Digital Object Identifier (DOI) 10.1007/s00205-002-0215-8 Arch. Rational Mech. Anal. 164 (2002) 287–309 Spectral Stability of Small Shock Waves H. Freist ¨ uhler & P. Szmolyan Communicated by T.-P. Liu Abstract This paper is the first in a series of articles that study the eigenvalue problem for small-amplitude shock waves in systems of conservation laws with viscos- ity or relaxation. The papers show in various contexts that, in the zero-amplitude limit, appropriately scaled versions of the associated “Evans bundles” converge to suspensions of Evans bundles for fixed shock waves in related reduced systems with lower-dimensional state space. In this article the new geometric framework establishing this connection is introduced in the simplest context, that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension. 1. Introduction and results We consider shock waves v φ (x,t) = φ(x - st), φ(±∞) = v ± , (1.1) briefly: φ, in systems of hyperbolic-parabolic viscous conservation laws v t + (f (v)) x = v xx , (1.2) f : V → R n smooth,V ⊂ R n convex and open, Df R-diagonalizable. (1.3) We study the “Evans bundles” associated with the corresponding eigenvalue prob- lem κp + ((Df (φ) - sI)p) ′ = p ′′ , (1.4)