TY - JOUR

T1 - Formation of Shocks for 2D Isentropic Compressible Euler

AU - Buckmaster, Tristan

AU - Shkoller, Steve

AU - Vicol, Vlad

N1 - Publisher Copyright:
© 2020 Wiley Periodicals LLC.

PY - 2022/9

Y1 - 2022/9

N2 - 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 (Formula presented.) amplitude, there exist smooth solutions to the Euler equations which form a shock in time (Formula presented.). 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 (Formula presented.) 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.

AB - 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 (Formula presented.) amplitude, there exist smooth solutions to the Euler equations which form a shock in time (Formula presented.). 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 (Formula presented.) 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.

UR - http://www.scopus.com/inward/record.url?scp=85093939173&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85093939173&partnerID=8YFLogxK

U2 - 10.1002/cpa.21956

DO - 10.1002/cpa.21956

M3 - Article

AN - SCOPUS:85093939173

SN - 0010-3640

VL - 75

SP - 2069

EP - 2120

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 9

ER -