Singular solutions for a 2X2 system in nonconservative form with incomplete set of eigenvectors

TL;DR

This paper investigates the formation of singular solutions in two first-order non-conservative hyperbolic systems, demonstrating existence and convergence of solutions, especially highlighting the effects of eigenvector completeness and parameter limits.

Contribution

It introduces a method to establish singular solutions for systems with incomplete eigenvector sets and analyzes the convergence of solutions as a parameter approaches zero.

Findings

Singular solutions exist for both systems using weak asymptotics.

The second system develops singular concentrations from Riemann data.

Solutions for the first system converge to those of the second as parameter k approaches zero.

Abstract

In this paper, we study the initial-value problem for two first order systems in non-conservative form. The first system arises in elastodynamics and belongs to the class of strictly hyperbolic, genuinely nonlinear systems. The second system has repeated eigenvalues and an incomplete set of right eigenvectors. Solutions to such systems are expected to develop singular con- centrations. Existence of singular solutions to both the systems have been shown using the method of weak asymptotics. The second system has been shown to develop singular concentrations even from Riemann-type initial data. The first system differing from the second in having an extra term containing a positive constant k, the solution constructed for the first system have been shown to converge to the solution of the second as k tends to 0.