Shock Formation and Vorticity Creation for 3d Euler

Tristan Buckmaster, Steve Shkoller, Vlad Vicol

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound 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 nonisentropic 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.

Original languageEnglish (US)
Pages (from-to)1965-2072
Number of pages108
JournalCommunications on Pure and Applied Mathematics
Issue number9
StatePublished - Sep 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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