{"title":"具有大扰动的三维粘性无阻力MHD系统的全局解和渐近行为","authors":"Youyi Zhao","doi":"10.1007/s00021-025-00949-9","DOIUrl":null,"url":null,"abstract":"<div><p>We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by <i>m</i>, is large compared to the perturbations. It was proved by Jiang–Jiang that the highest-order derivatives of the velocity increase with <i>m</i> and the convergence rate of the nonlinear system towards a linearized problem is of <span>\\(m^{-1/2}\\)</span> in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with <i>m</i>, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to <i>m</i> does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or <i>m</i> approaches infinity. Notably, our analysis reveals that the convergence rate in <i>m</i> is faster compared to the finding of Jiang–Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations\",\"authors\":\"Youyi Zhao\",\"doi\":\"10.1007/s00021-025-00949-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by <i>m</i>, is large compared to the perturbations. It was proved by Jiang–Jiang that the highest-order derivatives of the velocity increase with <i>m</i> and the convergence rate of the nonlinear system towards a linearized problem is of <span>\\\\(m^{-1/2}\\\\)</span> in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with <i>m</i>, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to <i>m</i> does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or <i>m</i> approaches infinity. Notably, our analysis reveals that the convergence rate in <i>m</i> is faster compared to the finding of Jiang–Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":\"27 3\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2025-06-07\",\"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/s00021-025-00949-9\",\"RegionNum\":3,\"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":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-025-00949-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations
We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by m, is large compared to the perturbations. It was proved by Jiang–Jiang that the highest-order derivatives of the velocity increase with m and the convergence rate of the nonlinear system towards a linearized problem is of \(m^{-1/2}\) in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with m, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to m does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or m approaches infinity. Notably, our analysis reveals that the convergence rate in m is faster compared to the finding of Jiang–Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.
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.