American Institute of Mathematical Sciences

December  2021, 29(6): 4243-4255. doi: 10.3934/era.2021082

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

 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

* Corresponding author: Wei-Xi Li

Received  June 2021 Published  December 2021 Early access  October 2021

Fund Project: The work was supported by NSF of China(Nos. 11961160716, 11871054, 11771342), the Natural Science Foundation of Hubei Province (No. 2019CFA007) and the Fundamental Research Funds for the Central Universities(No. 2042020kf0210)

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.

Citation: Wei-Xi Li, Rui Xu. Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity. Electronic Research Archive, 2021, 29 (6) : 4243-4255. doi: 10.3934/era.2021082
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