Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

By T. Buckmaster, Camillo De Lellis, L. Sz'ekelyhidi. 2014

158 citations recorded by the source corpus.

For any ɛ > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space Lt1(Cx1/3−ε) ; namely, x ↦ v (x,t) is ⅓−ε‐Hölder continuous in space at a.e. time t and the integral ∫[ υ(⋅,t) ]1/3−εdt is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class Lt∞(Cx1/3−ε) .© 2016 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/8a154ae3cc1bb089aa1d6e684bea6565bc1e1b95

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

In Computational Mechanics

Citations in the corpus

Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, background): “1) in the regularity class C t L q x ∩ L(2)tH(1) x , for any q ∈ [2, 3).”
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.”
Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “Also, the recent constructions of non-unique energy dissipative solutions of the 3D Euler equations [22, 23, 33, 17] are not known to have an analogue in the presence of dissipation [5], with or without boundaries.”
On Nonperiodic Euler Flows with Hölder Regularity (2014, methodology): “We also note the construction of solutions with compact time support in the class C0 t,x ∩ L(1)t C1/3−ε x by [3,5].”
Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, unknown): “1)) via convex integration methods [47, 52, 2, 17, 1, 3], leading to the resolution of the conjecture by Isett [25, 26], using a key ingredient by Daneri and Székelyhidi Jr.”
Convex integration and phenomenologies in turbulence (2019, methodology): “By taking advantage of this idea and using a delicate bookkeeping scheme, in [17] non-conservative solutions were constructed in the space L(1)tC (1)/3−.”
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.”