Radially symmetric shadow wave solutions to the system of pressureless gas dynamics in arbitrary dimensions

TL;DR

This paper introduces radially symmetric shadow wave solutions for multidimensional pressureless gas dynamics, transforming the problem into a one-dimensional system with source terms, and provides a complete solution to the pseudo-Riemann problem with spherical shock data.

Contribution

It presents a novel class of solutions capturing mass concentration and establishes boundary conditions, entropy criteria, and solutions for spherical shock initial data.

Findings

Successfully transforms multidimensional problem into 1D with source terms

Derives entropy conditions for physical and dissipative solutions

Provides complete solution to pseudo-Riemann problem with spherical shock

Abstract

Radially symmetric shadow wave solutions to the system of multidimensional pressureless gas dynamics are introduced, which allow one to capture concentration of mass. The transformation to a one-dimensional system with source terms is performed and physically meaningful boundary conditions at the origin are determined. Entropy conditions are derived and applied to single out physical (nonnegative mass) and dissipative (entropic) solutions. A complete solution to the pseudo-Riemann problem with initial data exhibiting a single shock on a sphere is obtained.