Kevin Zumbrun

The Gap Lemma and Geometric Criteria for Instability of Viscous Shock Profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for nn

Multidimensional Stability of Planar Viscous Shock Waves

Physical and mathematical considerations warrant the inclusion of regularizing effects such as diffusion, dissipation, and/or relaxation in the study of stability of shock waves, particularly in MHD, combustion, and multiphase flow. Indeed, in this generality, the “ideal shock” approximation has relevance only in the long-wave limit, that is, in the large, but not in the small scale. Likewise, multidimensional effects are known both from experiment and from the study of the inviscid (hyperbolic) case to play a key role in determining stability. Yet, until very recently, there were no rigorous analyses for systems taking into account both effects simultaneously.

Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems

Introduction Linear stability: the model case Pieces of paradifferential calculus

Stability of Large-Amplitude Shock Waves of Compressible Navier–Stokes Equations

L^2

Stability of rarefaction waves in viscous media

and conormal estimates near the boundary Linear stability Nonlinear stability Appendix A. Kreiss symmetrizers Appendix B. Para-differential calculus Appendix Bibliography.

Alternate Evans Functions and Viscous Shock Waves

Abstract We summarize recent progress on one-dimensional and multidimensional 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 multidimensions by a co-dimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions 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. 1

On nonlinear stability of general undercompressive viscous shock waves

AbstractWe study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w0 with small mass and localized as w0(x)=

Existence of relaxation shock profiles for hyperbolic conservation laws

\mathcal{O}(|x|^{ - 1} )