3D shear flows driven by Lévy noise at the boundary

Wai-tong (Louis) Fan; Ali Pakzad; Krutika Tawri; Roger Temam

doi:10.3934/puqr.2023004

Article Contents

2023, Volume 8, Issue 1: 75-94. Doi: 10.3934/puqr.2023004

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3D shear flows driven by Lévy noise at the boundary

1.
Department of Mathematics, Indiana University Bloomington, IN 47405, USA
2.
Department of Mathematics, California State University Northridge, CA 91330, USA
3.
Department of Mathematics, University of California Berkeley, CA 94720, USA

Corresponding author: Roger Temam
Corresponding author: Roger Temam

Received: May 31, 2022

Accepted: November 28, 2022

Early access: February 2023

Published: March 2023

This work was partially supported by the Research Fund of Indiana University.

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Abstract

This paper is concerned with the stochastic incompressible Navier–Stokes equations in a layer of fluid between two flat no-slip boundaries. The fluid is driven by the noisy movement of the bottom boundary, where the noise is given by a Lévy process. After establishing existence of a martingale solution, we use the background flow method to derive an upper bound on the turbulent energy dissipation rate. Our estimate recovers one of the basic scaling ideas of turbulence theory, namely, that the dissipation rate is independent of the viscosity at high Reynolds number.

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Mathematics Subject Classification: 35Q30, 76D05.

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Figure 1. The graph of $ x_3 \mapsto \phi(x_3, \mathbb{X}_{t}(\omega)) $, where $ \delta_t=\delta( \mathbb{X}_{t}(\omega)) $ is the boundary layer thickness

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