Viscous singular shock profiles for the Keyfitz-Kranzer system

TL;DR

This paper studies viscous shock profiles for the Keyfitz-Kranzer system, proving their existence, weak convergence, and that the maximum scales as , offering a more intuitive approach and stronger results than prior work.

Contribution

It provides a more intuitive proof of viscous profiles for the Keyfitz-Kranzer system and establishes their weak convergence and maximum order, improving upon previous methods.

Findings

Existence of viscous shock profiles for the system.

Weak convergence of solutions as  .

Maximum of solutions scales as  with .

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 $\exp(\epsilon^{-1})$.