On the Endpoint Regularity in Onsager's Conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866911964229271552 |
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| author | Isett, Philip |
| author_facet | Isett, Philip |
| contents | Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with Hölder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$.
Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of \cite{IOnonpd} to modify the convex integration scheme.
We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their singular supports have space-time Lebesgue measure zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1706_01549 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the Endpoint Regularity in Onsager's Conjecture Isett, Philip Analysis of PDEs Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with Hölder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their singular supports have space-time Lebesgue measure zero. |
| title | On the Endpoint Regularity in Onsager's Conjecture |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1706.01549 |