# Seminario di Equazioni Differenziali e Applicazioni: “Dissipative Hölder solutions to the incompressible Euler equations”

## Martedì 11 Ottobre 2016, ore 14:30 - Aula 1BC45 - Sara Daneri

ARGOMENTI: Seminari

Martedì 11 Ottobre 2016 alle ore 14:30 in Aula 1BC45, Sara Daneri (Universitat Erlangen) terrà un seminario dal titolo “Dissipative Holder solutions to the incompressible Euler equations”.

Abstract
We address the Cauchy problem for the incompressible Euler equations in a periodic setting. Our result aims at showing that, below the Onsager’s critical regularity of Holder $1/3$ in space, the Euler equations are ill-posed, and the kind of non-uniqueness one obtains is an instance of an h-principle phenomenon. Basing on the estimates developed by [1], we prove [2, 3] the existence of infinitely many Hölder $1/5 - \varepsilon$ initial data, each one admitting infinitely many Hölder $1/5 - \varepsilon$ solutions with preassigned total kinetic energy. Moreover, we prove that the set of non-uniqueness initial data so constructed is dense among $L^2$ solenoidal vector fields. This second step requires a new set of ideas which have been recently used to prove the full Onsager’s conjecture, namely non-uniqueness of Euler solutions up to exponent $1/3 - \varepsilon$ [4].

References
[1] Buckmaster, T., De Lellis, C., Isett, P. and Székelyhidi, Jr., L. Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of Math. (2) 182 (2015), no. 1, 127172.
[2] Daneri, S. Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phys. 329 (2014), no. 2, 745786.
[3] Daneri, S. and Székelyhidi, Jr., L. Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations, arXiv:1603.09714.
[4] Isett, P. A proof of Onsager’s conjecture, arXiv:1608.08301.

Rif. Int. M. Bardi, C. Marchi, F. Ancona.