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

par Camillo De Lellis (IAS, Princeton)

mardi 15 févr. 2022, 15:00 → 16:00 Europe/Paris

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Consider the 2d incompressible Euler equations. A celebrated sixties theorem of Yudovich guarantees that, if the initial data has bounded vorticity, there is a global unique solution to the Cauchy problem which has bounded vorticity at every subsequent time. If the initial vorticity is in $L^p$ for $p<\mathrm{\infty }$, the existence of global in time solutions with uniform $L^p$ bounds on the vorticity follows from a simple regularization scheme, however their uniqueness is a long-standing open problem. Yudovich's uniqueness result extends naturally if we add, to the vorticity equation, a forcing term $f$ which belongs to $L^1_t L^\infty_x$.Intwogroundbreakingworksof2018Vishikshowedthatthelattergeneralizationfailsfor$p<\infty$. His proof consists of two very subtle parts: the analysis of an appropriate spectral problem and a corresponding nonlinear instability proof. I will present the ideas of Vishik's proof and an alternative way of closing the nonlinear part, which we found during a reading seminar at IAS with Dallas Albritton, Elia Brué, Maria Colombo, Vikram Giri, Maximilian Janisch, and Hyunju Kwon. If time allows I will also mention a beautiful development by Dallas Albritton, Elia Brué, and Maria Colombo, who proved a result with a similar flavor for the 3-dimensional incompressible Navier-Stokes equations.