Journal of Mathematical Fluid Mechanics· 2026Q1
Beale–Kato–Majda-Type Continuation Criteria for Hall- and Electron-Magnetohydrodynamics
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- Q1SCImago
- 2026year
Short summary
New Beale–Kato–Majda (BKM) type continuation criteria are established for resistive electron-MHD and Hall-MHD, showing solutions can be extended as long as the time integral of the supremum of the current gradient (for electron-MHD) or vorticity, velocity gradient, and current gradient (for Hall-MHD) remains finite.
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Key points
- Established a BKM-type continuation criterion for resistive electron-MHD based on the time integral of the supremum of the current gradient.
- Developed a similar continuation criterion for resistive Hall-MHD involving the time integral of the supremum of vorticity, velocity gradient, and current gradient.
- These criteria are new for electron/Hall-MHD systems.
- The absence of a classical BKM criterion for non-resistive electron/Hall-MHD underscores the singular nature of the Hall term.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is a Beale–Kato–Majda (BKM) type of condition. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well. We emphasize that the BKM type criterion for electron/Hall-MHD was not previously known, and such criterion for the electron/Hall-MHD without resistivity still remains open. The absence of the classical BKM criterion for the non-resistive electron/Hall-MHD is an additional evidence of the very singular nature of the Hall term, besides what have been discovered on strong ill-posedness of these systems, for instance see [5, 10, 12].
The authors' abstract, as published at the source. Journal of Mathematical Fluid Mechanics, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics