arXiv subjects

Kevin Zumbrun

Publications and source records attributed to Kevin Zumbrun.

At least 73 records · Page 4Linked to original sources

arXiv2013

Viscous hyperstabilization of detonation waves in one space dimension

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 detonation waves and the presence of small but nonzero diffusive effects. While our calculations do show a modest delay in the onset of instability in agreement with recently reported calculations by direct numerical simulation of the physical equations, our approach, based on the Evans function, also provides additional spectral information. In particular, for each of the families of detonation waves in our computational domain, we find a completely unexpected kind of hysteresis in the limit of increasing activation energy; that is, our calculations suggest that, whenever diffusive effects are present, there is a return to stability as unstable eigenvalues return to the stable complex half plane.

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arXiv2013

Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

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arXiv2013

Nonlinear stability of source defects in the complex Ginzburg-Landau equation

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.

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arXiv2013

Stability of viscous weak detonation waves for Majda's model

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, weak detonation waves are undercompressive and the stability of these waves is delicate and has not been treated by standard weighted-norm techniques. The present analysis thus provides a case study illustrating the flexibility and power of the Evans-function- based approach to stability. As in the case of strong detonations, we find that all tested waves are spectrally stable, hence nonlinearly stable.

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arXiv2013

Stability of viscous detonations for Majda's model

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 results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J. Differential Equations 233 (2007), no. 2, 654-698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of the Majda model without a smallness assumption. Notably, our results indicate that, for the simplified Majda model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to "galloping" or "pulsating" solutions as is observed in the full reactive Navier-Stokes equations. This answers in the negative a question posed by Majda as to whether the scalar detonation model captures this aspect of detonation behavior.

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arXiv2012

Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability

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 of view, (iv) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (i) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (ii) and (iii) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (iv) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

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arXiv2012

Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations

We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated second-order formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate to initial data for this formal system, a task accomplished by detailed estimates on the linearized solution operator about the background wave. Notably, our approach gives both a common theoretical treatment and a complete classification in terms of "phase-coupling" or "-decoupling" of general systems of conservation or balance laws, encompassing cases that had previously been studied separately or not at all. At the same time, our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase-decoupled case, further distinguishing behavior in the different cases. An interesting technical aspect of our analysis is that for systems of conservation laws the Whitham modulation description is of system rather than scalar form, as a consequence of which renormalization methods such as have been used to treat the reaction-diffusion case in general do not seem to apply.

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arXiv2012

Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions

We initiate the study of noncharacteristic boundary layers in hyperbolic-parabolic problems with Neumann boundary conditions. More generally, we study boundary layers with mixed Dirichlet--Neumann boundary conditions where the number of Dirichlet conditions is fewer than the number of hyperbolic characteristic modes entering the domain, that is, the number of boundary conditions needed to specify an outer hyperbolic solution. We have shown previously that this situation prevents the usual WKB approximation involving an outer solution with pure Dirichlet conditions. It also rules out the usual maximal estimates for the linearization of the hyperbolic-parabolic problem about the boundary layer. Here we show that for linear, constant-coefficient, hyperbolic-parabolic problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed Dirichlet--Neumann rather than Dirichlet boundary conditions. When this hyperbolic problem can be solved, a unique formal boundary-layer expansion can be constructed. In the extreme case of pure Neumann conditions and totally incoming characteristics, we carry out a full analysis of the quasilinear case, obtaining a boundary-layer approximation to all orders with a rigorous error analysis. As a corollary we characterize the small viscosity limit for this problem. The analysis shows that although the associated linearized hyperbolic and hyperbolic--parabolic problems do not satisfy the usual maximal estimates for Dirichlet conditions, they do satisfy analogous versions with losses.

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arXiv2012

Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns

Extending the approach of Grillakis-Shatah-Strauss, Bronski-Johnson-Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $\min_{X:C(X)=c_0} \mathcal{E}(X)$, $c_0\in \RM^r$, and stability of solutions of a class of degenerate "quasi-gradient" systems $dX/dt=-M(X)\nabla \mathcal{E}(X)$ admitting constraints, including Cahn-Hilliard equations, one- and multi-dimensional viscoelasticity, and coupled conservation law-reaction diffusion systems arising in chemotaxis and related settings. Using the relation between variational stability and the signature of $\partial c/\partial ω\in \R^{r\times r}$, where $c(ω)=C(X^*_ω)\in \RM^r$ denote the values of the imposed constraints and $ω\in \RM^r$ the associated Lagrange multipliers at a critical point $X^*_ω$, we obtain as in the Hamiltonian case a general criterion for co-periodic stability of periodic waves, illuminating and extending a number of previous results obtained by direct Evans function techniques. More interestingly, comparing the form of the Jacobian arising in the co-periodic theory to Jacobians arising in the formal Whitham equations associated with modulation, we recover and substantially generalize a previously mysterious "modulational dichotomy" observed in special cases by Oh-Zumbrun and Howard, showing that co-periodic and sideband stability are incompatible. In particular, we both illuminate and extend to general viscosity/strain-gradient effects and multidimensional deformations the result of Oh-Zumbrun of universal modulational instability of periodic solutions of the equations of viscoelasticity with strain-gradient effects, considered as functions on the whole line. Likewise, we generalize to multi-dimensions corresponding results of Howard on periodic solutions of Cahn-Hilliard equations.

