  We derive new, localized geometric integral identities for solutions to the
$3D$ compressible Euler equations under an arbitrary equation of state when the
sound speed is positive. The identities are coercive in the first derivatives
of the specific vorticity and the second derivatives of the entropy, and the
error terms exhibit remarkable regularity and null structures. Our framework
allows one to simultaneously unleash the full power of the geometric
vectorfield method for both the wave- and transport- parts of the flow on
compact regions. In particular, the integral identities yield localized control
over one additional derivative of the vorticity and entropy compared to
standard results, assuming that the initial data enjoy the same gain. Similar
results hold for the solution's higher derivatives. We derive the identities in
detail for two classes of spacetime regions that frequently arise in PDE
applications: i) compact spacetime regions that are globally hyperbolic with
respect to the acoustical metric and ii) compact regions covered by
double-acoustically null foliations. Our results have implications for the
geometry and regularity of solutions, the formation of shocks, the structure of
the maximal classical development of the data, and for controlling solutions
whose state along a pair of intersecting characteristic hypersurfaces is known.
Our analysis relies on a recent new formulation of the compressible Euler
equations that splits the flow into a geometric wave-part coupled to a
div-curl-transport part. Our main new contribution is our analysis of the
positive co-dimension, spacelike boundary integrals that arise in the div-curl
identities. By exploiting interplay between the elliptic and hyperbolic parts
of the new formulation, we observe several crucial cancellations, which in
total show that the boundary terms have a good sign.
