Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves (original) (raw)

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.

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].

Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit

Archive for Rational Mechanics and Analysis, 2005

In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u 0 to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions u to a parabolic viscous perturbation of the hyperbolic system which converge to u 0 as viscosity → 0 and satisfy an appropriate (conormal) version of Majda's stability estimate.

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.

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.

Pointwise semigroup methods and stability of viscous shock waves

Indiana University Mathematics Journal, 1998

Considered as rest points of ODE on L p , stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case Sat , K.2 , resp., each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We o v ercome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard H.1 , Kapitula K.1-2 , and Zeng Ze,LZe . These techniques allow us to do hard" analysis in PDE within the dynamical systems semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications also in other situations of sensitive stability.

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.

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

Eprint Arxiv Math 0401054, 2004

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].