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

Abstract

We show a couple of typicality results for weak solutions $v\in C^{𝜃}$ of the Euler equations, in the case . 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 $e_v\in C^{2𝜃∕(1-𝜃)}$. 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 $e_v\in C^{2𝜃∕(1-𝜃)}$ but for any open $I\subset [0,T]$, 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.