Shock Development in Spherical Symmetry

TL;DR

This paper addresses the shock development problem in spherical symmetry for a barotropic fluid, providing a detailed description of singularities and extending the understanding of shock formation beyond initial blowup.

Contribution

It offers the first complete solution to the shock development problem under spherical symmetry, describing singularities with smooth functions.

Findings

Complete description of shock singularities in spherical symmetry

Extension of shock formation analysis beyond initial blowup

Framework for analyzing shock development in symmetric fluids

Abstract

The general problem of shock formation in three space dimensions was solved by D. Christodoulou in his 2007 monograph: 'The Formation of Shocks in 3-dimensional Fluids'. In this work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but possesses the additional property of having singular initial data due to the behavior of the solution at the blowup surface. The present work delivers the solution to this problem in the case of spherical symmetry for a barotropic fluid. A complete description of the singularities associated to the development of shocks in terms of smooth functions is given.