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Abstract The three-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation arises in the study of reconnecting plasmas. When the spatial domain is the whole space $$\mathbb R^3$$, the small data global well-posedness remains an extremely challenging open problem. The one-directional dissipation is simply not sufficient to control the nonlinearity in $$\mathbb R^3$$. This paper solves this open problem when the spatial domain is the strip $$\Omega := \mathbb R^2\times [0,1]$$ with Dirichlet boundary conditions. By invoking suitable Poincaré type inequalities and designing a multi-step scheme to separate the estimates of the horizontal and the vertical derivatives, we are able to establish the global well-posedness in the Sobolev setting $H^3$ as long as the initial horizontal derivatives are small. We impose no smallness condition on the vertical derivatives of the initial data. Furthermore, the $H^3$-norm of the solution is shown to decay exponentially in time. This exponential decay is surprising for a system with no horizontal dissipation. This large-time behavior reflects the smoothing and stabilizing phenomenon due to the interaction within the MHD system and with the boundary.
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Blowup of Dyadic MHD Models with Forward Energy Cascade
Abstract A particular type of dyadic model for the magnetohydrodynamics (MHD) with dominating forward energy cascade is studied. The model includes intermittency dimension $$\delta $$ in the nonlinear scales. It is shown that when $$\delta $$ is small, positive solution with large initial data for either the dyadic MHD or the dyadic Hall MHD model develops blowup in finite time. Moreover, for a class of positive initial data with large velocity components and small magnetic field components, we prove that there exists a positive solution that blows up at a finite time.
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- Award ID(s):
- 2009422
- PAR ID:
- 10477827
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 24
- ISSN:
- 1073-7928
- Format(s):
- Medium: X Size: p. 21805-21837
- Size(s):
- p. 21805-21837
- Sponsoring Org:
- National Science Foundation
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