含散度和旋度算子的方程组与端点估计

发布时间：2018-03-13 14:15

本文选题：热电模型　切入点：Beltrami场　出处：《华东师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本文研究几类来自数学物理中的含散度和旋度算子的方程组,得到了一些解的存在性、正则性、Liouville型结果;建立了向量场的Hardy型不等式,在适当的空间之下,得出了一类分数次积分算子端点情形的有界性.在第一章绪论中,我们简要地介绍了本文的研究背景与主要结果.在第二章中我们考察了一个稳态的热电模型.该模型是由一个非线性Maxwell方程组和一个椭圆方程耦合而成.我们对一般边值得到了弱解的存在性与正则性结果,并在小边值情形下给出了唯一性结论.同时,我们也研究了几类相关的模型.第三章由两部分组成.在第一部分中,我们得到了无界区域中的Beltrami流的Liouville型结果,对于无界区域情形,在无穷远提衰减性条件,当区域是星形区域时在边界上切向为零,和以及当区域是星形区域之外时在边界上法向为零,Beltrami场都是平凡的.运用同样的研究技巧,我们还研究了Maxwell和Stokes第一特征值以及第一特征函数的性质.在第二部分中,在外力小的条件下,运用Schauder不动点定理得到Hall-MHD方程组其磁场Holder连续的弱解的存在性.在第四章中,首先我们考虑了在L1和加权L1向量场空间中的分数次积分算子.利用分数次积分算子的有界性结果和Stein-Weiss不等式,我们给出一类Caffarelli-Kohn-Nirenberg不等式的新证明,并建立了新的div-curl不等式.其次,我们对Bourgain和Brezis关于L1向量场的不等式给出了一个初等证明.最后,我们对有界区域中的向量场建立了Hardy型不等式.
[Abstract]:In this paper, we study some kinds of equations with divergence and curl operators in mathematics and physics, obtain the existence of some solutions, regularity and Liouville type results, and establish Hardy type inequalities for vector fields in proper spaces. The boundedness of the extreme case of a class of fractional integral operators is obtained. We briefly introduce the research background and main results of this paper. In chapter 2, we investigate a steady-state thermoelectric model, which consists of a nonlinear Maxwell system coupled with an elliptic equation. The existence and regularity of weak solutions for general edges are worth obtaining. At the same time, we study some related models. Chapter 3 is composed of two parts. In the first part, we obtain the Liouville type results of Beltrami flows in unbounded regions. For the case of unbounded region, the attenuation condition is proposed at infinity. When the region is star-shaped, the tangent direction is zero on the boundary. And the Beltrami field is trivial when the region is outside the star-shaped region. Using the same technique, we also study the properties of the first eigenvalues and the first eigenfunctions of Maxwell and Stokes. In the second part, Under the condition of small external force, Schauder fixed point theorem is used to obtain the existence of the weak solution of the magnetic field Holder continuity for the Hall-MHD equations. In Chapter 4th, Firstly, we consider fractional integral operators in L1 and weighted L1 vector field spaces. By using boundedness of fractional integral operators and Stein-Weiss inequality, we give a new proof of Caffarelli-Kohn-Nirenberg inequality. A new div-curl inequality is established. Secondly, we give an elementary proof of Bourgain's and Brezis's inequalities on L1 vector fields. Finally, we establish Hardy type inequalities for vector fields in bounded domains.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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