The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

Abstract

We derive a new formulation of the relativistic Euler equations that exhibitsremarkable properties. This new formulation consists of a coupled system of geometric wave,transport, and transport-div-curl equations, sourced by nonlinearities that are null formsrelative to the acoustical metric. Our new formulation is well-suited for various applications,in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover,using the new formulation presented here, we establish a local well-posedness result showingthat the vorticity and the entropy of the fluid are one degree moredifferentiable comparedto the regularity guaranteed by standard estimates (assuming that the initial data enjoy theextra differentiability). This gain in regularity is essential for the study of shock formationwithout symmetry assumptions. Our results hold for an arbitrary equation of state, notnecessarily of barotropic type.