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We consider the 3D isentropic compressible Euler equations with the ideal gaslaw. We provide a constructive proof of shock formation from smooth initialdatum of finite energy, with no vacuum regions, with nontrivial vorticitypresent at the shock, and under no symmetry assumptions. We prove that for anopen set of Sobolev-class initial data which are a small $L^ \infty $perturbation of a constant state, there exist smooth solutions to the Eulerequations which form a generic stable shock in finite time. The blow up timeand location can be explicitly computed, and solutions at the blow up time aresmooth except for a single point, where they are of cusp-type with H\"{o}lder$C^ {\frac{1}{3}}$ regularity. Our proof is based on the use of modulatedself-similar variables that are used to enforce a number of constraints on theblow up profile, necessary to establish the stability in self-similar variablesof the generic shock profile.

Formation of point shocks for 3D compressible Euler