Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation

Shihu Li; Wei Liu; Yingchao Xie

doi:10.3934/cpaa.2019113

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2019, Volume 18, Issue 5: 2491-2509. Doi: 10.3934/cpaa.2019113

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Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation

1.
School of Mathematical Sciences, Nankai University, 300071 Tianjin, China
2.
School of Mathematics and Statistics, Jiangsu Normal University, 221116 Xuzhou, China

^* Corresponding author
^* Corresponding author

Received: June 2018

Revised: December 2018

Published online: September 2019

The research of W. Liu is supported by NSFC (No. 11571147, 11822106, 11831014), NSF of Jiangsu Province (No. BK20160004) and the Qing Lan Project; the research of Y. Xie is supported by NSFC (No. 11771187) and PAPD of Jiangsu Higher Education Institutions.

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Abstract

In this paper we establish the Freidlin-Wentzell's large deviation principle for stochastic 3D Leray-$ \alpha $ model with general fractional dissipation and small multiplicative noise. This model is the stochastic 3D Navier-Stokes equations regularized through a smoothing kernel of order $ \theta_1 $ in the nonlinear term and a $ \theta_2 $-fractional Laplacian. The main result generalizes the corresponding LDP result of the classical stochastic 3D Leray-$ \alpha $ model ($ \theta_1 = 1 $, $ \theta_2 = 1 $), and it is also applicable to the stochastic 3D hyperviscous Navier-Stokes equations ($ \theta_1 = 0 $, $ \theta_2\geq\frac{5}{4} $) and stochastic 3D critical Leray-$ \alpha $ model ($ \theta_1 = \frac{1}{4} $, $ \theta_2 = 1 $).

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Mathematics Subject Classification: Primary: 60H15, 60F10; Secondary: 35Q30, 35R11.

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