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We derive a new formulation of the $3D$ compressible Euler equations withdynamic entropy exhibiting remarkable null structures and regularityproperties. Our results hold for an arbitrary equation of state (which yieldsthe pressure in terms of the density and the entropy) in non-vacuum regionswhere the speed of sound is positive. Our work is an extension of our priorjoint work with J. Luk, in which we derived a similar new formulation in thespecial case of a barotropic fluid, that is, when the equation of state dependsonly on the density. The new formulation comprises covariant wave equations forthe Cartesian components of the velocity and the logarithmic density coupled toa transport equation for the specific vorticity (defined to be vorticitydivided by density), a transport equation for the entropy, and some additionaltransport-divergence-curl-type equations involving special combinations of thederivatives of the solution variables. The good geometric structures in theequations allow one to use the full power of the vectorfield method in treatingthe "wave part" of the system. In a forthcoming application, we will use thenew formulation to give a sharp, constructive proof of finite-time shockformation, tied to the intersection of acoustic "wave" characteristics, forsolutions with nontrivial vorticity and entropy at the singularity. In thepresent article, we derive the new formulation and overview the central rolethat it plays in the proof of shock formation.

A New Formulation of the $3D$ Compressible Euler Equations With Dynamic Entropy: Remarkable Null Structures and Regularity Properties