Work

Existence and Blow-Up of Compressible Spherically Symmetric Euler Equations with Vacuum Free Boundary

Year: 2026

Type: article

Abstract: This paper studies the compressible spherically symmetric Euler equations with a vacuum free boundary, a fundamental model for astrophysical gas dynamics. We rigorously resolve an open problem by proving that nontrivial homogeneous linear velocity solutions exist if, and only if, the adiabatic exponent γ=4/3, the critical value for monatomic gases and radiative stellar atmospheres. Using qualitative analysis of the reduced planar dynamical system, we characterize the flow’s global existence and finite-time blow-up behavior, establish a sharp existence threshold, and derive an explicit upper bound for the blow-up time. Quantitative energy estimates via Bernoulli’s head verify the physical consistency of solutions in both regimes. Our results complete the classification of self-similar solutions in this class, laying a rigorous theoretical foundation for planetary atmospheric gas flows and providing a practical criterion for predicting blow-up.

Source: Mathematics

DOI: 10.3390/math14101775

Authors: Lijun Zhang, Junmei Shi, Chaudry Masood Khalique

Institutions: Zhejiang University of Science and Technology, North-West University, Shandong University of Science and Technology

Language: English

Topic: Navier-Stokes equation solutions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: National Natural Science Foundation of China