Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

Xuhui Peng; Jianhua Huang; Yan Zheng

doi:10.3934/cpaa.2020204

Article Contents

2020, Volume 19, Issue 9: 4479-4506. Doi: 10.3934/cpaa.2020204

This issue Previous Article Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system Next Article Motion of interfaces for a damped hyperbolic Allen–Cahn equation

Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

1.
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China, Key Laboratory of Applied Statistics and Data Science, Hunan Normal University, College of Hunan Province, Changsha, 410081, China
2.
College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

^* Corresponding author
^* Corresponding author

Received: November 30, 2019

Revised: February 29, 2020

Published: June 2020

The first author was supported by Hunan Provincial Natural Science Foundation of China (No. 2019JJ50377), NSFC (No.11871476) and the Construct Program of the Key Discipline in Hunan Province. The last two were supported by the NSF of China(No.11771449)

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Abstract

We establish the existence, uniqueness and exponential attraction properties of an invariant measure for the MHD equations with degenerate stochastic forcing acting only in the magnetic equation. The central challenge is to establish time asymptotic smoothing properties of the associated Markovian semigroup corresponding to this system. Towards this aim we take full advantage of the characteristics of the advective structure to discover a novel Hörmander-type condition which only allows for several noises in the magnetic direction.

Keywords:

Exponential mixing,
Malliavin calculus,
ergodic,
fractional Magneto-Hydrodynamic equations.

Mathematics Subject Classification: Primary:60H15, 60H07.

Citation:

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Figure 1. An illustration of how the new directions generated from the existing directions via the iterations of the chain of bracket computations. In this figure, $ m,m'\in \{0,1\}, \ell\in \mathcal{Z}_0. $ Solid arrows mean that the new function is generated from a Lie bracket, with the type of bracket indicated above the arrow. Dashed arrows with green color signify that the new element is generated as a linear combination of elements from the previous position. The dotted arrows with red color shows that the process is iterative. The doubled arrow with yellow color (→) shows that $ k\pm \ell $ is a element belongs to $ \mathcal{Z}_{2n+1} $ or $ \mathcal{Z}_{2n+2} $ actually.

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Figure 2. An illustration of the structure of the lemmas that leads to the proof of Proposition 2. The solid arrows indicate that if one term is "small" then the other one "small" on a set of large measure(displayed up or left of the arrow), where the meaning of "smallness" is made precise in each lemma. The dashed arrows shows that the process is iterative. In this figure, $ m,m'\in \{0,1\}, \ell\in \mathcal{Z}_0. $ One may notice the close relationship between Figure 1 and Figure 1.

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