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We prove a series of results tied to the regularity and geometry of solutionsto the $3D$ compressible Euler equations with vorticity and entropy. Ourframework exploits and reveals additional virtues of a recent new formulationof the equations, which decomposed the flow into a geometric "(sound)wave-part" coupled to a "transport-div-curl-part" (transport-part for short),with both parts exhibiting remarkable properties. Our main result is that thetime of existence can be controlled in terms of the$H^{2^+}(\mathbb{R}^3)$-norm of the wave-part of the initial data and variousSobolev and H\"{o}lder norms of the transport-part of the initial data, thelatter comprising the initial vorticity and entropy. The wave-part regularityassumptions are optimal in the scale of Sobolev spaces: shocks can instantlyform if one only assumes a bound for the $H^2(\mathbb{R}^3)$-norm of thewave-part of the initial data. Our proof relies on the assumption that thetransport-part of the initial data is more regular than the wave-part, and weshow that the additional regularity is propagated by the flow, even though thetransport-part of the flow is deeply coupled to the rougher wave-part. Toimplement our approach, we derive several results of independent interest: i)sharp estimates for the acoustic geometry, i.e., the geometry of sound cones;ii) Strichartz estimates for quasilinear sound waves coupled to vorticity andentropy; and iii) Schauder estimates for the transport-div-curl-part. Comparedto previous works on low regularity, the main new features of the paper arethat the quasilinear PDE systems under study exhibit multiple speeds ofpropagation and that elliptic estimates for various components of the fluid areneeded, both to avoid loss of regularity and to gain space-time integrability.

Rough sound waves in $3D$ compressible Euler flow with vorticity