On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

Yat Tin Chow; Ali Pakzad

doi:10.3934/dcdsb.2021270

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2022, Volume 27, Issue 9: 5181-5203. Doi: 10.3934/dcdsb.2021270

This issue Previous Article Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments Next Article Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

Yat Tin Chow^1, and
Ali Pakzad^2, ,

1.
Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA
2.
Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA

^*Corresponding author: Ali Pakzad
^*Corresponding author: Ali Pakzad

We dedicate this paper to the late Charlie Doering. He was an inspiring scientist and a wonderful person.

Received: March 31, 2021

Revised: August 31, 2021

Early access: November 2021

Published: September 2022

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We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.

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Mathematics Subject Classification: Primary: 76F02; Secondary: 35Q30, 60H30.

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