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


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”.

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].

[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.

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