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Shock formation for 2d Euler

Mathematical Physics & Probability

Speaker: Steve Shkoller, UC Davis
Location: 1147 MSB
Start time: Wed, Oct 2 2019, 3:10PM

For the 2d Euler equations, we provide an elementary constructive proof of shock formation for smooth solutions, having smooth initial datum of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data prescribed at time $t=t_0$, which has minimum slope $-1/\epsilon$, there exist smooth solutions to the Euler equations which form a shock at time $t=T_*$ with $T_*-t_0=O(\epsilon)$. The blowup time and location can be explicitly computed. The solution at the blowup time has Holder regularity $C^{1/3}$.

Our objective is the construction of solutions with large 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 therefore 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. This is joint work with Tristan Buckmaster and Vlad Vicol.