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
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328742 |
| Date | January 2013 |
| Contributors | Mei, Yu., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (3, 88 leaves) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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