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 FriedrichsLax-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.