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研究生： 蘇承芳
Cheng-Fang Su
論文名稱： The well-posedness of Navier-Stokes equations and the incompressible limit of rotational compressible magnetohydrodynamic flows
指導教授： 鄭經斅
Ching-Hsiao Cheng
口試委員:
學位類別： 博士
Doctor
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 123
中文關鍵詞： 不可壓縮極限 、磁流體 、準地轉方程 、適定性 、表面張力
外文關鍵詞： incompressible limits, magnetohydrodynamic flows, relative entropy method, well-posedness
相關次數： 點閱：13  下載：0 
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    • 本論文包含了兩個研究子題。在第一個研究子題裡，我們考慮了磁流體動力學流動的可壓縮模型。給予足夠好的初始值，證明出可壓縮且具有由柯氏力所造成的旋轉項的磁流體之不可壓縮極限，在不同的參數的調整之下，使用相對熵方法，分別以弱收斂的方式推導出帶有黏性項與不帶有黏性項的準地轉方程，且不可壓縮極限即為這些準地轉方程的解，然後我們在這個問題裡證明不同的準地轉方程分別具有全域強解及局部強解。

    在第二個研究子題裡，我們討論帶有表面張力的 Navier-Stokes 方程在自由邊界下的解存在性。為了處理與時間有關的邊界，我們引進 ALE formulation，找到某個良好的微分映射，將原本隨著時間而改變的邊界利用此映射變換後固定，然後經過光滑化、線化性的處理並使用固定點定理，最後證明出此系統的解存在性。

    There are two sub-projects in this dissertation: one is the problem of the incompressible limits for rotational compressible MHD flows and another is the well-posedness of Navier-Stokes equations with surface tension in an optimal Sobolev space.
    　　
    In the first sub-project, we consider the compressible models of magnetohydrodynamic flows which gives rise to a variety of mathematical problems in many areas. We introduce the asymptotic limit for the compressible rotational magnetohydrodynamic flows with the well-prepared initial data such that a rigorous quasi-geostrophic equation with diffusion term governed by the magnetic field from a compressible rotational magnetohydrodynamic flows is derived. After that, we show the two results: the existence of the unique global strong solution of quasi-geostrophic equation with good regularity on the velocity and magnetic field and the derivation of quasi-geostrophic equation with diffusion.
    　　
    In the second sub-project, we establish the existence of a solution to the 2-dimensional Navier-Stokes equations with surface tension on a moving domain. To deal with the free boundary problem, we adopt the ALE formulation which transform the moving domain into a fixed domain. Next, we show the well-posedness of this systems in an optimal sobolev space and no compatibility conditions are required to guarantee the existence of a solution.

    1 Derivation of inviscid quasi-geostrophic equation from rotational compressible magnetohydrodynamic flows 1 1.1 Introduction　1 1.2 Preliminary results　5 1.2.1 Gronwall inequality - Integral form　 5 1.2.2 Interpolation　6 1.2.3 Commutator estimate　6 1.2.4 Polynomial-type inequality 　7 1.2.5 The Aubin-Lions lemma　7 1.3 Main results　7 1.4 Proof of Theorem 1.8　8 1.5 Proof of Theorem 1.9 　24 2 Other Results of the incompressible limits 34 2.1 Introduction　 34 2.2 Main results　35 2.3 Proof of Theorem 2.1 　36 2.4 Proof of Theorem 2.2　47 2.4.1 Uniform bounds 47 2.4.2 Relative entropy inequality　49 2.4.3 Convergence of viscosity and velocity terms　50 2.4.4 Convergence of pressure terms　51 2.4.5 Convergence of quasi-geostropic equation　53 2.4.6 Convergence of magnetic field　56 2.4.7 Convergence of initial data and conclusion　59 3 The Existence of Solutions of 2D Incompressible Navier-Stokes Equations with Surface Tension 61 3.1 Introduction　61 3.1.1 The equations　61 3.1.2 Some prior results　62 3.1.3 The difficulties　63 3.1.4 Outlines 63 3.2 The ALE formulation　64 3.2.1 A map that maps from a fixed reference domain to Ω(t)　64 3.2.2 The representation of some geometric quantities　64 3.2.3 Some basic identities concerning the map　65 3.2.4 The equations in ALE coordinate　66 3.2.5 The evolution equation of h　66 3.2.6 A modification of the ALE formulation　67 3.3 Notation and preliminary results　68 3.3.1 The energy spaces V(T), H(T), W(T), H1(T)　68 3.3.2 A useful lemma　69 3.3.3 The horizontal convolution-by-layers operator and a commutator type estimate　69 3.3.4 The generalized Gronwall inequality 71 3.3.5 The Lagrangian multiplier lemma　71 3.3.6 Elliptic regularity　73 3.4 Main Theorem　75 3.5 The construction of solutions　76 3.5.1 The regularized problem　76 3.5.2 The linearization of the regularized problem　76 3.5.3 Some a priori estimates　78 3.5.4 The construction of solutions to the regularized problem　79 3.6 The "\epsilon-independent estimates　92 3.6.1 Key elliptic estimate　93 3.6.2 The estimate for v＿t in L^2L^2　94 3.6.3 The estimate for v in L^2H^{1.5}　98 3.6.4 The implication of the Stokes regularity　105 3.7 The existence of a solution to the problem　106 3.7.1 The continuation argument　106 3.7.2 The existence of a solution to equation (3.2.10) 107 Bibliography 109

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