Shock formation and vorticity creation for 3d Euler

2020/06/26 by Buckmaster, Tristan, Shkoller, Steve, Vicol, Vlad · 3 citations
#35L67 #35Q31 #76L05 #76N15 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn)

paper · doi:10.48550/arxiv.2006.14789

Abstract

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.

Cited by

• Singularity formation for the supersonic inward wave of compressible Euler equations with radial symmetry
• Increased lifespan for 3D compressible Euler flows with rotation
• Global well-posedness of the inviscid resistive isentropic compressible MHD system

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• Formation of point shocks for 3D compressible Euler
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• Shock formation for 2D Isentropic Euler equations with self-similar variables