与磁流体相关的几类方程解的正则性

发布时间：2018-03-16 20:19

本文选题：椭圆型方程　切入点：抛物型方程　出处：《云南民族大学》2012年硕士论文　论文类型：学位论文

【摘要】：随着现代科学技术的发展，有关偏微分方程方面的各种问题已经引起了人们的广泛重视，所以当前一个重要的研究方向，就是应用偏微分方程来研究物理、化学、经济等领域的各种现象。本文考虑了几个与磁流体相关的方程解的正则性问题：其一，考虑一类椭圆型方程边值问题的弱解的内部正则性。其二，考查一类抛物型方程初边值问题的弱解的正则性。其三，研究三维不可压缩的磁流体动力学方程组(MHD),建立了有关磁流体方程组解的两个正则性准则。第1章，叙述与本文研究内容相关的背景知识与发展概况，以及本文的主要工作。第2章，考查在一定条件假设下，通过一些技巧和方法，讨论一类椭圆型方程Lu=f边值问题的弱解的内部正则性。这些技巧和方法包括：截断函数，边界的展平，有限覆盖定理，Sobolev不等式，内插不等式等等。第3章，通过Galerkin逼近法，解的弱收敛，，差商法，Sobolev不等式，内插不等式等技巧和方法，讨论一类抛物型方程初边值问题弱解的正则性。第4章，研究三维的不可压缩的磁流体动力学方程组解的正则性问题，首先通过对速度场的某个方向上的导数项的估计，建立了一个正则性准则。其次对压力场在某个方向上的导数项的有界性估计，建立了另一个正则性准则。
[Abstract]:With the development of modern science and technology, people have paid more and more attention to various problems related to partial differential equations, so an important research direction is to apply partial differential equations to study physics and chemistry. In this paper, we consider the regularity of solutions of some equations related to magnetic fluid. Firstly, we consider the internal regularity of weak solutions of boundary value problems for a class of elliptic equations. The regularity of weak solutions of initial boundary value problems for a class of parabolic equations is investigated. Thirdly, the incompressible three dimensional magnetohydrodynamic equations are studied and two regularity criteria are established for the solutions of the magnetohydrodynamic equations. Chapter 1 describes the background knowledge and development of this paper, and the main work of this paper. In chapter 2, we discuss the internal regularity of weak solutions for Lu=f boundary value problems of elliptic equations under some conditions and methods. These techniques and methods include: truncation functions, boundary flattening, The finite covering theorem includes Sobolev inequality, interpolation inequality and so on. In chapter 3, we discuss the regularity of weak solutions for a class of parabolic equations by means of Galerkin approximation, weak convergence of solutions, differential quotient Sobolev inequality, interpolation inequality and other techniques and methods. In chapter 4, we study the regularity of the solutions of three-dimensional incompressible magnetohydrodynamic equations. First, we estimate the derivative term in a certain direction of the velocity field. A regularity criterion is established and another regularity criterion is established for estimating the boundedness of derivative term of pressure field in a certain direction.
【学位授予单位】：云南民族大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O241.82

【共引文献】