Global existence of smooth solutions to spherically symmetric relativistic Euler equations

This paper studies classical solutions to three-dimensional (3D) relativistic Euler equations with radial initial data. We identify some sufficient conditions on the initial data to obtain the global existence of bounded expanding smooth solutions. The crucial point for the global existence is to establish a priori estimates for the C1-norm of solutions. To this end, we derive a group of suitable characteristic decompositions for the spherically symmetric relativistic Euler equations. Owing to the good structure of these characteristic decompositions, we use “maximum principle” to construct some invariant regions for the derivatives of the sound speed. Using these invariant regions we establish a priori estimates for the C1-norm of the solutions. The asymptotic behavior of the solutions is also discussed. We show that the mass-energy densities of the solutions scatter to zero as t → +∞.