The Global Well-Posedness and Decay Estimates for the 3D Incompressible MHD Equations With Vertical Dissipation in a Strip

被引:5
|
作者
Lin, Hongxia [1 ,2 ]
Suo, Xiaoxiao [3 ,4 ]
Wu, Jiahong [5 ]
机构
[1] Chengdu Univ Technol, Geomath Key Lab Sichuan Prov, Chengdu 610059, Peoples R China
[2] Chengdu Univ Technol, Coll Math & Phys, Chengdu 610059, Peoples R China
[3] Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China
[4] Minist Educ, Lab Math & Complex Syst, Beijing 100875, Peoples R China
[5] Oklahoma State Univ, Dept Math, Stillwater, OK 74078 USA
基金
美国国家科学基金会; 中国国家自然科学基金; 中国博士后科学基金;
关键词
BACKGROUND MAGNETIC-FIELD; MAGNETOHYDRODYNAMICS EQUATIONS; LOCAL EXISTENCE; SYSTEM; REGULARITY; STOKES; UNIQUENESS;
D O I
10.1093/imrn/rnac361
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The three-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation arises in the study of reconnecting plasmas. When the spatial domain is the whole space R-3, the small data global well-posedness remains an extremely challenging open problem. The one-directional dissipation is simply not sufficient to control the nonlinearity in R-3. This paper solves this open problem when the spatial domain is the strip Omega := R-2 x [0, 1] with Dirichlet boundary conditions. By invoking suitable Poincare type inequalities and designing a multi-step scheme to separate the estimates of the horizontal and the vertical derivatives, we are able to establish the global well-posedness in the Sobolev setting H-3 as long as the initial horizontal derivatives are small. We impose no smallness condition on the vertical derivatives of the initial data. Furthermore, the H-3-norm of the solution is shown to decay exponentially in time. This exponential decay is surprising for a system with no horizontal dissipation. This large-time behavior reflects the smoothing and stabilizing phenomenon due to the interaction within the MHD system and with the boundary.
引用
收藏
页码:19115 / 19155
页数:41
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