{"title":"薄三维域中各向异性 MHD 方程的水平磁性基元方程近似值","authors":"Jie Zhang and Wenjun Liu","doi":"10.1088/1361-6544/ad5131","DOIUrl":null,"url":null,"abstract":"In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"9 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain\",\"authors\":\"Jie Zhang and Wenjun Liu\",\"doi\":\"10.1088/1361-6544/ad5131\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .\",\"PeriodicalId\":54715,\"journal\":{\"name\":\"Nonlinearity\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":1.6000,\"publicationDate\":\"2024-06-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinearity\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1088/1361-6544/ad5131\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinearity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6544/ad5131","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain
In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .
期刊介绍:
Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest.
Subject coverage:
The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal.
Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena.
Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.