Planar stability critera for viscous shock waves of systems with real viscosity (original) (raw)

Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity

Springer eBooks, 2007

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].

The gap lemma and geometric criteria for instability of viscous shock profiles

Communications on Pure and Applied Mathematics, 1998

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.

Viscous and inviscid stability of multidimensional planar shock fronts

Indiana University Mathematics Journal, 1999

We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock pro le. Our main result, generalizing earlier one-dimensional calculations, is that the Evans function reduces in the long-wave limit to the Kreiss{Sakamoto{ Lopatinski determinant obtained by Majda in the inviscid case, multiplied by a constant measuring transversality of the shock connection in the underlying (viscous) traveling wave ODE. Remarkably, this result holds independently of the nature of the viscous regularization, or the type of the shock connection. Indeed, the analysis is more general still: in the overcompressive case, we obtain a simple long-wave stability criterion even in the absence of a sensible inviscid problem.

Transition to Instability of Planar Viscous Shock Fronts: the Refined Stability Condition

Zeitschrift für Analysis und ihre Anwendungen, 2008

Classical inviscid stability analysis determines stability of shock waves only up to a region of neutral stability occupying an open set of physical parameters. To locate a precise transition point within this region, it has been variously suggested that nonlinear and or viscous effects should be taken into account. Recently, Zumbrun and Serre showed that transition under localized (L 1 ∩ H s) perturbations is in fact entirely decided by viscous effects, and gave an abstract criterion for transition in terms of an effective viscosity coefficient β determined by second derivatives of the Evans function associated with the linearized operator about the wave. Here, generalizing earlier results of Kapitula, Bertozzi et al, and Benzoni-Gavage et al., we develop a simplified perturbation formula for β, applicable to general shock waves, that is convenient for numerical and analytical investigation.

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

Handbook of Mathematical Fluid Dynamics, 2005

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 largeamplitude 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.

Stability of Viscous Shocks on Finite Intervals

Archive for Rational Mechanics and Analysis, 2007

Consider the Cauchy problem for a system of viscous conservation laws with a solution consisting of a thin, viscous shock layer connecting smooth regions. We expect the time-dependent behavior of such a solution to involve two processes. One process consists of the large-scale evolution of the solution. This process is well modeled by the corresponding inviscid equations. The other process is the adjustment in shape and position of the shock layer to the large-scale solution. The time scale of the second process is much faster than the first, 1/ν compared to 1. The second process can be divided into two parts, adjustment of the shape and of the position. During this adjustment the end states are essentially constant.

On nonlinear stability of general undercompressive viscous shock waves

Communications in Mathematical Physics, 1995

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds and L p behavior of the solution for all p, 1 ^ p ^ oo.

Conditional stability of unstable viscous shocks

Journal of Differential Equations, 2009

Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translationinvariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L 1 ∩ H 2 perturbatoins, converging timeasymptotically to a translate of the unperturbed wave. That is, for a shock with p unstable eigenvalues, we establish conditional stability on a codimension-p manifold of initial data, with sharp rates of decay in all L p . For p = 0, we recover the result of unconditional stability obtained by Howard, Mascia, and Zumbrun.