Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations
Luigi De Rosa, Riccardo Tione
TL;DR
This work proves typicality and sharp energy regularity results for Hölder continuous weak solutions of the incompressible Euler equations in the subcritical regime $\theta<1/3$. By blending a refined convex integration scheme with a Baire-category argument, the authors show that within a natural complete metric space $X_\theta$ of weak solutions, the energy profile $e_v$ generically lies in $C^{\theta^*}$ with $\theta^* = \frac{2\theta}{1-\theta}$ but fails to lie in any $W^{\theta^*+\varepsilon,p}$ on open time intervals, demonstrating highly irregular energy dynamics. They also establish that smooth solutions are nowhere dense in $C^\theta$, indicating the prevalence of wild, energy-nonconserving behavior in this regime. The construction relies on a careful Euler–Reynolds framework, a three-step mollify-glue-perturb strategy, and Mikado-flow-based corrections to enforce the necessary energy and stress constraints, yielding a convergent $C^\theta$-solution with prescribed energy. Overall, the results address conjectures on energy regularity and typify energy behavior for dissipative weak solutions, advancing the understanding of Onsager-critical turbulence in the subcritical regime.
Abstract
This paper is devoted to show a couple of typicality results for weak solutions $v\in C^θ$ 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 $e_v$. We show that those solutions are typical in the Baire category sense. From [8], it is know that the kinetic energy $e_v$ of $θ$-Hölder continuous weak solution $v$ of the Euler equations satisfy $ e_v\in C^{\frac{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 $e_v \in C^{\frac{2θ}{1-θ}}$ but not to $\bigcup_{p\ge 1,\varepsilon>0}W^{\frac{2θ}{1-θ} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_θ$. 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 $C^θ$ 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.