Onsager's Conjecture for Admissible Weak Solutions

By T. Buckmaster, Camillo De Lellis, L. Sz'ekelyhidi, Vlad Vicol. Communications on Pure and Applied Mathematics, 2017

352 citations recorded by the source corpus.

We prove that given any $\beta<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v \in C^{\beta}([0,T]\times \mathbb{T}^3)$, with $e(t) = \int_{\mathbb{T}^3} |v(x,t)|^2 dx$ for all $t\in [0,T]$. Moreover, we show that a suitable $h$-principle holds in the regularity class $C^\beta_{t,x}$, for any $\beta<1/3$. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/bad1e66d00b17a927dd5d7d6dca122ec2052946c

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Citations in the corpus

Convex integration and phenomenologies in turbulence (2019, methodology): “In later papers [44, 102, 18, 101], Mikado waves were employed (see Section 6.”
Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, neutral): “3 shows that the nonconservative weak solutions to the Euler equations obtained in [25, 4] arise in the vanishing viscosity limit of weak solutions to the Navier-Stokes equations.”
Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data (2018, background): “For details see [15], [9] for one side of the conjecture and [14], [1], [2], [3], [4], [11], [18] and [19] for the other side.”
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2021, methodology): “Their proof introduces the important tool of intermittency and builds on the fundamental works for the Euler equations and the proof of the Onsager conjecture [31, 9] (see [11, 10] for further connections to fluid turbulence).”
Beyond Kolmogorov cascades (2019, methodology): “Some work is needed to make the link between these observations, and abstract analytical or numerical construction of such solutions using e.g. the mikado technique of Buckmaster et al. (2018).”