【作者】 陈伟；

【导师】 肖跃龙；

【作者基本信息】 湘潭大学 ， 基础数学， 2010， 硕士

【摘要】 本篇硕士学位论文主要是应用Galerkin方法和Hodge分解理论研究满足一类特殊Navier ?Slip型边界条件的MHD ?α方程.我们获得了对任意初值H1解的整体存在性,并且还讨论了弱解的正则性,在本论文的最后我们还讨论了MHD ?α方程的弱解当α→0+时收敛于MHD方程的解.本文分五章叙述.第一章介绍本文所讨论的主要问题和研究思想,以及国内外研究的动态.第二章给出了本文的记号约定和基本概念,以及我们在论文中将要用到的重要方法和经典结果.第三章讨论MHD ?α方程.在第一节,我们应用Hodge分解理论证明非线性项与Navier?Slip型边值条件的相容性.在第二节,我们给出了一些先验估计.在第三节,我们主要讨论了在三维情况下,运用Galerkin方法证明了弱解的存在性.在第四节,我们主要讨论了解的正则性.在第五节,我们主要讨论了MHD?α方程当α→0+时的收敛性问题.第四章总结我们所做的工作,指出以后还可以继续研究的问题.更多还原

【Abstract】 In this thesis, we study the Magnetohydrodynamic-αmodels in a bounded smoothdomain of R3,with slip boundary condition by Galerkin method and Hodge Decomposi-tion theory. We obtain the global weak solution of H1 solution for arbitrary initial dataand the regularity of the weak solution. Finally we discuss that as the length scaleαtendsto zero, a subsequence of solutions of the MHD-αequations converges to a certain so-lution of the three-dimensional MHD equations. In the light of contents, this thesis isdivided into four chapters.The first chapter is to introduce the main problem that we are concerned and thedevelopment of the problem in the in domestic and foreign.In the second chapter, We introduce notations, some important theorems and severalclassical results that will be used in the following proofs.In the third chapter, We consider the MHD-αequations. In§3.1 We show thenonlinearity in MHD-αequations to match with the boundary condition smoothly;In§3.2 we prove some priori estimates; In§3.3 we discuss the existence of H1 solution forarbitrary initial data; In§3.4 we discuss the regularity of the above weak solution; In§3.5 we discuss that as the length scaleαtends to zero, a subsequence of solutions ofthe MHD-αequations converges to a certain solution of the three-dimensional MHDequations.The fourth chapter summaries the work done by us and points out the continue workwe may study.更多还原