Dissipative Euler Flows and Onsager's Conjecture

By Camillo De Lellis, L. Sz'ekelyhidi. 2012

219 citations recorded by the source corpus.

For any \theta<1/10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Holder-continuous with exponent \theta. A famous conjecture of Onsager states the existence of such dissipative solutions with any Holder exponent \theta<1/3. Our theorem is the first result in this direction.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/7e97b355b8d2df61d27e005214390507453e6b3d

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, methodology): “It follows from [8, 23] that no additional constraints on R̄ exist for L∞ weak solutions, but for Cθ weak solutions this was open and in fact there were indications that constraints might exist [4, 11].”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, methodology): “2 relies on a modification of the convex integration scheme employed by [10, 44, 45] to study the Onsager conjecture for the 3D Euler equations.”
Onsager's Conjecture for Admissible Weak Solutions (2017, background): “Indeed, in [21,22] they were able to give the first examples of, respectively, continuous and Höldercontinuous solutions that dissipate the energy, reaching the threshold exponent 1=10 (see also [7, 28] for constructions of Hölder-continuous weak solutions for other hydrodynamic models such as the I”
h-Principles for the Incompressible Euler Equations (2012, methodology): “5 in [10] for details) that R̊1 ∈ Sd×d 0 and that div R̊1 = ∂tv1 + div (v1 ⊗ v1)+∇ p1.”
On Nonperiodic Euler Flows with Hölder Regularity (2014, methodology): “2 builds upon work of [2,12,14] for prescribing smooth energy profiles the periodic setting and on the organizational framework developed in [23].”
Dissipative Euler Flows with Onsager‐Critical Spatial Regularity (2014, methodology): “In line with previous works [1, 2, 12, 13, 18], the solution will be constructed as a limit of a rather complicated convex integration scheme.”
Anomalous dissipation for 1/5-Hölder Euler flows (2015, methodology): “will be based on the constructions given in [11, 9].”
Convex integration and phenomenologies in turbulence (2019, methodology): “In earlier papers [48, 49, 16, 14, 103, 17], Beltrami waves were used as the building blocks of the convex”