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In this paper, we study the Riemann problem of relativistic Euler system forrarefied monatomic and diatomic gases when the constitutive equation for theenergy is the Synge equation that is the only one compatible with therelativistic kinetic theory. The Synge equation is involved with modifiedBessel functions of the second kind and this makes the relativistic Eulersystem quite complex. Based on delicate estimates of the modified Besselfunctions of the second kind, we provide a detailed investigation of basichyperbolic properties and the structure of elementary waves, especially for thestructure of shock waves and in this way, the mathematical theory of theRiemann problem for these relativistic Euler system, which is analogous to thecorresponding theory of the classical ones, is rigorously provided.

The Riemann problem of relativistic Euler system with Synge energy