dorsal/arxiv
View SchemaSharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence
| Authors | Peter Constantin, Boris Levant, Edriss S. Titi |
|---|---|
| Categories | |
| ArXiv ID | physics/0611004 |
| URL | https://arxiv.org/abs/physics/0611004 |
| DOI | 10.1007/s10955-007-9317-x |
Abstract
In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for sufficiently small viscosity term $\nu$, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to $\log_\lambda \nu^{-1}$ for all values of the governing parameter $\epsilon$, except for $\epsilon=1$. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, we show different scenarios of the transition to chaos for different parameters regime and for specific forcing. In the ``three-dimensional'' regime of parameters this scenario changes when the parameter $\epsilon$ becomes sufficiently close to 0 or to 1. We also show that in the ``two-dimensional'' regime of parameters for a certain non-zero forcing term the long-time dynamics of the model becomes trivial for any value of the viscosity.
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"abstract": "In this work we derive a lower bounds for the Hausdorff and fractal\ndimensions of the global attractor of the Sabra shell model of turbulence in\ndifferent regimes of parameters. We show that for a particular choice of the\nforcing and for sufficiently small viscosity term $\\nu$, the Sabra shell model\nhas a global attractor of large Hausdorff and fractal dimensions proportional\nto $\\log_\\lambda \\nu^{-1}$ for all values of the governing parameter\n$\\epsilon$, except for $\\epsilon=1$. The obtained lower bounds are sharp,\nmatching the upper bounds for the dimension of the global attractor obtained in\nour previous work. Moreover, we show different scenarios of the transition to\nchaos for different parameters regime and for specific forcing. In the\n``three-dimensional\u0027\u0027 regime of parameters this scenario changes when the\nparameter $\\epsilon$ becomes sufficiently close to 0 or to 1. We also show that\nin the ``two-dimensional\u0027\u0027 regime of parameters for a certain non-zero forcing\nterm the long-time dynamics of the model becomes trivial for any value of the\nviscosity.",
"arxiv_id": "physics/0611004",
"authors": [
"Peter Constantin",
"Boris Levant",
"Edriss S. Titi"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1007/s10955-007-9317-x",
"title": "Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence",
"url": "https://arxiv.org/abs/physics/0611004"
},
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