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On the Existence and Temporal Asymptotics of Solutions for the Two and Half Dimensional Hall MHD

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Abstract

In this paper, we deal with the \(2\frac{1}{2}\) dimensional Hall MHD by taking the velocity field u and the magnetic field B of the form \(u(t,x,y)=\left( \nabla ^{\perp }\phi (t,x,y), W(t,x,y)\right) \) and \(B(t,x,y)=\left( \nabla ^{\perp }\psi (t,x,y), Z(t,x,y)\right) \). We begin with the Hall equations (without the effect of the fluid part). In this case, we provide several results such as the long time behavior of weak solutions, weak-strong uniqueness, the existence of local and global in time strong solutions, decay rates of \((\psi ,Z)\), the asymptotic profiles of \((\psi ,Z)\), and the perturbation around harmonic functions. In the presence of the fluid field, the results, by comparison, fall short of the previous ones in the absence of the fluid part and we show the existence of local and global in time strong solutions.

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Acknowledgements

The authors acknowledge the referees for his/her valuable comments and suggestions that highly improved the manuscript. H.B. and R.K acknowledge Prof. Danchin for letting us know his recent paper in [17]. H.B. was supported by NRF-2018R1D1A1B07049015. K.K. was supported by NRF-2019R1A2C1084685 and NRF-20151009350.

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  1. Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, Korea

    Hantaek Bae

  2. Department of Mathematics, Yonsei University, Seoul, Korea

    Kyungkeun Kang

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  1. Hantaek Bae
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Bae, H., Kang, K. On the Existence and Temporal Asymptotics of Solutions for the Two and Half Dimensional Hall MHD. J. Math. Fluid Mech. 25, 24 (2023). https://doi.org/10.1007/s00021-022-00755-7

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