A Proof of Onsager's Conjecture
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arXiv
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| Format: | Preprint |
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2016
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| _version_ | 1866914881598390272 |
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| author | Isett, Philip |
| author_facet | Isett, Philip |
| contents | For any $α< 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^α$ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $α> 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $α= 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^α$. The previous best results were solutions in the class $C_tC_x^α$ for $α< 1/5$, due to the author, and in the class $L_t^1 C_x^α$ for $α< 1/3$ due to Buckmaster, De Lellis and Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Székelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author's previous work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1608_08301 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A Proof of Onsager's Conjecture Isett, Philip Analysis of PDEs Mathematical Physics Fluid Dynamics For any $α< 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^α$ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $α> 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $α= 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^α$. The previous best results were solutions in the class $C_tC_x^α$ for $α< 1/5$, due to the author, and in the class $L_t^1 C_x^α$ for $α< 1/3$ due to Buckmaster, De Lellis and Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Székelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author's previous work. |
| title | A Proof of Onsager's Conjecture |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics |
| url | https://arxiv.org/abs/1608.08301 |