文献类型：期刊文献

英文题名：GLOBAL SOLUTIONS TO 3D INCOMPRESSIBLE MHD SYSTEM WITH DISSIPATION IN ONLY ONE DIRECTION

作者：Lin, Hongxia[1]; Wu, Jiahong[2]; Zhu, Yi[3]

机构：[1] College of Mathematics and Physics, Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu, 610059, China; [2] Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, United States; [3] Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

年份：2022

外文期刊名：arXiv

收录：EI(收录号：20220422719)

语种：英文

外文关键词：Control nonlinearities - Navier Stokes equations

摘要：The small data global well-posedness of the 3D incompressible Navier-Stokes equations in 3 with only one-directional dissipation remains an outstanding open problem. The dissipation in just one direction, say ?12u is simply insufficient in controlling the nonlinearity in the whole space 3. The beautiful work of Paicu and Zhang [39] solved the case when the spatial domain is bounded in the x1-direction by observing a crucial Poincaré type inequality. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, this paper intends to understand the global well-posedness and stability of a special 3D magnetohydrodynamic (MHD) system near a background magnetic field. The spatial domain is 3 and the velocity in this MHD system obeys the 3D Navier-Stokes with only one-directional dissipation. With no Poincaré type inequality, this problem appears to be impossible. By discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several innovative techniques to deal with the derivative loss difficulties, we are able to bound the Navier-Stokes nonlinearity and solve the desired global well-posedness and stability problem.MSC Codes 35A01, 35B35, 35B65, 76D03, 76E25 Copyright ? 2022, The Authors. All rights reserved.