Formation of singularities in one-dimensional Chaplygin gas

TL;DR

This paper studies the formation of novel 'Delta-like' singularities in solutions to the one-dimensional Chaplygin gas equations, revealing new blowup behaviors due to envelope interactions of characteristic families.

Contribution

It introduces a new type of singularity called 'Delta-like' solutions for the Chaplygin gas system, expanding understanding of blowup phenomena in linearly degenerate hyperbolic systems.

Findings

Identification of 'Delta-like' singularities in the system

Analysis of blowup behavior near singularity points

Different initial data lead to various singularity shapes

Abstract

In this paper we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas. In particular, under suitable assumptions we construct a physical solution with a new type of singularities called "Delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By careful analysis, we study the behavior of the solution in a neighborhood of a blowup point. The formation of this new kind of singularities is due to the envelope of the different families of characteristics instead of the same family of characteristics in the traditional situation. This shows that the blowup phenomenon of solution for the system with linearly degenerate characteristics is quite different from the situation of shock formation for the system with genuinely nonlinear characteristics. Different initial data can lead to kinds of different Delta-like singularities: the Delta-like singularity with point-shape and Delta-like singularity with line-shape.