On the local and global existence of the Hall equations with fractional Laplacian and related equations

Hantaek Bae; Hantaek Bae

doi:10.3934/nhm.2022021

Networks and Heterogeneous Media

2022, Volume 17, Issue 4: 645-663. doi: 10.3934/nhm.2022021

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On the local and global existence of the Hall equations with fractional Laplacian and related equations

Hantaek Bae ^,

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

Received: 01 November 2021 Revised: 01 February 2022 Published: 20 May 2022
Primary: 35K55; Secondary: 35Q85, 35Q86

In this paper, we deal with the Hall equations with fractional Laplacian
$ B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda B = 0. $
We begin to prove the existence of unique global in time solutions with sufficiently small initial data in $ H^{k} $, $ k>\frac{5}{2} $. By correcting $ \Lambda B $ logarithmically, we then show the existence of unique local in time solutions. We also deal with the two dimensional systems closely related to the $ 2\frac{1}{2} $ dimensional version of the above Hall equations. In this case, we show the existence of unique local and global in time solutions depending on whether the damping term is present or not.
- Hall equations,
- fractional Laplacian,
- well-posedness
Citation: Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations[J]. Networks and Heterogeneous Media, 2022, 17(4): 645-663. doi: 10.3934/nhm.2022021

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Abstract

In this paper, we deal with the Hall equations with fractional Laplacian

$ B_{t}+{\rm{curl}} \left(({\rm{curl}} \;B)\times B\right)+\Lambda B = 0. $

We begin to prove the existence of unique global in time solutions with sufficiently small initial data in $ H^{k} $, $ k>\frac{5}{2} $. By correcting $ \Lambda B $ logarithmically, we then show the existence of unique local in time solutions. We also deal with the two dimensional systems closely related to the $ 2\frac{1}{2} $ dimensional version of the above Hall equations. In this case, we show the existence of unique local and global in time solutions depending on whether the damping term is present or not.

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