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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> 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.</p>

The aim of this paper is to consider the asymptotic dynamics of solutions to 2D MHD equations when the external forces contain some hereditary characteristics. First, we establish, respectively, the well-posedness of strong solutions and weak solutions; then, the process Ũ(⋅,⋅) generated by the weak solutions is constructed in MH2(=H×LH2); and finally, we analyze the long-time behavior of the weak solutions by proving the existence of a compact pullback attractor.

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