A proof of Onsager’s conjecture. (English) Zbl 1416.35194
The author of this works presents his proof for Onsager’s conjecture, negative direction, which states that when the speed of incompressible liquid \(v \in C_tC_x^\alpha\) instead of \(v\in C^1\), then for every \(\alpha<1/3\) there exist periodic weak solutions of the 3D Euler equation such that the conservation of energy fails. It is shown here that there is a non-zero solution such that \(v\) is identically zero outside a finite time interval that fails to conserve energy for any \(a<1/3\).
Reviewer: Eugene Postnikov (Kursk)
MSC:
| 35Q31 | Euler equations |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35D30 | Weak solutions to PDEs |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76F02 | Fundamentals of turbulence |
| 76F05 | Isotropic turbulence; homogeneous turbulence |
Keywords:
Euler equations; incompressible; weak solutions; energy conservation; energy dissipation; regularity; anomalous dissipation; turbulence; fluid dynamics; convex integration; nonuniquenessReferences:
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