Calculus of Variations and Geometric Measure Theory

D. Albritton - E. Bruè - M. Colombo - C. De Lellis - V. Giri - M. Janisch - H. Kwon

Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishik

created by bruè on 30 Dec 2021
modified on 13 Dec 2022


Accepted Paper

Inserted: 30 dec 2021
Last Updated: 13 dec 2022

Journal: Annals of Mathematics Studies
Year: 2021


In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, with initial vorticity in $L^p$ and body force in $L^1_tL^p_x$, $p<\infty$. His theorem demonstrates, in particular, the sharpness of the Yudovich class. An important intermediate step is the rigorous construction of an unstable vortex, which is of independent physical and mathematical interest. We follow the strategy of Vishik but allow ourselves certain deviations in the proof and substantial deviations in our presentation, which emphasizes the underlying dynamical point of view.