arXiv subjects

Kevin Zumbrun

Publications and source records attributed to Kevin Zumbrun.

At least 37 records · Page 2Linked to original sources

arXiv2022

Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato-Lefever wave

In an interesting recent analysis, Haragus-Johnson-Perkins-de Rijk have shown modulational stability under localized perturbations of steady periodic solutions of the Lugiato-Lefever equation (LLE), in the process pointing out a difficulty in obtaining standard "nonlinear damping estimates" on modulated perturbation variables to control regularity of solutions. Here, we point out that in place of standard "inverse-modulated" damping estimates, one can alternatively carry out a damping estimate on the "forward-modulated" perturbation, noting that norms of forward- and inverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classical argument structure of Johnson-Noble-Rodrigues-Zumbrun for parabolic systems. This observation seems of general use in situations of delicate regularity. Applied in the context of (LLE) it gives the stronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.

math.AP↗

arXiv2022

Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

We establish an instantaneous smoothing property for decaying solutions on the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of such equations, showing that $L^2_{loc}$ solutions that remain sufficiently small in $L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hence lie eventually in the $H^1$ stable manifold constructed by Pogan and Zumbrun

math.AP↗

arXiv2022

Large-amplitude modulation of periodic traveling waves

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and valitidy to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

math.AP↗

arXiv2020

Transverse bifurcation of viscous slow MHD shocks

We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model and setting) an open question of Zumbrun and Serre.

math.AP↗

arXiv2020

Stability of strong detonation waves for Majda's model with general ignition functions

For strong detonation waves of the inviscid Majda model, spectral stability was established by Jung and Yao for waves with step-type ignition functions, by a proof based largely on explicit knowledge of wave profiles. In the present work, we extend their stability results to strong detonation waves with more general ignition functions where explicit profiles are unknown. Our proof is based on reduction to a generalized Sturm-Liouville problem, similar to that used by Sukhtayev, Yang, and Zumbrun to study spectral stability of hydraulic shock profiles of the Saint-Venant equations.

math.AP↗

arXiv2019

A Sturm Liouville theorem for quadratic operator pencils

We establish a Sturm{Liouville theorem for quadratic operator pencils counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

math.CA↗

arXiv2019

Stability of hydraulic shock profiles

We establish nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans-Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.

math.AP↗

arXiv2019

Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions

We study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a finite interval with non characteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possesses (i) a spectral gap between neutral and growing/decaying modes, and (ii) an even number of nonstable eigenvalues ? (with a nonnegative real part). In the case that there are no nonstable eigenvalues, i.e., of spectral stability, we show this solution to be nonlinearly exponentially stable in H2 X H3. Using "Goodman-type" weighted energy estimates, we establish spectral stability for small-amplitude data. For large amplitude data, we obtain high-frequency stability, reducing stability investigations to a bounded frequency regime. On this remaining, bounded-frequency regime, we carry out a numerical Evans function study, with results again indicating universal stability of solutions.

math.AP↗

arXiv2018

Spectral stability of hydraulic shock profiles

By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear and nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.

math.AP↗

arXiv2018

Spectral Stability of Inviscid Roll Waves

We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint Venant equations, based on a periodic Evans-Lopatinski determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but not implied by the associated formal first-order Whitham modulation equations.

math.AP↗

arXiv2018

Nonlinear stability of source defects in oscillatory media

In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined with an instantaneous phase-tracking technique to obtain detailed pointwise estimates describing perturbations to lowest order as a phase-shift radiating outward at a linear rate plus a pair of localized approximately Gaussian excitations along the phase-shift boundaries; we show that in the wake of these outgoing waves the perturbed solution converges time-exponentially to a space-time translate of the original source pattern.

math.AP↗

arXiv2018

Convergence as period goes to infinity of spectra of periodic traveling waves toward essential spectra of a homoclinic limit

We revisit the analysis by R.A. Gardner of convergence of spectra of periodic traveling waves in the homoclinic, or infinite-period limit, extending his results to the case of essential rather than point spectra of the limiting homoclinic wave. Notably, convergence to essential spectra is seen to be of algebraic rate with respect to period as compared to the exponential rate of convergence to point spectra. In the course of the analysis, we show not only convergence of spectrum but also convergence of an appropriate renormalization of the associated periodic Evans function to the Evans function for the limiting homoclinic wave, a fact that is useful for numerical investigations

math.AP↗

arXiv2017

A singular local minimizer for the volume constrained minimal surface problem in a nonconvex domain

It has recently been established byWang and Xia [WX] that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of the convexity condition by exhibiting a simple example, given by the Simons cone, of a singular volume-constrained locally area-minimizing surface within a nonconvex domain that is arbitrarily close to the unit ball.

math.AP↗

arXiv2017

Euler vs. Lagrange: The role of coordinates in practical Evans-function computations

The Evans function has become a standard tool in the mathematical study of nonlinear wave stability. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). We report on an unexpected complication that frustrates this computation for viscous shock profiles in gas dynamics. Although this phenomenon---related to the choice of Eulerian or Lagrangian coordinate system used to describe the gas---is present already in the one-dimensional setting, its implications are especially important in the multidimensional case where no computationally viable Lagrangian description of the gas is readily available. We introduce new "pseudo-Lagrangian" coordinates that allow us to overcome this difficulty, and we illustrate the utility of these coordinates in the setting of isentropic gas dynamics in two space dimensions.

math.AP↗

arXiv2017

On nonlinear stabilization of linearly unstable maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.

math.DS↗

arXiv2017

A calculus proof of the Cramér-Wold theorem

We present a short, elementary proof not involving Fourier transforms of the theorem of Cramér and Wold that a Borel probability measure is determined by its values on half-spaces.

math.PR↗

arXiv2017

Balanced flux formulations for multidimensional Evans function computations for viscous shocks

The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman's integrated coordinates [G1]; this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.

math.AP↗

arXiv2017

Reverse norms and L infinity exponential decay for a class of degenerate evolution systems arising in kinetic theory

We consider the question of exponential decay to equilibrium of solutions of an abstract class of degenerate evolution equations on a Hilbert space modeling the steady Boltzmann and other kinetic equations. Specifically, we provide conditions suitable for construction of a stable manifold in a particular "reverse L infinity norm" and examine when these do and do not hold.

math.AP↗