Sharp energy regularity and typicality results for H\"older solutions of incompressible Euler equations

Abstract

This paper is devoted to show a couple of typicality results for weak solutions $v\in C^\theta$ of the Euler equations, in the case $\theta<1/3$. 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 [8], it is know that the kinetic energy $e_v$ of $\theta$-H\"older continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{2\theta}{1-\theta}}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_\theta$, that is contained in the space of all $C^\theta$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_\theta$ with $e_v \in C^{\frac{2\theta}{1-\theta}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2\theta}{1-\theta} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_\theta$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^\theta$ 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.