### Burgers' Equation with Vanishing Hyper-Viscosity

Abstract: We prove that bounded solutions of the vanishing hyper-viscosity equation, $u_t+f(u)_x+(-1)^{s}\eps\partial^{2s}_x u=0$ converge to the entropy solution of the corresponding convex conservation law $u_t+f(u)_x=0, \ f'' >0$. The hyper-viscosity case, $s>1$, 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.

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