Stable viscous shocks in elliptic conservation laws

We study quadratic systems of viscous conservation laws which arise as long-wavelength modulation equations near planar, modulated traveling waves. In [3], we showed that the conservation law is either elliptic or hyperbolic in a full neighborhood of the origin. Moreover, in parameter space the ill-posed, elliptic inviscid limit coincides with the robust occurrence of localized degenerate viscous shocks that correspond to localized spikes in the profile of the traveling wave. We refer to these localized degenerate shock waves as holes. In this paper, we study a special case, where the effective viscosity in the conservation law is a scalar. Although the ellipticity of the underlying inviscid conservation law creates an instability of the linearized transport equation at every single point of the degenerate shock wave, which moreover is absolute in a region near the shock location, holes and accompanying overcompressive shocks turn out to be asymptotically stable. We conclude with an example of a reaction-diffusion system with a planar modulated wave where our results predict the existence of families of stable holes in the planar front. Indiana University Mathematics Journal