{"title":"The Landau–Lifshitz–Bloch Equation in the Thin Film","authors":"Yuxun He, Huaqiao Wang","doi":"10.1007/s00021-023-00801-y","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the initial-boundary value problem of the Landau–Lifshitz–Bloch equation in three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by the Maxwell equation, and the exchange field contains exchange constant. Firstly, we establish the existence of weak solutions of the equation by using the Faedo–Galerkin approximation. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends towards zero under appropriate compactness conditions. Moreover, we obtain an equation that can better describe the magnetic dynamic behavior of ferromagnetic films with negligible thickness at high temperatures.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-023-00801-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the initial-boundary value problem of the Landau–Lifshitz–Bloch equation in three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by the Maxwell equation, and the exchange field contains exchange constant. Firstly, we establish the existence of weak solutions of the equation by using the Faedo–Galerkin approximation. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends towards zero under appropriate compactness conditions. Moreover, we obtain an equation that can better describe the magnetic dynamic behavior of ferromagnetic films with negligible thickness at high temperatures.
期刊介绍:
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.