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

Chu Shun Yin. / "August 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 55-60). / 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.

This thesis investigates questions of uniqueness in the theory of Compressible flow. First, various uniqueness theorems for compressible flow are reviewed in an expository manner. Roughly, these theorems state that fluid motion in a bounded region Ω = Ω(t) is uniquely determined by its initial data together along with certain boundary conditions. Next, this analysis is extended to magnetohydrodynamic flows and uniqueness theorems are given for a variety of possible cases. The basic question in all these theorems is the determination of appropriate boundary conditions. The proofs are by energy estimates. / Science, Faculty of / Mathematics, Department of / Graduate

Chan Man-fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 40-41). / Abstracts in English and Chinese. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Equation of motion and main results --- p.3 / Chapter 3 --- The space K(x) --- p.9 / Chapter 4 --- Proof of the main theorem --- p.17 / Chapter 4.1 --- The passage to the limit as ε →0 --- p.18 / Chapter 4.2 --- The passage to the limit as δ→ 0 --- p.26 / Chapter 4.3 --- Properties of the solution --- p.29 / Chapter 5 --- Conclusion and comments on future works --- p.36 / Appendix --- p.38 / Bibliography --- p.40

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A thesis submitted to the Faculty of Science, University of the Witwatersrand in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 3 April 2017. / The effect of suction and blowing at the base on the horizontal spreading under gravity of a two-dimensional thin fluid film and an axisymmetric liquid drop is in- vestigated. The velocity vn which describes the suction/injection of fluid at the base is not specified initially. The height of the thin film satisfies a nonlinear diffusion equation with vn as a source term. The Lie group method for the solution of partial differential equations is used to reduce the partial differential equations to ordinary differential equations and to construct group invariant solutions. For a group invari- ant solution to exist, vn must satisfy a first order linear partial differential equation. The two-dimensional spreading of a thin fluid film is first investigated. Two models for vn which give analytical solutions are analysed. In the first model vn is propor- tional to the height of the thin film at that point. The constant of proportionality is β (−∞ < β < ∞). The half-width always increases to infinity as time increases even for suction at the base. The range of β for the thin fluid film approximation to be valid is determined. For all values of suction and a small range of blowing the maximum height of the film tends to zero as time t → ∞. There is a value of β corresponding to blowing for which the maximum height remains constant with the blowing balancing the effect of gravity. For stronger blowing the maximum height tends to infinity algebraically, there is a value of β for which the maximum height tends to infinity exponentially and for stronger blowing, still in the range for which the thin film approximation is valid, the maximum height tends to infinity in a finite time. For blowing the location of a stagnation point on the centre line is determined by solving a cubic equation approximately by a singular perturbation method and then exactly using a trigonometric solution. A dividing streamline passes through the stagnation point which separates the flow into two regions, an upper region consisting of fluid descending due to gravity and a lower region consisting of fluid rising due to blowing. For sufficiently strong blowing the lower region fills the whole of the film. In the second model vn is proportional to the spatial gradient of the height with constant of proportionality β∗ (−∞ < β∗ < ∞). The maximum height always decreases to zero as time increases even for blowing. The range of β∗ for the thin fluid film approximation to be valid is determined. The half-width tends to infinity algebraically for all blowing and a small range of weak suction. There is a value of β∗ corresponding to suction for which the half-width remains constant with the suction balancing the spreading due to gravity. For stronger suction the half-width tends to zero as t → ∞. For even stronger suction there is a value of β∗ for which the half-width tends to zero exponentially and a range of β∗ for which it tends to zero in a finite time but these values lie outside the range for which the thin fluid film approximation is valid. For blowing there is a stagnation point on the centre line at the base. Two dividing streamlines passes through the stagnation point which separate fluid descending due to gravity from fluid rising due to blowing. An approximate analytical solution is derived for the two dividing streamlines. A similar analysis is performed for the axisymmetric spreading of a liquid drop and the results are compared with the two-dimensional spreading of a thin fluid film. Since the two models for vn are still quite general it can be expected that general results found will apply to other models. These include the existence of a divid- ing streamline separating descending and rising fluid for blowing, the existence of a strength of blowing which balances the effect of gravity so the maximum height remains constant and the existence of a strength of suction which balances spreading due to gravity so that the half-width/radius remains constant. / MT 2017

Xie Chunjing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 65-71). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Stability of Shock Waves in Viscous Conservation Laws --- p.10 / Chapter 1.1 --- Cauchy Problem for Scalar Viscous Conservation Laws and Viscous Shock Profiles --- p.10 / Chapter 1.2 --- Stability of Shock Waves by Energy Method --- p.15 / Chapter 1.3 --- Nonlinear Stability of Shock Waves by Spectrum Anal- ysis --- p.20 / Chapter 1.4 --- L1 Stability of Shock Waves in Scalar Viscous Con- servation Laws --- p.26 / Chapter 2 --- Propagation of a Viscous Shock in Bounded Domain and Half Space --- p.35 / Chapter 2.1 --- Slow Motion of a Viscous Shock in Bounded Domain --- p.36 / Chapter 2.1.1 --- Steady Problem and Projection Method --- p.36 / Chapter 2.1.2 --- Projection Method for Time-Dependent Prob- lem --- p.40 / Chapter 2.1.3 --- Super-Sensitivity of Boundary Conditions --- p.43 / Chapter 2.1.4 --- WKB Transformation Method --- p.45 / Chapter 2.2 --- Propagation of a Stationary Shock in Half Space --- p.50 / Chapter 2.2.1 --- Asymptotic Analysis --- p.50 / Chapter 2.2.2 --- Pointwise Estimate --- p.51 / Chapter 3 --- Nonlinear Stability of Viscous Transonic Flow Through a Nozzle --- p.58 / Chapter 3.1 --- Matched Asymptotic Analysis --- p.58 / Bibliography --- p.65