Shock formation and vorticity creation for 3d Euler

We analyze the shock formation process for the 3d non-isentropic Euler equations with the ideal gas law, in which sounds waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3,4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the non-isentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces.

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