Small scale creation of the Lagrangian flow in 2d perfect fluids
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909071126298624 |
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| author | Said, Ayman Rimah |
| author_facet | Said, Ayman Rimah |
| contents | In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small scale creation of the Lagrangian flow in 2d perfect fluids Said, Ayman Rimah Analysis of PDEs 35Q31, 35S99 In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman. |
| title | Small scale creation of the Lagrangian flow in 2d perfect fluids |
| topic | Analysis of PDEs 35Q31, 35S99 |
| url | https://arxiv.org/abs/2401.06476 |