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

De Rosa, Luigi; Tione, Riccardo

doi:10.2140/apde.2022.15.405

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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

DOI: 10.2140/apde.2022.15.405

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 $e_{v} \in C^{2 𝜃 ∕ (1 - 𝜃)}$ . 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 $e_{v} \in C^{2 𝜃 ∕ (1 - 𝜃)}$ but $e_{v} \notin ⋃_{p \geq 1, 𝜀 > 0} W^{2 𝜃 ∕ (1 - 𝜃) + 𝜀, p} (I)$ 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.

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Keywords

incompressible Euler equations, Hölder solutions, energy regularity, convex integration, Baire category

Mathematical Subject Classification

Primary: 35Q31

Secondary: 35D30, 76B03, 26A21

Milestones

Received: 14 April 2020

Accepted: 6 October 2020

Published: 12 April 2022

Authors

Luigi De Rosa
Institute of Mathematics École Polytechnique Fédérale de Lausanne Lausanne Switzerland

Riccardo Tione
Institut für Mathematik Universität Zürich Zürich Switzerland

Vol. 15, No. 2, 2022