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Exact solutions to the three-dimensional incompressible magnetohydrodynamics equations without viscosity

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Abstract

The incompressible magnetohydrodynamics (MHD) equations have been widely used to describe many physical systems in geophysics, astrophysics, cosmology and engineering. In this paper, we construct two types of exact global solutions with elementary functions to the three-dimensional incompressible MHD equations without viscosity. The first type of solutions is expressed by exponential functions that are nonstationary and correspond to a generalization of the well-known Arnold–Beltrami–Childress (ABC) flow for the three-dimensional MHD system. The second type of solutions has rational forms that are rotational and are similar to the ABC flow. Both types of solutions can exhibit interesting local behaviors with infinite energy. Under special parameter values, these solutions can be reduced to those of the incompressible Euler equations.

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Acknowledgements

This paper was partially supported by the Small Grant for Academic Staff (MIT/SGA03/2019-20) from the Department of Mathematics and Information Technology, the Education University of Hong Kong.

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Authors and Affiliations

  1. Department of Mathematics, Lishui University, Lishui, 323000, People’s Republic of China

    Junchao Chen

  2. Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo ** Road, Tai Po, New Territories, Hong Kong

    Manwai Yuen

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  1. Junchao Chen
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Correspondence to Manwai Yuen.

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Chen, J., Yuen, M. Exact solutions to the three-dimensional incompressible magnetohydrodynamics equations without viscosity. Nonlinear Dyn 106, 919–926 (2021). https://doi.org/10.1007/s11071-021-06881-7

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