文献类型：期刊文献

英文题名：GLOBAL WELL-POSEDNESS OF 2D INCOMPRESSIBLE MHD EQUATIONS WITHOUT MAGNETIC DIFFUSION

作者：Ding, Shijin[1]; Pan, Ronghua[2]; Zhu, Yi[3]

机构：[1] School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China; [2] School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, United States; [3] Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

年份：2025

外文期刊名：arXiv

收录：EI(收录号：20250226623)

语种：英文

外文关键词：Magnetic levitation - Nonlinear equations - Sobolev spaces - Wave equations

摘要：In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in [38, 48], under the assumption that initial data is close to equilibrium states with nontrivial magnetic field, and the perturbation is small in some suitable spaces, say for instance, the Sobolev spaces with negative exponents. It leads to an interesting open question: Can one establish the global existence of classical solutions without the extra help from Sobolev spaces with negative exponents like its counterparts of ideal MHD (i.e. without viscosity and magnetic diffusion), and fully dissipative MHD (i.e. with both viscosity and magnetic diffusion)? This paper offers an affirmative answer to this question. In fact, we will establish the existence of a global unique solution for initial perturbations being small in H2(2). The key idea is further exploring the structure of system, using dispersive effects of Alfvén waves in the direction which is transversal to the dissipation favorable direction. This motivates our key strategy to treat the wildest nonlinear terms as an artificial linear term. These observations help us to construct some interesting quantities which improves the nonlinearity order for the wildest terms, and to control them by terms with better properties. Copyright ? 2025, The Authors. All rights reserved.