Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

Vanishing viscosity limit for compressible magnetohydrodynamic equations with transverse background magnetic field

Pages: 1363 – 1392

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n5.a9

Authors

Xiufang Cui (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Shengxin Li (School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China)

Feng Xie (School of Mathematical Sciences, CMA-Shanghai, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China)

Abstract

We are concerned with the uniform regularity estimates and vanishing viscosity limit of the solution to two-dimensional viscous compressible magnetohydrodynamic (MHD) equations with transverse background magnetic field. When the magnetic field is assumed to be transverse to the boundary and the tangential component of magnetic field satisfies zero Neumann boundary condition, even though the the no-slip velocity boundary condition is imposed, the uniform regularity estimates of the solution and its derivatives still can be achieved in suitable conormal Sobolev spaces in the half plane $\mathbb{R}^2_+$, and then the vanishing viscosity limit is justified in $L^\infty$ sense based on these uniform regularity estimates and some compactness arguments. At the same time, together with $\href{https://dx.doi.org/10.1088/1361-6544/aca511}{[\textrm{X. Cui, S. Li, and F. Xie, } Nonlinearity, 36(1):354–400, 2022]}$, our results show that the transverse background magnetic field can prevent the strong boundary layer from occurring for compressible magnetohydrodynamics whether there is magnetic diffusion or not.

Keywords

magnetohydrodynamics, initial-boundary value problem, inviscid limit, conormal Sobolev space

2010 Mathematics Subject Classification

35M13, 35Q35, 76N10, 76W05

Received 21 April 2022

Received revised 21 September 2022

Accepted 18 October 2022

Published 30 August 2023