  We prove a series of results tied to the regularity and geometry of solutions
to the $3D$ compressible Euler equations with vorticity and entropy. Our
framework exploits and reveals additional virtues of a recent new formulation
of 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 the
time of existence can be controlled in terms of the
$H^{2^+}(\mathbb{R}^3)$-norm of the wave-part of the initial data and various
Sobolev and H\"{o}lder norms of the transport-part of the initial data, the
latter comprising the initial vorticity and entropy. The wave-part regularity
assumptions are optimal in the scale of Sobolev spaces: shocks can instantly
form if one only assumes a bound for the $H^2(\mathbb{R}^3)$-norm of the
wave-part of the initial data. Our proof relies on the assumption that the
transport-part of the initial data is more regular than the wave-part, and we
show that the additional regularity is propagated by the flow, even though the
transport-part of the flow is deeply coupled to the rougher wave-part. To
implement 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 and
entropy; and iii) Schauder estimates for the transport-div-curl-part. Compared
to previous works on low regularity, the main new features of the paper are
that the quasilinear PDE systems under study exhibit multiple speeds of
propagation and that elliptic estimates for various components of the fluid are
needed, both to avoid loss of regularity and to gain space-time integrability.
