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)$(\lambda ,r)$ of a system _ut+f_(u)x=_uxx$u_t+f{(u)}_x=u_{xx}$ of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ   in state space at whose points r⋅∇λ=0$r\cdot \mathrm{\nabla }\lambda =0$ and (r⋅∇)²λ≠0${(r\cdot \mathrm{\nabla })}^2\lambda \ne 0$. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law _ut+_(u3)x=_uxx$u_t+{(u^3)}_x=u_{xx}$, 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.