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We derive new, localized geometric integral identities for solutions to the$3D$ compressible Euler equations under an arbitrary equation of state when thesound speed is positive. The identities are coercive in the first derivativesof the specific vorticity and the second derivatives of the entropy, and theerror terms exhibit remarkable regularity and null structures. Our frameworkallows one to simultaneously unleash the full power of the geometricvectorfield method for both the wave- and transport- parts of the flow oncompact regions. In particular, the integral identities yield localized controlover one additional derivative of the vorticity and entropy compared tostandard results, assuming that the initial data enjoy the same gain. Similarresults hold for the solution's higher derivatives. We derive the identities indetail for two classes of spacetime regions that frequently arise in PDEapplications: i) compact spacetime regions that are globally hyperbolic withrespect to the acoustical metric and ii) compact regions covered bydouble-acoustically null foliations. Our results have implications for thegeometry and regularity of solutions, the formation of shocks, the structure ofthe maximal classical development of the data, and for controlling solutionswhose state along a pair of intersecting characteristic hypersurfaces is known.Our analysis relies on a recent new formulation of the compressible Eulerequations that splits the flow into a geometric wave-part coupled to adiv-curl-transport part. Our main new contribution is our analysis of thepositive co-dimension, spacelike boundary integrals that arise in the div-curlidentities. By exploiting interplay between the elliptic and hyperbolic partsof the new formulation, we observe several crucial cancellations, which intotal show that the boundary terms have a good sign.

Remarkable localized integral identities for $3D$ compressible Euler flow and the double-null framework