Related Problems On Three Dimensional Relativistic Euler Equations

In this dissertation, we study the following two problems on three dimensional relativistic Euler equations:Firstly, the local existence of classical solutions of the Cauchy problem to three dimensional isentropic relativistic Euler equations (Chapter 2) for both non-vacuum and vacuum cases. For the non-vacuum case, we use the method of Godunov[20] to solve out the strictly convex entropy of this system, and symmetrize the system with this convex entropy, then based on the theory of Friedrich-Lax-Kato ([47,35,29]) we establish the local-in-time existence result. For the vacuum case, the symmetrized system based on the method of convex entropy will be degenerated near the vacuum, so we use the generalized Riemann invariants and normalized velocity instead of the convex entropy to symmetrize the system, then we also deduce a local existence of classical solutions.Secondly, the special relativistic effect of Riemann problems (Chapter 3). This effect implies the smooth transitions of wave patterns with respect to the change of initial tangential velocity. This effect has no analogue in Newtonian hydrodynamics [66] and one-dimensional relativistic system. By vigorous mathematical reasoning and analysis we will prove the monotonicity of relative normal velocity with respect to the intermediate pressure. We will compute precisely the changes of limiting relative nor-mal velocity and the intermediate states with respect to the initial tangential velocity. Therefore, the special relativistic effect can be clearly explained from the mathematical point of view.Moreover, in Chapter 4, we also consider the non-relativistic global limits of en-tropy solutions to the extremely relativistic Euler equations.