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We derive a new formulation of the relativistic Euler equations that exhibitsremarkable properties. This new formulation consists of a coupled system ofgeometric wave, transport, and elliptic equations, sourced by nonlinearitiesthat are null forms relative to the acoustical metric. Our new formulation iswell-suited for various applications, in particular for the study of stableshock formation, as it is surveyed in the paper. Moreover, using the newformulation presented here, we establish a local well-posedness result showingthat the vorticity and the entropy of the fluid are one degree moredifferentiable compared to the regularity guaranteed by standard estimates(assuming that the initial data enjoy the extra differentiability). This gainin regularity is essential for the study of shock formation without symmetryassumptions. Our results hold for an arbitrary equation of state, notnecessarily of barotropic type.

The relativistic Euler equations: Remarkable null structures and regularity properties