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A Geometric Cascade for the Spectral Approximation of the Navier-Stokes Equations

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Probability and Partial Differential Equations in Modern Applied Mathematics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 140))

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Abstract

We explain some ideas contained in some recent papers, concerning the statistical long time behaviour of the spectral approximation of the Navier-Stokes equations, driven by a highly degenerate white noise forcing. The analysis highlights that the ergodicity of the stochastic system is obtained by a geometric cascade. Such a cascade can be interpreted as the mathematical counterpart of th e energy cascade, a well-known phenomenon in turbulence.

In the second part of the paper, we analyse the results of some numerical simulations. Such simulations give a hint on the behaviour of the system in the case where the white noise forcing fails the assumptions of the main theorem.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Universita di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italia

    M. Romito

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  1. M. Romito
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Editors and Affiliations

  1. Department of Mathematics, Oregon State University, Covallis, OR, 97331, USA

    Edward C. Waymire

  2. Department of Applied Mathematics, Illionis Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

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Romito, M. (2005). A Geometric Cascade for the Spectral Approximation of the Navier-Stokes Equations. In: Waymire, E.C., Duan, J. (eds) Probability and Partial Differential Equations in Modern Applied Mathematics. The IMA Volumes in Mathematics and its Applications, vol 140. Springer, New York, NY. https://doi.org/10.1007/978-0-387-29371-4_13

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