Pullback attractors for 2D MHD equations on time-varying domains

Daomin Cao; Xiaoya Song; Chunyou Sun

doi:10.3934/dcds.2021132

Article Contents

2022, Volume 42, Issue 2: 643-677. Doi: 10.3934/dcds.2021132

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Pullback attractors for 2D MHD equations on time-varying domains

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China
3.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

^*Corresponding author: Xiaoya Song (songxy29math@163.com)
^*Corresponding author: Xiaoya Song (songxy29math@163.com)

Received: April 30, 2020

Revised: March 31, 2021

Early access: September 2021

Published: February 2022

D. Cao was supported by NSFC (grant No. 11831009) and Chinese Academy of Sciences by grant QYZDJ-SSW-SYS021; X. Song was supported by China Postdoctoral Science Foundation (grant No. 2020M672566); C. Sun was supported by NSFC (grant No. 11522109, 11871169)

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Abstract

In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any $ n $-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure $ p $ and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.

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Mathematics Subject Classification: Primary: 35Q35, 35B40, 35B41, 76W05.

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