#### 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
##### Abstract

We show a couple of typicality results for weak solutions $v\in {C}^{𝜃}$ of the Euler equations, in the case $𝜃<\frac{1}{3}$. It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy ${e}_{v}$. 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 ${e}_{v}$ of a $𝜃$-Hölder continuous weak solution $v$ of the Euler equations satisfies ${e}_{v}\in {C}^{2𝜃∕\left(1-𝜃\right)}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space ${X}_{𝜃}$ that is contained in the space of all ${C}^{𝜃}$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in {X}_{𝜃}$, with ${e}_{v}\in {C}^{2𝜃∕\left(1-𝜃\right)}$ but ${e}_{v}\notin {\bigcup }_{p\ge 1,𝜀>0}{W}^{2𝜃∕\left(1-𝜃\right)+𝜀,p}\left(I\right)$ for any open $I\subset \left[0,T\right]$, are a residual set in ${X}_{𝜃}$. 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 ${C}^{𝜃}$ 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.

##### Keywords
incompressible Euler equations, Hölder solutions, energy regularity, convex integration, Baire category
##### Mathematical Subject Classification
Primary: 35Q31
Secondary: 35D30, 76B03, 26A21