Global solutions of the relativistic Euler equations

Abstract

We demonstrate the existence of solutions with shocks for the equations describing a perfect fluid in special relativity, namely, divT=0, whereT ^ij=(p+ρc ²)u ⁱ u ^j+pη ^ij is the stress energy tensor for the fluid. Here,p denotes the pressure,u the 4-velocity, φ the mass-energy density of the fluid,η ^ij the flat Minkowski metric, andc the speed of light. We assume that the equation of state is given byp=σ ² ρ, whereσ ², the sound speed, is constant. For these equations, we construct bounded weak solutions of the initial value problem in two dimensional Minkowski spacetime, for any initial data of finite total variation. The analysis is based on showing that the total variation of the variable ln(ρ) is non-increasing on approximate weak solutions generated by Glimm's method, and so this quantity, unique to equations of this type, plays a role similar to an energy function. We also show that the weak solutions (ρ(x ⁰,x ¹),v(x ⁰,x ¹)) themselves satisfy the Lorentz invariant estimates Var{ln(ρ(x ⁰,·)}<V ₀ and$\left\{ {In\frac{{c + v(x^0 , \cdot )}}{{c - v(x^0 , \cdot )}}} \right\}< V_1$ for allx ⁰≧0, whereV ₀ andV ₁ are Lorentz invariant constants that depend only on the total variation of the initial data, andv is the classical velocity. The equation of statep=(c ²/3)ρ describes a gas of highly relativistic particles in several important general relativistic models which describe the evolution of stars.