Vol. 15, No. 2, 2022

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Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations

Luigi De Rosa and Riccardo Tione

Vol. 15 (2022), No. 2, 405–428

We show a couple of typicality results for weak solutions v C𝜃 of the Euler equations, in the case 𝜃 < 1 3. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy ev. We show that those solutions are typical in the Baire category sense. From work of Isett (2013, arXiv:1307.0565), it is know that the kinetic energy ev of a 𝜃-Hölder continuous weak solution v of the Euler equations satisfies ev C2𝜃(1𝜃). As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space X𝜃 that is contained in the space of all C𝜃 weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions v X𝜃, with ev C2𝜃(1𝜃) but ev p1,𝜀>0W2𝜃(1𝜃)+𝜀,p(I) for any open I [0,T], are a residual set in X𝜃. This, in particular, partially solves Conjecture 1 of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016), 725–804). We also show that smooth solutions form a nowhere dense set in the space of all the C𝜃 weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one.

incompressible Euler equations, Hölder solutions, energy regularity, convex integration, Baire category
Mathematical Subject Classification
Primary: 35Q31
Secondary: 35D30, 76B03, 26A21
Received: 14 April 2020
Accepted: 6 October 2020
Published: 12 April 2022
Luigi De Rosa
Institute of Mathematics
École Polytechnique Fédérale de Lausanne
Riccardo Tione
Institut für Mathematik
Universität Zürich