boundary layers) involves technical and philosophical difficulties beyond the scope of the present analysis, as yet unresolved even for nonreacting, incompressible flows: for example, even the construction of a background traveling profile becomes problematic in this setting. We point out that viscous boundary effects (since also viscosity) are also neglected in the ZND setting of [9], [52].

A second natural direction for future investigation is the question of the stability of the periodic waves whose existence we have established here. In the absence of a spectral gap, our method of analysis does not directly yield stability as in the case of center manifold reduction, but at best partial information on the location of the point spectrum associated with oscillatory modes, with stability presumably corresponding to the standard condition $d\epsilon /da>0$. The hope is that we could combine such information with an analysis like that carried out for stationary waves in [97], [73], adapted from the autonomous to the time-periodic setting: that is, a generalized Floquet analysis in the PDE setting and in the absence of a spectral gap. We consider this a quite exciting direction for further development of the theory.

Bifurcation in the absence of a spectral gap has been considered by a number of authors in different settings; in particular, it has been studied systematically by Ioss et al. [45], [46], [47], [48], [49], [83], [84] in various contexts using an alternative “spatial dynamics” approach. It would be very interesting to investigate what results could be obtained by this technique in the context of viscous shock waves: more generally, to relate it at a technical level to the one used here. Considered in this larger context, the interest of the present approach is that it gives a simple and explicit connection between stability and bifurcation, in the spirit of center-manifold reduction and formal asymptotics, but adapted to the boundary case of zero spectral gap and time-algebraic decay. On the other hand, the price of this “direct” approach is that the needed estimates may in practice (as here) be rather delicate to obtain.

We mention also a recent work of Kunze and Schneider [60] in which they analyze pitchfork bifurcation in the absence of a spectral gap using Sattinger’s weighted-norm method, as described in Section 1.3, but in a situation where convection is outward, away from the profile layer. This entails the use of “wrong-way” exponentially decaying weights at the linearized level, introducing a spatially-exponentially growing multiplier in quadratic-order source terms, in combination with separate, compensating estimates at the nonlinear level. For similar arguments in the context of nonlinear stability, see, e.g., [80], [15]. This interesting approach has been used successfully in the shock-wave context to treat the scalar undercompressive case [15]; however, it does not appear to generalize to the system case.

Remarks 1.9

• Birtea et al. in [7] successfully carry out a rather complicated bifurcation analysis in the ODE setting without explicit knowledge of the background solution. Similar techniques might perhaps be useful in treating flow in a duct with physical, nonslip boundary conditions, for which description of the background flow is itself problematic.

• In carrying out our nonstandard Implicit Function construction for shock waves, we faced the problem of non-uniform convergence of series (1.35) due to lack of spatial localization. We remedied this problem by additional weighted-norm estimates. However, the example of another famous nonstandard Implicit Function construction, namely, Nash–Moser iteration, suggests the alternative approach of introducing a “localizing” step dual to the smoothing step in the Nash–Moser scheme. It would be interesting to see if this approach could also be carried out, thus avoiding the need for additional analysis associated with pointwise bounds. We note that Nash–Moser iteration combining temporal with the usual frequency cutoffs has been carried out by Klainerman [55] in the context of a nonlinear wave equation.

Plan of the paper. In Section 2, we formalize the reduction procedure set out in Section 1.6.4 as an abstract bifurcation framework suitable for application to general discrete dynamical systems in the absence of a spectral gap. In Section 3, we recall the pointwise estimates furnished by the methods of [97], [73] on the Green function $\tilde{G}$ associated with the transverse linearized solution operator $e^{\tilde{L}t}$, then, in Sections 4 Return map construction, 5 Proof of use these to verify that the shock bifurcation problem indeed fits the hypotheses of our abstract framework. This establishes a Lipschitz version of Main Theorem 1.3; we improve this to $C^1$ by a bootstrap argument in Section 5.4. In Section 6, we describe the extension to multidimensions, verifying Theorem 1.4.

Notes. Since the completion of this work, there have been several further developments. In [85], Sandstede and Scheel recover and somewhat sharpen our results using spatial dynamics techniques, answering the question posed in Section 1.7 of what results may be obtained by these methods, obtaining the additional information of exponential localization of solutions and exchange of spectral stability. In [89], [90], we extend our results to shock and detonation waves of systems with physical viscosity, at the same time greatly sharpening and simplifying the basic cancellation estimate. The latter yields also exponential localization and appears to shed light on the more general question of the technical relation between spatial dynamics methods and the “temporal dynamics” method used here; see Remark 1.10, [89].