Global Entropy Solutions to Multi-Dimensional Isentropic Gas Dynamics with Spherical Symmetry

TL;DR

This paper establishes the existence of global entropy solutions for spherically symmetric multi-dimensional isentropic gas dynamics, addressing the singularity at the origin and analyzing solution behavior near it.

Contribution

It proves two new existence theorems for entropy solutions in spherical symmetry, including cases with and without the origin, using compensated compactness and vanishing viscosity methods.

Findings

Existence of global entropy solutions with singularity at the origin.

Solutions remain uniformly bounded over time.

Densities and velocities decay near the origin under certain initial conditions.

Abstract

We are concerned with spherically symmetric solutions to the Euler equations for the multi-dimensional compressible fluids, which have many applications in diverse real physical situations. The system can be reduced to one dimensional isentropic gas dynamics with geometric source terms. Due to the presence of the singularity at the origin, there are few papers devoted to this problem. The present paper proves two existence theorems of global entropy solutions. The first one focuses on the case excluding the origin in which the negative velocity is allowed, and the second one is corresponding to the case including the origin with non-negative velocity. The $L^\infty$ compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rates near origin, then the densities and velocities tend to zero with the same rates near the origin for any positive time. In particular, the entropy solutions in two existence theorems are uniformly bounded with respect to time.