Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

Wei-Xi Li; Rui Xu; Wei-Xi Li; Rui Xu

doi:10.3934/era.2021082

Electronic Research Archive

2021, Volume 29, Issue 6: 4243-4255. doi: 10.3934/era.2021082

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Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

Wei-Xi Li ^{1
,
,},
Rui Xu ^{2
,}

1.
School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, 430072 Wuhan, China
2.
School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China

Received: 01 June 2021 Published: 08 October 2021
Primary: 35Q35, 76D10; Secondary: 76W05

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.
- MHD boundary layer,
- well-posedness,
- Sobolev space
Citation: Wei-Xi Li, Rui Xu. Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity[J]. Electronic Research Archive, 2021, 29(6): 4243-4255. doi: 10.3934/era.2021082

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Abstract

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.

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