本文選題：熱電模型　切入點(diǎn)：Beltrami場　出處：《華東師范大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：本文研究幾類來自數(shù)學(xué)物理中的含散度和旋度算子的方程組,得到了一些解的存在性、正則性、Liouville型結(jié)果;建立了向量場的Hardy型不等式,在適當(dāng)?shù)目臻g之下,得出了一類分?jǐn)?shù)次積分算子端點(diǎn)情形的有界性.在第一章緒論中,我們簡要地介紹了本文的研究背景與主要結(jié)果.在第二章中我們考察了一個穩(wěn)態(tài)的熱電模型.該模型是由一個非線性Maxwell方程組和一個橢圓方程耦合而成.我們對一般邊值得到了弱解的存在性與正則性結(jié)果,并在小邊值情形下給出了唯一性結(jié)論.同時,我們也研究了幾類相關(guān)的模型.第三章由兩部分組成.在第一部分中,我們得到了無界區(qū)域中的Beltrami流的Liouville型結(jié)果,對于無界區(qū)域情形,在無窮遠(yuǎn)提衰減性條件,當(dāng)區(qū)域是星形區(qū)域時在邊界上切向?yàn)榱?和以及當(dāng)區(qū)域是星形區(qū)域之外時在邊界上法向?yàn)榱?Beltrami場都是平凡的.運(yùn)用同樣的研究技巧,我們還研究了Maxwell和Stokes第一特征值以及第一特征函數(shù)的性質(zhì).在第二部分中,在外力小的條件下,運(yùn)用Schauder不動點(diǎn)定理得到Hall-MHD方程組其磁場Holder連續(xù)的弱解的存在性.在第四章中,首先我們考慮了在L1和加權(quán)L1向量場空間中的分?jǐn)?shù)次積分算子.利用分?jǐn)?shù)次積分算子的有界性結(jié)果和Stein-Weiss不等式,我們給出一類Caffarelli-Kohn-Nirenberg不等式的新證明,并建立了新的div-curl不等式.其次,我們對Bourgain和Brezis關(guān)于L1向量場的不等式給出了一個初等證明.最后,我們對有界區(qū)域中的向量場建立了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.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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