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author:

Lin, Xue-yun (Lin, Xue-yun.) [1] (Scholars:林雪云) | Liu, Cheng-jie (Liu, Cheng-jie.) [2] | Zhang, Ting (Zhang, Ting.) [3]

Indexed by:

Scopus SCIE

Abstract:

In this paper, we study the well-posedness theory of the magneto-micropolar boundary layer and justify the high Reynolds numbers limit for the magneto-micropolar system with Prandtl boundary layer expansion. If the initial tangential magnetic field is nondegenerate, we obtain the local-in-time existence, uniqueness of solutions for the incompressible magneto-micropolar boundary layer equations with the lower regularity initial data in H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_3$$\end{document}. This work is inspired by the recent progresses by Liu et al. (Commun Pure Appl Math 72(1):63-121, 2019) for 2D MHD boundary layer equations theory in Sobolev spaces without monotonicity. There are some differences between this work and Liu et al. (2019). First, the model has the familar form of the MHD equations but it coupled with the equation of the microrotation field, which essentially describes the motion inside the macrovolume as they undergo mocro-rotational effects represented by the micro-rotational velocity vector. Second, the lack of higher-order boundary condition at y=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} is one of the main difficulties to solve the Prandtl type equations. We present a reconstruction argument for higher-order boundary conditions to fix this technical difficulty. Third, our greatest difference is the application of a stratified energy estimates instead of using space-time differential multi-indices as in Liu et al. (2019), so we can work in a functional framework of lower regularity. Our analysis is based on direct Sobolev spaces and allows us to give an L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>\infty $$\end{document} estimate on the error by multiscale analysis under the assumption that the kinematic, spin viscosity coefficients and the resistivity coefficient are of the same order and the initial tangential magnetic field on the boundary is not degenerate.

Keyword:

35B30 35M33 35Q35 76N20 76W05

Community:

  • [ 1 ] [Lin, Xue-yun]Fuzhou Univ, Key Lab Operat Res & Cybernet Fujian Univ, Coll Math & Comp Sci, Ctr Appl Math Fujian Prov, Fuzhou 350108, Fujian, Peoples R China
  • [ 2 ] [Liu, Cheng-jie]Shanghai Jiao Tong Univ, Sch Math Sci, LSC MOE, Shanghai 200240, Peoples R China
  • [ 3 ] [Liu, Cheng-jie]Shanghai Jiao Tong Univ, Inst Nat Sci, Shanghai 200240, Peoples R China
  • [ 4 ] [Zhang, Ting]Zhejiang Univ, Sch Math Sci, Hangzhou 310027, Zhengjiang, Peoples R China

Reprint 's Address:

  • [Zhang, Ting]Zhejiang Univ, Sch Math Sci, Hangzhou 310027, Zhengjiang, Peoples R China

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Source :

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

ISSN: 0944-2669

Year: 2024

Issue: 3

Volume: 63

2 . 1 0 0

JCR@2023

Cited Count:

WoS CC Cited Count: 2

SCOPUS Cited Count: 2

ESI Highly Cited Papers on the List: 0 Unfold All

WanFang Cited Count:

Chinese Cited Count:

30 Days PV: 1

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