The well-posedness of Navier-Stokes equations and the incompressible limit of rotational compressible magnetohydrodynamic flows

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姓名	蘇承芳(Cheng-Fang Su) 查詢紙本館藏	畢業系所	數學系
論文名稱	(The well-posedness of Navier-Stokes equations and the incompressible limit of rotational compressible magnetohydrodynamic flows)
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摘要(中)	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。
摘要(英)	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.
關鍵字(中)	★ 不可壓縮極限 ★ 磁流體 ★ 準地轉方程 ★ 適定性 ★ 表面張力	關鍵字(英)	★ incompressible limits ★ magnetohydrodynamic flows ★ relative entropy method ★ Navier-Stokes equations ★ well-posedness ★ surface tension
論文目次	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
參考文獻	[1] C Grandmont, Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate, SIAM J. Math. Anal., 40(2), (2007), 716–737. [2] Xinfu Chen and Fernando Reitich, Local existence and uniqueness of solutions of the Stefan Problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164(2), (1992), pp. 350–362 3.1.2. [3] C.H. A. Cheng, D. Coutand and S. Shkoller Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., 39, (2007), 742–800 3.1.2 [4] C.H. A. Cheng, D. Coutand and S. Shkoller The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., 42, (2010), pp. 1094–1155 3.1.2 [5] D. Coutand and S. Shkoller. Well-posedness of the free-surface incompressible Euler equations with or without surface. Amer. Math. Soc. 20, 829–930 2007. 1.2.4 [6] D. Coutand and S. Shkoller Unique Solvability of the Free Boundary Navier-Stokes Equations with Surface Tension, preprint 3.1.2 [7] D. Coutand and S. Shkoller Motion of an elastic solid inside of an incompressible viscous fluid, Arch. Rational Mech. Anal. 176(1), (2005), pp. 25–102 [8] D. Coutand and S. Shkoller On the interaction between quasilinear elasto-dynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal. 179(3), (2006), pp. 303–352. [9] B. Desjardins and E. Grenier. Low Mach Number Limit of Viscous Compressible Flows in the Whole Space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. ,455(1986): 2271–2279 1999. 1.5, 2.4.1 [10] K. Ecker and G. Huisken. Interior estimates for hypersurfaces moving by mean curvature. Inventiones mathematicae. ,105(3):547–570 1991. [11] E. Feireisl and Novotny. Dynamics of viscous compressible fluids. [12] E. Feireisl and Novotny. Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Comm. Math. Phys. ,321(3):605–628 2013. 1.1 [13] E. Feireisl and Novotny. The low Mach number limit for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal.,186(1):77–107 2007. 1.1 [14] E. Feireisl and Novotny. Singular Limits in Thermodynamics of Viscous Fluids. [15] E. Feireisl, B. Jin, and Novotny. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech.,14(4):717–730 2012. 2.4.2 [16] X. Hu, and D. Wang. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Ration. Mech. Anal. , 197:203–238, 2010. 1.1 [17] X. Hu, and D. Wang. Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. , 41(3):1272–1294, 2009. 2.1 [18] S. Jiang, Q. Ju, and F. Li. Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Comm. Math. Phys. ,297(2): 371–400 2010. 2.1 [19] S. Jiang, Q. Ju, and F. Li. Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients. SIAM J. Math. Anal. ,42(6):2539–2553 2010. 2.1 [20] S. Jiang, Q. Ju, and F. Li. Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations. Nonlinearity,25(5):1351–1365 2012. 2.1 [21] S. Jiang, Q. Ju, F. Li, and Z. Xin. Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data. Adv. Math. ,259:384–420 2014. 2.1 [22] Y.-S. Kwon, and K. Trivisa. On the incompressible limits for the full magnetohydrody-namics flows. J. Differential Equations ,251(7):1990–2023 2011. 2.1 [23] J.L. Lions. Quelque methodes de resolution des problemes aux limites non lineaires. Dunod-Gauth, Paris, France, 1969. 1.2.5 [24] A. J‥ungel, C.K, Lin, and K.C. Wu. An Asymptotic Limit of a Navier–Stokes System with Capillary Effects. Comm. Math. Phys. (2)329, 725–744 2014. 1.1, 1.5 [25] M. E. Bogovskii. Solution of some vector analysis problems connected with operators Trudy Sem. S.L. obolev, 80(1):5–40, 1980. [26] P.-L. Lions, and N. Masmoudi. Incompressible limit for a viscous compressible fluid J. Math. Pures Appl. (9), 77 585–627, 1998. 1.1 [27] O.A. Ladyzenskaja, V.A. Solonnikov, and N. N. Uralceva. Linear and quasi-linear equations of parabolic type. American Mathematical Society. 1968. 1.2.2 [28] T.R. Bose. High temperature gas dynamics. Springer-Verlag, Berlin, 2004. [29] N. Masmoudi. Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincare Anal. Non Lineaire. (2), 18 199–224, 2001. 1.1 [30] V. A. Solonnikov and A. Tani, Free boundary problem for the Navier–Stokes equations for a compressible fluid with a surface tension, Boundary-value problems of mathematical physics and related problems of function theory. Part 21, Zap. Nauchn. Sem. LOMI, 182, ”Nauka”, Leningrad. Otdel., Leningrad, 1990, pp. 142–148 3.1.2 [31] V.A. Solonnikov and A. Tani, Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid, The Navier-Stokes equations II-theory and numerical methods (Oberwolfach, 1991), Lecture Notes in Math., 1530, Springer, Berlin, (1992), pp. 30-55.
指導教授	鄭經?(Ching-Hsiao Cheng)	審核日期	2018-7-26
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