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On Energy Cascades in the Forced 3D Navier–Stokes Equations

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Abstract

We show—in the framework of physical scales and \((K_1,K_2)\)-averages—that Kolmogorov’s dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier–Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.

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Authors and Affiliations

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

    R. Dascaliuc

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Z. Grujić

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  1. R. Dascaliuc
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  2. Z. Grujić
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Corresponding author

Correspondence to Z. Grujić.

Additional information

Communicated by Peter Constantin.

R.D. was supported in part by National Science Foundation Grant DMS-1211413. Z.G. acknowledges support of the Research Council of Norway via Grant 213474/F20 and the NSF via Grant DMS-1212023.

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Dascaliuc, R., Grujić, Z. On Energy Cascades in the Forced 3D Navier–Stokes Equations. J Nonlinear Sci 26, 683–715 (2016). https://doi.org/10.1007/s00332-016-9287-8

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  • DOI: https://doi.org/10.1007/s00332-016-9287-8

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