# On the Ideal Magnetohydrodynamics in Three-Dimensional Thin Domains: Well-Posedness and Asymptotics

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## Abstract

- (i)
We construct the global solutions (Alfvén waves) to MHD in the thin domain \(\Omega _\delta \) with \(\delta >0\). In addition, the uniform energy estimates are obtained with respect to the parameter \(\delta \).

- (ii)
We justify the asymptotics of the MHD equations from the thin domain \(\Omega _\delta \) to the plane \(\mathop {{\mathbb {R}}}\nolimits ^2\). More precisely, we prove that the 3D Alfvén waves in \(\Omega _\delta \) will converge to the 2D Alfvén waves in \(\mathop {{\mathbb {R}}}\nolimits ^2\) in the limit that \(\delta \) goes to zero provided that the leading part (horizontal component) of the initial 3D Alfvén waves on \(\Omega _\delta \) converges to a 2D vector filed on \(\mathop {{\mathbb {R}}}\nolimits ^2\) as \(\delta \rightarrow 0\). This shows that Alfvén waves propagating along the horizontal direction of the 3D thin domain \(\Omega _\delta \) are stable and can be approximated by the 2D Alfvén waves when \(\delta \) is sufficiently small. Moreover, the control of the 2D Alfvén waves can be obtained from the control of 3D Alfvén waves in the thin domain \(\Omega _\delta \) with aid of the uniform bounds.

## Notes

### Acknowledgements

The author would like to thank the anonymous referees for many helpful suggestions and comments. Part of this work was done when the author was working in LSEC, Academy of Mathematics and Systems Science, CAS. The author is partially supported by NSF of China under Grant 11671383 and by an innovation grant from the National Center for Mathematics and Interdisciplinary Science.

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