by Cheng Ting. / "August 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 103-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

本論文討論了粘性依賴於密度的等熵可壓縮MHD 方程具有一般數據的整體通定性問題。它是關於可壓縮Navier-Stokes 方程整體通定性的相應推廣(參見38，48 ，73) 。具體而言，我們得到了以下新的結果。 / I:我們證明了具有形如 μ=const. >0, λ(ρ)= ρ^β, β>4/3的粘性係數的等熵可壓縮MHD 方程在二維週期域上整體光滑解的存在唯一性。其中初始密度可以包含真空，並且初始教據可以任意大。 / II 對於全空間上的初值問題，無論初始密度具有具空或者非具空遠場，在具有和I中粘性係教相同的限制條件下，我們都能證明其整體光滑解的存在唯一性。 / 這些結果基於磁場H 的任意的 Lt^∞ Lx^p先驗估計和H▽H ủ的L¹ 估計的一個但等式，它們是處理搞合的磁場和速度場的關鍵。我們充分利用了這兩點觀察和(42 ， 46 ， 73) 中針對NavierStokes 方程提出的框架獲得了密度的一致上界并進一步得到了高階估計。 / In this thesis, we study the global well-posedness of solutions to the compressible MHD equations with density-dependent viscosity coefficients with general initial data. These results are the generalization of the corresponding ones for the compressible Navier-Stokes equations [42, 56, 83]. We obtain the following new results. / I.We show that the global existence and uniqueness of classical solutions to the isentropic compressible MHD equations with the viscosity coefficients satisfying μ=const. >0, λ(ρ)= ρ^β, β>4/3 on the two-dimensional torus. The initial density is allowed to vanish and the initial data can be arbitrary large. / II. We establish the same result for the Cauchy problem of the compressible MHD equations under the same assumptions, whenever the initial density with vacuum or nonvacuum as far fields. / These results based on the arbitrary Lt^∞ Lx^p a priori estimates of magnetic field H and a new identity for the L¹ estimates of H▽H ủ which are crucial to deal with the strongly coupled magnetic field with the velocity field. We take full advantage of these two key observations and framework proposed in [42, 56, 83] for the compressible Navier-Stokes equations to obtain the uniform upper bound of the density and further derive higher order estimates. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Mei, Yu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 79-88). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 2 --- Global Classical Solutions to the 2D Compressible MHD Equations with Large Data and Vacuum on T² --- p.13 / Chapter 2.1 --- Main Results --- p.14 / Chapter 2.2 --- Preliminaries --- p.15 / Chapter 2.3 --- A priori estimates --- p.18 / Chapter 2.4 --- Higher order estimates --- p.36 / Chapter 2.5 --- Proof of the Theorem --- p.54 / Chapter 3 --- Global Classical Solutions to the 2D Compressible MHD Equations with Large Data and Vacuum on R² --- p.57 / Chapter 3.1 --- Main Results --- p.58 / Chapter 3.2 --- Preliminaries --- p.60 / Chapter 3.3 --- A priori estimates --- p.64 / Chapter 3.4 --- Proof of main results --- p.73 / Chapter 4 --- Discussions and Future Work --- p.77 / Bibliography --- p.78

In this thesis we discuss the theory of symmetric MHD equilibria with anisotropic pressure. More specifically, we focus on gyrotropic pressures, where the pressure tensor can be split into components along and across the magnetic field. We first explore 2D solutions, which can be found using total field type formalisms. These formalisms rely on treating quantities as functions of both the magnetic flux function and the magnetic field strength, and reduce the equilibrium equations to a single Grad-Shafranov equation that can be solved to find the magnetic flux function. However, these formalisms are not appropriate when one includes a shear field component of magnetic flux, since they lead to a set of equations which are implicitly coupled. Therefore, in order to solve the equilibrium problem with a magnetic shear field component, we introduce the poloidal formalism. This new formalism considers quantities as functions of the poloidal magnetic field strength (instead of the total magnetic field strength), and yields a set of two equations which are not coupled, and can be solved to find the magnetic flux function and the shear field. There are some situations where the poloidal formalism is difficult to use, however, such as in rotationally symmetric systems. Thus we require a further formalism, which we call the combined approach, which allows a more general use of the poloidal formalism. One finds that the combined formalism leads to multi-valued functions, which must be dealt with appropriately. Finally, we present some numerical examples of MHD equilibria, which have been found using each of the three formalisms mentioned above.