Anomalous dissipation and Euler flows
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912032357351424 |
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| author | Burczak, Jan Székelyhidi Jr., László Wu, Bian |
| author_facet | Burczak, Jan Székelyhidi Jr., László Wu, Bian |
| contents | We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\sfrac{1}{3})^-}$ regularity, for an arbitrary initial datum in $\dot H^1 (\T^3)$. This is the first rigorous derivation of zeroth law of scalar turbulence, where the scalar is advected by solution to an equation of hydrodynamics (unforced and deterministic). As a byproduct of our method, we provide a typicality statement for the drift, and recover certain desired properties of turbulence, including a lower bound on scalar variance commensurate with the Richardson pair dispersion hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02934 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anomalous dissipation and Euler flows Burczak, Jan Székelyhidi Jr., László Wu, Bian Analysis of PDEs 76B03, 35Q31 We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\sfrac{1}{3})^-}$ regularity, for an arbitrary initial datum in $\dot H^1 (\T^3)$. This is the first rigorous derivation of zeroth law of scalar turbulence, where the scalar is advected by solution to an equation of hydrodynamics (unforced and deterministic). As a byproduct of our method, we provide a typicality statement for the drift, and recover certain desired properties of turbulence, including a lower bound on scalar variance commensurate with the Richardson pair dispersion hypothesis. |
| title | Anomalous dissipation and Euler flows |
| topic | Analysis of PDEs 76B03, 35Q31 |
| url | https://arxiv.org/abs/2310.02934 |