Well-posedness of the MHD boundary layer equations with small initial data in Sobolev space

Xiaolei Dong; Xiaolei Dong

doi:10.3934/era.2024309

Electronic Research Archive

2024, Volume 32, Issue 12: 6618-6640. doi: 10.3934/era.2024309

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Research article

Well-posedness of the MHD boundary layer equations with small initial data in Sobolev space

Xiaolei Dong ^,

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

Received: 10 November 2024 Revised: 02 December 2024 Accepted: 04 December 2024 Published: 09 December 2024

The purpose of this paper is to prove the well-posedness of the 2D magnetohydrodynamic (MHD) boundary layer equations for small initial data in Sobolev space of polynomial weight and low regularity. Our proofs are based on the paralinearization method and an abstract bootstrap argument. We first obtain the systems (3.3)–(3.6) by paralinearizing and symmetrizing the system (1.2). Then, we establish the estimates of the solution in horizontal direction and vertical direction, respectively. Finally, we prove the well-posedness of the 2D MHD boundary layer equations by an abstract bootstrap argument.
- 2D MHD boundary layer equations,
- polynomial weight,
- well-posedness,
- bootstrap argument,
- paralinearization method
Citation: Xiaolei Dong. Well-posedness of the MHD boundary layer equations with small initial data in Sobolev space[J]. Electronic Research Archive, 2024, 32(12): 6618-6640. doi: 10.3934/era.2024309

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Abstract

The purpose of this paper is to prove the well-posedness of the 2D magnetohydrodynamic (MHD) boundary layer equations for small initial data in Sobolev space of polynomial weight and low regularity. Our proofs are based on the paralinearization method and an abstract bootstrap argument. We first obtain the systems (3.3)–(3.6) by paralinearizing and symmetrizing the system (1.2). Then, we establish the estimates of the solution in horizontal direction and vertical direction, respectively. Finally, we prove the well-posedness of the 2D MHD boundary layer equations by an abstract bootstrap argument.

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