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Kevin Zumbrun

Publications and source records attributed to Kevin Zumbrun.

137 records · Page 8Linked to original sources
arXiv2004

Hyperbolic Boundary Value Problems for Symmetric Systems with Variable Multiplicities

We extend the Kreiss--Majda theory of stability of hyperbolic initial--boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a ``totally nonglancing'' or a ``nonglancing and linearly splitting'' condition. At the same time, we give a simple characterization of the block structure condition as ``geometric regularity'' of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner--Kruskal and Blokhin--Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin--Trakhinin by direct ``dissipative integral'' methods, of stability in the zero-magnetic field limit. Our methods apply also to the viscous case.

math.AP↗
arXiv2004

Stability of large-amplitude shock waves of compressible Navier--Stokes equations

We summarize recent progress on one- and multi-dimensional stability of viscous shock wave solutions of compressible Navier--Stokes equations and related symmetrizable hyperbolic--parabolic systems, with an emphasis on the large-amplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multi-dimensions by a codimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multi-dimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently small-amplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently large-amplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory.

math-ph↗
arXiv2004

Planar stability critera for viscous shock waves of systems with real viscosity

We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with ``real,'' or partial viscosity. The main result is the establishment of necessary, or ``weak'', and sufficient, or ``strong'', conditions for nonlinear stability analogous to those established by Majda [Ma.1--3] in the inviscid case but (generically) separated by a codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stability and instability is thereby determined, lying on the boundary set between the open regions of strong stability and strong instability (the latter defined as failure of weak stability). Strong stability holds always for small-amplitude shocks of classical ``Lax'' type [PZ.1--2, FreS]; for large-amplitude shocks, however, strong instability may occur [ZS, Z.3].

math.AP↗
arXiv2003

A stability index for detonation waves in Majda's model for reacting flow

Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier-Stokes equations of reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [CMR] and analytical results of [Sz,LY] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, ``bump-type'' ignition function) deflagration profiles, as discussed in [GS.1-2] for the full equations.

math.AP↗
arXiv2003

On the One-dimensional Stability of Viscous Strong Detonation Waves

Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting Navier-Stokes equations. The primary result, following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting Navier-Stokes equations are included

math.AP↗
arXiv2002

An Evans function approach to spectral stability of small-amplitude shock profiles

We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers equation. In multidimensions, the canonical behavior is described, rather by a 2x2 viscous conservation law; this case will be treated in a companion paper by similar techniques.

math.AP↗
arXiv2002

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.

math.AP↗
arXiv2001

Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems

Combining pointwise Green's function bounds obtained in a companion paper [MZ.2] with earlier, spectral stability results obtained in [HuZ], we establish nonlinear orbital stability of small amplitude viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [Kaw], notably including compressible Navier--Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in $L^p$ with respect to small $L^1\cap H^3$ perturbations, $2\le p\le \infty$. Our analysis follows the approach introduced in [MZ.1] to treat stability of relaxation profiles.

math.AP↗
arXiv1998

The Obstacle Problem for Functions of Least Gradient

For a given domain $Ω\subset \Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $Ω$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower obstacle condition $u\ge Ψ$ inside $Ω$. Under the assumption of strictly positive mean curvature of the boundary $\partialΩ$, we show existence of a continuous solution, with Hölder exponent half of that of data and obstacle. This generalizes previous results obtained for the unconstrained and double-obstacle problems. The main new feature in the present analysis is the need to extend various maximum principles from the case of two area-minimizing sets to the case of one sub- and one superminimizing set. This we accomplish subject to a weak regularity assumption on one of the sets, sufficient to carry out the analysis. Interesting open questions include the uniqueness of solutions and a complete analysis of the regularity properties of area superminimizing sets. We provide some preliminary results in the latter direction, namely a new monotonicity principle for superminimizing sets, and the existence of ``foamy'' superminimizers in two dimensions.

math.AP↗