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arXiv2012

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we include in an appendix the corresponding treatment of the Swift-Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.

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arXiv2012

Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit

We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_x v+\partial_x^3 v+δ(\partial_x^2 v +\partial_x^4 v)=0$, $δ>0$, in the Korteweg-de Vries limit $δ\to 0$, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem to the evaluation for each Bloch parameter $ξ\in [0,2π]$ of certain elliptic integrals derived formally (on an incomplete set of frequencies/Bloch parameters, hence as necessary conditions for stability) and numerically evaluated by Bar and Nepomnyashchy \cite{BN}, thus obtaining, up to machine error, complete conclusions about stability. The main technical difficulty is in treating the large-frequency and small Bloch-parameter regimes not studied by Bar and Nepomnyashchy \cite{BN}, which requires techniques rather different from classical Fenichel-type analysis. The passage from small-$δ$ to small-$ξ$ behavior is particularly interesting, using in an essential way an analogy with hyperbolic relaxation at the level of the Whitham modulation equations.

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arXiv2011

Toward nonlinear stability of sources via a modified Burgers equation

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a modified Burgers equation as a model problem that captures some of the essential features of coherent structures, we show how this phenomenon can be analysed and nonlinear stability be established in this simpler context.

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arXiv2011

Nash--Moser iteration and singular perturbations

We present a simple and easy-to-use Nash--Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter $\eps\to 0.$ The novel feature is to allow loss of powers of $\eps$ as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems.

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arXiv2011

Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability

By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data $\bar u'(x)h_0(x)$, where $\bar u$ is the background profile and $h_0$ is the initial modulation

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arXiv2011

The Erpenbeck high frequency instability theorem for ZND detonations

The rigorous study of spectral stability for strong detonations was begun by J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-Döring (ZND) model, which assumes a finite reaction rate but ignores effects like viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function $V(τ,\eps)$ whose zeros in $\Re τ>0$ correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper [Er3] he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of $V$ exist for perturbations of sufficiently large transverse wavenumber $\eps$, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly twenty years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of $V(τ,\eps)$, the paper \cite{Er3} remains the only work we know of that presents a detailed and convincing theoretical argument for detecting them. The analysis in [Er3] points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in [Er3] and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as $\eps\to \infty$ of solutions to a linear system of ODEs in $x$, depending on $\eps$ and a complex frequency $τ$ as parameters, with turning points $x_*$ on the half-line $[0,\infty)$.

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arXiv2010

Stability of Periodic Kuramoto-Sivashinsky Waves

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second- and higher-order parabolic equations, for example, the Cahn-Hilliard equation or more general thin film models.

math.AP↗

arXiv2010

$2$-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardne

Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic Fredholm determinant of an associated Birman--Schwinger system, together with a Volterra integral computation introduced by Gesztesy--Makarov, we give an explicit connection between the generalized Birman--Schwinger-type periodic Evans function and the standard Jost function-type periodic Evans function defined by Gardner in terms of the fundamental solution of the eigenvalue equation written as a first-order system. This extends to a large family of operators the results of Gesztesy--Makarov for scalar Schrödinger operators and of Gardner for vector-valued second-order elliptic operators, in particular recovering by independent argument the fundamental result of Gardner that the zeros of the Evans function agree in location and (algebraic) multiplicity with the periodic eigenvalues of the associated operator

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arXiv2010

Nonlinear stability of viscous roll waves

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 Lagrangian coordinates, for which one can obtain large-amplitude nonlinear damping estimates similar to those carried out by Mascia--Zumbrun in the related shock wave case, assuming only symmetrizability of the hyperbolic part. The second is resolved by the observation that, similarly as in the relaxation and detonation cases, sources occurring in nonconservative components experience greater than expected decay, comparable to that experienced by a differentiated source.

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