Global Solutions to the Ultra-Relativistic Euler Equations

We show that when entropy variations are included and special relativity is imposed, the thermodynamics of a perfect fluid leads to two distinct families of equations of state whose relativistic compressible Euler equations are of Nishida type. (In the non-relativistic case there is only one.) The first corresponds exactly to the StefanBoltzmann radiation law, and the other, emerges most naturally in the ultra-relativistic limit of a γ -law gas, the limit in which the temperature is very high or the rest mass very small. We clarify how these two relativistic equations of state emerge physically, and provide a unified analysis of entropy variations to prove global existence in one space dimension for the two distinct 3 × 3 relativistic Nishida-type systems. In particular, as far as we know, this provides the first large data global existence result for a relativistic perfect fluid constrained by the Stefan-Boltzmann radiation law. It was shown in [6,10] that for non-relativistic perfect fluids a unique equation of state of the form p = a2ρ emerges from a γ -law gas in the (appropriately re-scaled) limit γ → 1. A global existence theorem for the 3 × 3 non-relativistic compressible Euler equations was then proven for this model equation of state. This non-relativistic equation of state is unique, but has questionable physical interpretation as an isothermal gas, cf. [8]. Surprisingly, in contrast with the classical γ → 1 limit, the equation of state p = a2ρ emerges in two fundamental limits, not one, when special relativity is imposed: it is exact in the case of the Stefan-Boltzmann radiation law, and also emerges in a most natural ultra-relativistic limit of a γ -law gas, the limit in which the temperature is very high or the rest mass very small [2], (not the awkward limit γ → 1). Our results clarify how these two relativistic equations of state emerge physically, and provide a unified analysis of the entropy variations for the resulting two distinct relativistic Nishida systems that leads to a large data global existence theorem for both. In particular, as far as we know, this provides the first large data global existence result for a relativistic perfect fluid constrained by the Stefan-Boltzmann radiation law.