Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime

In this paper, we investigate the higher regularity and asymptotic behavior for the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Due to the degeneracy of horizontal velocity near boundary, the higher regularity of solution is a tricky problem. By constructing suitable approximated syst...

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Bibliographic Details
Published in:Journal of Differential Equations 2024-03, Vol.386, p.294-367
Main Authors: Gao, Jincheng, Li, Minling, Yao, Zheng-an
Format: Article
Language:English
Subjects:
Asymptotic behavior
Higher regularity
Magnetic Prandtl equation
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Summary:In this paper, we investigate the higher regularity and asymptotic behavior for the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Due to the degeneracy of horizontal velocity near boundary, the higher regularity of solution is a tricky problem. By constructing suitable approximated system and establishing closed energy estimate for a good quantity (called “quotient” in [19]), our first result is to solve this higher regularity problem. Furthermore, we show the global well-posedness and global-in-x asymptotic behavior when the initial data is small perturbation of the classical Hartmann layer in Sobolev space. By using the energy method to establish closed estimate for the quotient, we overcome the difficulty arising from the degeneracy of horizontal velocity near boundary. Due to the damping effect, we also point out that this global solution will converge to the equilibrium state (called Hartmann layer) with exponent decay rate.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.12.030