Shock Development in Spherical Symmetry

The general problem of shock formation in three space dimensions was solved by D. Christodoulou in [2]. 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 in addition has singular initial data because of 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.