  We derive a new formulation of the $3D$ compressible Euler equations with
dynamic entropy exhibiting remarkable null structures and regularity
properties. Our results hold for an arbitrary equation of state (which yields
the pressure in terms of the density and the entropy) in non-vacuum regions
where the speed of sound is positive. Our work is an extension of our prior
joint work with J. Luk, in which we derived a similar new formulation in the
special case of a barotropic fluid, that is, when the equation of state depends
only on the density. The new formulation comprises covariant wave equations for
the Cartesian components of the velocity and the logarithmic density coupled to
a transport equation for the specific vorticity (defined to be vorticity
divided by density), a transport equation for the entropy, and some additional
transport-divergence-curl-type equations involving special combinations of the
derivatives of the solution variables. The good geometric structures in the
equations allow one to use the full power of the vectorfield method in treating
the "wave part" of the system. In a forthcoming application, we will use the
new formulation to give a sharp, constructive proof of finite-time shock
formation, tied to the intersection of acoustic "wave" characteristics, for
solutions with nontrivial vorticity and entropy at the singularity. In the
present article, we derive the new formulation and overview the central role
that it plays in the proof of shock formation.
