Burgers' Equation with Vanishing Hyper-Viscosity

Abstract

We prove that bounded solutions of the vanishing hyper-viscosity equation, converge to the entropy solution of the corresponding convex conservation law. The hyper-viscosity case lacks the monotonicity which underlines the Krushkov BV theory in the viscous case s = 1. Instead we show how to adapt the Tartar-Murat compensated compactness theory together with a weaker entropy dissipation bound to conclude the convergence of the vanishing hyper-viscosity.