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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 \(a<1/3\).

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
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References:

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