doi: 10.3934/cpaa.2021073

Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with transverse magnetic field

1. 

School of Mathematical Sciences, Institute of Natural Sciences, Center of Applied Mathematics, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, China

2. 

Hong Kong Institute for Advanced Study, City University of Hong Kong, Hong Kong, China

3. 

School of Mathematical Sciences and LSC-MOE, Shanghai Jiao Tong University, Shanghai, 200240, China

4. 

Department of Mathematics, City University of Hong Kong, Hong Kong, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  January 2021 Revised  March 2021 Published  April 2021

Fund Project: The research of C.-J. Liu was supported by National Natural Science Foundation of China (Grant No. 11743009, 11801364), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010402), Shanghai Sailing Program (Grant No. 18YF1411700), a startup grant from Shanghai Jiao Tong University (Grant No. WF220441906), and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. F. Xie was supported by National Natural Science Foundation of China No.11831003 and Shanghai Science and Technology Innovation Action Plan No. 20JC1413000. The research of T. Yang was supported by the General Research Fund of Hong Kong Project No 11304419. The first author's research was also supported by Hong Kong Institute for Advanced Study, No. 9360157

This paper is concerned with the vanishing viscosity limit for the incompressible MHD system without magnetic diffusion effect in the half space under the influence of a transverse magnetic field on the boundary. We prove that the solution to the incompressible MHD system is uniformly bounded in both conormal Sobolev norm and $ L^\infty $ norm in a fixed time interval independent of the viscosity coefficient. As a direct consequence, the inviscid limit from the viscous MHD system to the ideal MHD system is established in $ L^\infty $-norm. In addition, the analysis shows that the boundary layer effect is weak because of the transverse magnetic field.

Citation: Cheng-Jie Liu, Feng Xie, Tong Yang. Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with transverse magnetic field. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021073
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