Local smooth solutions to the 3-dimensional isentropic relativistic Euler equations

The authors consider the local smooth solutions to the isentropic relativistic Euler equations in (3+1)-dimensional space-time for both non-vacuum and vacuum cases. The local existence is proved by symmetrizing the system and applying the Friedrichs-Lax-Kato theory of symmetric hyperbolic systems. For the non-vacuum case, according to Godunov, firstly a strictly convex entropy function is solved out, then a suitable symmetrizer to symmetrize the system is constructed. For the vacuum case, since the coefficient matrix blows-up near the vacuum, the authors use another symmetrization which is based on the generalized Riemann invariants and the normalized velocity.