Viscous Profiles for Traveling Waves of Scalar Balance Laws: The Uniformly Hyperbolic Case

Abstract

We consider a scalar hyperbolic conservation law with a nonlinear source term and viscosity ɛ. For ɛ = 0, there exist in general different types of heteroclinic entropy traveling waves. It is shown that for ɛ positive and sufficiently small the viscous equation possesses similar traveling wave solutions and that the profiles converge in exponentially weighted L1-norms as ɛ ↘ zero. The proof is based on a careful study of the singularly perturbed second-order equation that arises from the traveling wave ansatz.