Viscous singular shock profiles for the Keyfitz-Kranzer system

Abstract

It was shown by Schecter (2004, J. Differential Equations, 205, 185-210), using the methods of Geometric Singular Perturbation Theory, that the Dafermos regularization $u_t+f(u)_x= \epsilon tu_{xx}$ for the Keyfitz-Kranzer system admits an unbounded family of solutions. Inspired by that work, in this paper we provide a more intuitive approach which leads to a stronger result. In addition to the existence of viscous profiles, we also prove the weak convergence and show that the maximum of the solution is of order $\epsilon^{-2}$. This asymptotic behavior is distinct from that obtained in the author's recent work (arXiv:1512.00394) on a system modeling two-phase fluid flow, for which the maximum of the viscous solution is of order $\exp(\epsilon^{-1})$.