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Kevin Zumbrun

Publications and source records attributed to Kevin Zumbrun.

At least 55 records · Page 3Linked to original sources
arXiv2017

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.

math.AP↗
arXiv2016

Stable manifolds for a class of degenerate evolution equations and exponential decay of kinetic shocks

We construct stable manifolds for a class of degenerate evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

math.DS↗
arXiv2016

L infinity resolvent bounds for steady Boltzmann's equation

We derive lower bounds on the resolvent operator for the linearized steady Boltzmann equation over weighted L1 Banach spaces in velocity, comparable to those derived by Pogan & Zumbrun in an analogous weighted L2 Hilbert space setting. These show in particular that the operator norm of the resolvent kernel is unbounded in Lp(R) for all $1<p \leq \infty$, resolving an apparent discrepancy in behavior between the two settings suggested by previous work.

math.AP↗
arXiv2016

Center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium ow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.

math.AP↗
arXiv2016

Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves

Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analytically-varying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semi-separable integral kernels, which include in particular the general one-dimensional case, and for sums thereof, which latter possibility appears to offer applications in the multi-dimensional case. A second main result is to show that the multi-dimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the one-dimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration.

math.AP↗
arXiv2016

Recent results on stability of planar detonations

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.

math.AP↗
arXiv2016

Diffusive stability of spatially periodic solutions of the Brusselator model

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg Landau amplitude equations, rigorously validating the standard weakly unstable approximation and Eckhaus criterion.

math.AP↗
arXiv2016

Note on the stability of viscous roll-waves

In this note, we announce a complete classification of stability of periodic roll-wave solutions of the viscous shallow-water equations, from their onset at Froude number $F\approx 2$ up to the infinite-Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between $F$ and the boundaries of the region of stable periods, that appears potentially useful in hydraulic engineering applications. In the asymptotic regime $F\to 2$ (onset), we provide an analytic expression of the stability boundaries whereas in the limit $F\to\infty$, we show that roll-waves are always unstable.

physics.flu-dyn↗
arXiv2016

Pointwise nonlinear stability of nonlocalized modulated periodic reaction-diffusion waves

In this paper, extending previous results of \cite{J1}, we obtain pointwise nonlinear stability of periodic traveling reaction-diffusion waves, assuming spectral linearized stability, under nonlocalized perturbations. More precisely, we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically.

math.AP↗
arXiv2016

Multidimensional stability of large-amplitude Navier-Stokes shocks

Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $μ$, $μ+η$, and $ν=κ/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any $γ$-law gas, $γ>1$, and arbitrary $μ>|η|\ge 0$, $ν>0$ or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).

math.AP↗
arXiv2016

Numerical proof of stability of viscous shock profiles

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 verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.

math.AP↗
arXiv2015

Block-diagonalization of ODEs in the semiclassical limit and $C^ω$ vs. $C^\infty$ stationary phase

Motivated by issues in detonation stability, we study existence of block-diagonalizing transformations for ordinary differential semiclassical limit problems arising in the study of high-frequency eigenvalue problems. Our main results are to (i) establish existence of block-diagonalizing transformations in a neighborhood of infinity for analytic-coefficient ODE, and (ii) establish by a series of counterexample sharpness of hypotheses and conclusions on existence of block-diagonalizing transformations near a finite point. In particular, we show that, in general, bounded transformations exist only locally, answering a question posed by Wasow in the 1980's, and, under the minimal condition of spectral separation, for ODE with analytic rather than $C^\infty$ coefficients. The latter issue is connected with quantitative comparisons of $C^ω$ vs. $C^\infty$ stationary phase estimates

math.CA↗
arXiv2015

Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow

A technical obstruction preventing the conclusion of nonlinear stability of large-Froude number roll waves of the St. Venant equations for inclined thin film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial H s nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 \textless{} F 3.5, but to fail for 3.5 F. As hydraulic engineering applications typically involve Froude number 3 F 5, this issue is indeed relevant to practical considerations. Here, we show that the pointwise slope condition can be replaced by an averaged version which holds always, thereby completing the nonlinear theory in the large-F case. The analysis has potentially larger interest as an extension to the periodic case of a type of weighted "Kawashima-type" damping estimate introduced in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves.

math.AP↗
arXiv2015

Stability of viscous St. Venant roll-waves: from onset to infinite-Froude number limit

We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general values of the Froude number $F$, including the limit $F\to +\infty$. In the former, $F\to 2^+$, limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $F\to 2^+$ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson--Noble--Rodrigues--Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around $F=2.3$ from weakly unstable to different, large-$F$ behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $2.5\leq F\leq 6.0$.

math.AP↗
arXiv2014

Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations

math.AP↗
arXiv2014

O(2) Hopf bifurcation of viscous shock waves in a channel

Extending work of Texier and Zumbrun in the semilinear non-re ection symmetric case, we study O(2) transverse Hopf bifurcation, or \cellular instability," of viscous shock waves in a channel, for a class of quasilinear hyperbolic{parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fr'echet differentiability of the time-T solution operator by appropriate hyperbolic{parabolic energy estimates, and (ii) handle O(2) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyperbolic-parabolic form) the requisite framework for treatment by spatial dynamics on the space of time-periodic functions, the two standard treatments for this problem. The latter issue is resolved by Lyapunov{Schmidt reduction of the time-T map, yielding a four-dimensional problem with O(2) plus approximate S1 symmetry, which we treat \by hand" using direct Implicit Function Theorem arguments. The former is treated by balancing information obtained in Lagrangian coordinates with that from an augmented system. Interestingly, this argument does not apply to gas dynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangian symmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.

math.AP↗
arXiv2013

High-frequency stability of multidimensional ZND detonations

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 magnitude, while type D detonations were stable provided the frequency magnitude was sufficiently high. For type D detonations Erpenbeck's methods did not allow him to obtain a cutoff magnitude for stability that was \emph{uniform} with respect to frequency direction. Thus, he left open the question whether the cutoff magnitude for stability might approach $+\infty$ as certain frequency directions were approached. In this paper we show by quite different methods that for type D detonations there exists a uniform cutoff magnitude for stability independent of frequency direction. By reducing the search for unstable frequencies to a bounded frequency set, the uniform cutoff obtained here is a key step toward the rigorous validation of a number of results in the computational detonation literature. The main difficulty in the analysis is to treat "turning points at infinity".

math.AP↗