Shock Formation for the 3d Euler Equations

Abstract: Together with Tristan Buckmaster and Vlad Vicol, we give a constructive proof of the shock formation process for the 3d Euler equations with vorticity. Specifically, we prove that there exist smooth solutions which form a generic stable shock with explicitly computable blow up time, location, and direction. The cusp-type solution at blow up is obtained by proving stability of a generic blowup profile in modulated self-similar variables. The stability analysis controls the delicate interaction of wave families using pointwise bounds along Lagrangian trajectories, geometric vorticity structure, and high-order energy estimates in Sobolev spaces.

analysis of PDEs

Audience: researchers in the topic

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