Formation of Shocks for 2D Isentropic Compressible Euler

• Published in: Communications on pure and applied mathematics Vol. 75; no. 9; pp. 2069 - 2120
• Main Authors: Buckmaster, Tristan, Shkoller, Steve, Vicol, Vlad
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.09.2022; John Wiley and Sons, Limited
• Subjects: Compressibility; Euler-Lagrange equation; Transport equations; Vorticity; Waves
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21956

We consider the 2D isentropic compressible Euler equations, with pressure law p(ρ) = (1γ)ργ, with γ > 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1ε, for ε > 0 taken sufficiently small relative to the O1 amplitude, there exist smooth solutions to the Euler equations which form a shock in time Oε. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp‐type, with Hölder C13 regularity. Our objective is the construction of solutions with inherent O1 vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann‐type variables to obtain a system of forced transport equations. Using a transformation to modulated self‐similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self‐similar time, of a smooth blowup profile. © 2020 Wiley Periodicals LLC.