Spectral stability of shock waves associated with not genuinely nonlinear modes

We study viscous shock waves that are associated with a simple mode (λ,r)(λ,r) of a system ut+f(u)x=uxxut+f(u)x=uxx of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ   in state space at whose points r⋅∇λ=0r⋅∇λ=0 and (r⋅∇)2λ≠0(r⋅∇)2λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law ut+(u3)x=uxxut+(u3)x=uxx, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves.