Averaging principle for stochastic 3D fractional Leray-α model with a fast oscillation

Abstract

This article investigates the multiscale stochastic 3D fractional Leray-α model. By using the Khasminskii technique, we establish the strong average principle for stochastic 3D fractional Leray-α model with a fast oscillation. This model is the stochastic 3D Navier–Stokes equations regularized through a smoothing kernel of order θ₁ in the nonlinear term and a θ₂-fractional Laplacian. The main result is applicable to the classical stochastic 3D Leray-α model (${\theta }_1=1,{\theta }_2=1$), stochastic 3D hyperviscous Navier–Stokes equations (${\theta }_1=0,{\theta }_2\ge \frac{5}{4}$) and stochastic 3D critical Leray-α model (${\theta }_1=\frac{1}{4},{\theta }_2=1$).

Keywords:

• Stochastic Leray-α model
• averaging principle
• multi-scaling limit
• strong convergence
• fractional Laplacian

MATHEMATICS SUBJECT CLASSIFICATION:

• 60H15
• 35Q10
• 70K70

Acknowledgment

The authors would like to thank the referee for his/her very valuable suggestions and useful remarks which helped to considerably improve the quality of the article.

Additional information

Funding

This work is supported by the NNSF of China (No. 11771187, 11931004) and the PAPD of Jiangsu Higher Education Institutions.