Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations
By Sara Daneri, L. Sz'ekelyhidi. 2016
169 citations recorded by the source corpus.
In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. We prove that the set of Hölder $${1/5 - \varepsilon}$$1/5-ε wild initial data is dense in $${L^{2}}$$L2, where we call an initial datum wild if it admits infinitely many admissible Hölder $${1/5 - \varepsilon}$$1/5-ε weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows in order to show that a general form of the h-principle applies to Hölder-continuous weak solutions of the Euler equations. Our result indicates that in a deterministic theory of three dimensional turbulence the Reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation.
Subfields: Applied Mathematics, Computational Mechanics
View Paper: https://www.semanticscholar.org/paper/282b4c97606442539c1988de98541e6f84bbf4a6
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
- Vlad Vicol
- Maria Colombo
- T. Buckmaster
- Steve Shkoller
- Philip Isett
- Martina Hofmanov'a
- Xiangchan Zhu
- Rongchan Zhu
In Computational Mechanics
- Vlad Vicol
- T. Buckmaster
- Maria Colombo
- Philip Isett
- Steve Shkoller
- Martina Hofmanov'a
- Elia Brué
- Xiangchan Zhu
Citations in the corpus
- Nonuniqueness of Weak Solutions to the SQG Equation (2016, methodology): “We remark that recently, [Ise16] proved the full Onsager’s conjecture for the 3-D Euler equations, employing a novel technique involving gluing exact solutions to the Euler equations, along with the use of Mikado flows, introduced in [DSJ16] as a replacement to Beltrami waves.”
- Convex integration and phenomenologies in turbulence (2019, methodology): “We discuss the papers on which Isett’s construction relies (e.g. the Mikado flows by Daneri-Székelyhidi Jr. [44]) and the main ideas in Isett’s work [102].”
- A Proof of Onsager's Conjecture (2016, methodology): “The first results towards the negative direction of Onsager’s conjecture came in a breakthrough series of papers by De Lellis and Székelyhidi [DLS13b, DLS14] wherein the authors proved that the failure of energy conservation in Conjecture 2 is possible for solutions in L∞t C α x if α 1/10.”
- Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, background): “Let us recall the following result from [9]:”