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Consider a $1$D simple small-amplitude solution $(\rho_{(bkg)}, v^1_{(bkg)})$to the isentropic compressible Euler equations which has smooth initial data,coincides with a constant state outside a compact set, and forms a shock infinite time. Viewing $(\rho_{(bkg)}, v^1_{(bkg)})$ as a plane-symmetricsolution to the full compressible Euler equations in $3$D, we prove that theshock-formation mechanism for the solution $(\rho_{(bkg)}, v^1_{(bkg)})$ isstable against all sufficiently small and compactly supported perturbations. Inparticular, these perturbations are allowed to break the symmetry and havenon-trivial vorticity and variable entropy.  Our approach reveals the full structure of the set of blowup-points at thefirst singular time: within the constant-time hypersurface of first blowup, thesolution's first-order Cartesian coordinate partial derivatives blow upprecisely on the zero level set of a function that measures the inversefoliation density of a family of characteristic hypersurfaces. Moreover,relative to a set of geometric coordinates constructed out of an acousticeikonal function, the fluid solution and the inverse foliation density functionremain smooth up to the shock; the blowup of the solution's Cartesiancoordinate partial derivatives is caused by a degeneracy between the geometricand Cartesian coordinates, signified by the vanishing of the inverse foliationdensity (i.e., the intersection of the characteristics).

The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy