Chu Shun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 63-65). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Preliminaries --- p.8 / Chapter 2 --- Singular Integrals --- p.15 / Chapter 2.1 --- Marcinkiewicz Integral --- p.15 / Chapter 2.2 --- Decomposition in cubes of open sets in Rn --- p.17 / Chapter 2.3 --- Interpolation Theorem for Lp --- p.18 / Chapter 2.4 --- Singular Integrals on homogeneous of degree 0 --- p.25 / Chapter 3 --- Solutions to the Euler Equations --- p.36 / Chapter 3.1 --- Existence and Uniqueness of smooth solutions for Euler Equations --- p.36 / Chapter 3.2 --- Rate of Convergence and Decay in Time --- p.43 / Chapter 3.2.1 --- Rate of Convergence --- p.43 / Chapter 3.2.2 --- Lp Decay for Solutions of the Navier-Stokes Equations --- p.46 / Chapter 3.3 --- Weak Solution to the Euler Equations --- p.48 / Chapter 3.3.1 --- Weak Solution to the Velocity Formulation --- p.49 / Chapter 3.3.2 --- Weak Solution to the Vorticity Formulation --- p.52 / Bibliography --- p.63

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

Résumé : Cette thèse est consacrée à l'étude des équations de Stokes et de Navier-Stokes avec des conditions aux limites de Navier dans un ouvert borné de . Le manuscrit ici est composé de trois chapitres. Dans le premier, nous considérons les équations de Stokes stationnaires avec des conditions aux limites de Navier. Nous démontrons l'existence, l'unicité et la régularité de la solution d'abord dans un cadre hilbertien puis dans le cadre de la théorie . Nous traitons aussi le cas de solutions très faibles. Dans le deuxième chapitre, nous nous intéressons aux équations de Navier-Stokes avec la condition de Navier. Sous certaines hypothèses sur les données, nous démontrons l'existence de solution faible dans , avec en utilisant un théorème du point fixe appliqué à un problème d'Oseen. Nous démontrons examinons ensuite les questions de régularité des solutions en particulier dans . Dans le dernier chapitre, nous étudions le problème d'évolution de Stokes avec la condition de Navier. La résolution de ce problème se fait au moyen de la théorie des semi-groupes analytiques qui jouent un rôle important pour établir l'existence et l'unicité de la solution dans le cas homogène. Nous traitons le cas du problème non homogène par le biais des puissances imaginaires de l'opérateur de Stokes. / This thesis is devoted to the study of the Stokes equations and Navier-Stokes equations with Navier boundary conditions in a bounded domain of . The work contains three chapters: In the first chapter, we consider the stationary Stokes equations with Navier boundary condition. We show the existence, uniqueness and regularity of the solution in the Hilbert case and in the -theory. We prove also the case of very weak solutions. In the second chapter, we focus on the Navier-Stokes equations with the Navier boundary condition. We show the existence of the weak solution in , with by a fixed point theorem over the Oseen equation. We show also the existence of the strong solution in . In chapter three, we study the evolution Stokes problem with Navier boundary condition. For this, we apply the analytic semi-groups theory, which plays a crucial role in the study of existence and uniqueness of solution in the case of the homogeneous evolution problem. We treat the case of non-homogeneous problem through imaginary powers of the Stokes operator.

In chapter 1, we focus on the noncharacteristic boundary layers for the parabolic regularization of quasi-linear hyperbolic problems, where the viscosity matrix is positive definite, with the zero Dirichlet boundary conditions. We adapt the method developed by Grenier and Gues [?] where the center-stable manifold theorem is used to prove the existence and exponential decay property of the leading boundary layer profile under suitable conditions on the boundary x = 0. With this boundary condition we prove the well-posedness of the initial boundary value problem of the inviscid flow. Then we prove the stability of the boundary layer by an energy estimate, where exponential decay property of the boundary layer profile plays an important role. Finally, we can specify the limit of the viscous solutions to the corresponding inviscid solution. / In chapter 2, we consider the noncharacteristic one-dimensional compressible full Navier-Stokes equations for the ideal gas with outflow boundary condition on the velocity and suitable initial conditions, which make all the three characteristics to the corresponding Euler equations negative up to some local time, especially on the boundary. By the aymptotic analysis, we derive an algebraic-differential equation for the leading boundary layer functions. The center-stable manifold theorem helps to prove the existence and exponential decay property of the leading boundary layer function. The outflow boundary condition makes it possible to estimate the normal derivatives. Combining this with the tangential derivative estimate, we can recover the H1 estimate of the error term. Thus we establish the stability of the boundary layers which satisfy an algebraic-differential equation in this case. With this stability result, we obtain the relation between the solutions to Navier-Stokes and Euler equations. / In chapter 3, we concentrate on the existence and nonlinear stability of the totally characteristic boundary layer for the quasi-linear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x = 0. We carry out a weighted estimate to the boundary layer equations---Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions. / In this thesis we study three kinds of asymptotic limiting behavior of the solutions to the initial boundary value problem of one-dimensional quasilinear equations with viscosity by carrying out the boundary layer analysis. / Wang, Jing. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 71-01, Section: B, page: 0407. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 107-112). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese.

Finally, we prove that weak solutions to the compressible Navier-Stokes equations with the Navier boundary condition stabilize to static equilibrium states under a fair condition. / First, we show that the most general class of weak solutions to one-dimensional full compressible Navier-Stokes equations do not exhibit vacuum states in a finite time provided that no vacuum is present initially with the minimum physical assumptions on the data. Moreover, two initially non interacting vacuum regions will never meet each other in the future. / Secondly, we construct the local classical solutions to the compressible Navier-Stokes equations for initial vacuum far fields. In this case, we describe the blow-up phenomena of two-dimensional compact support smooth spherically symmetric solutions. When the far field of the initial state is away from vacuum, we obtain the global classical solutions and show the large time blow-up behavior of the gradient of the density. / This thesis deals with some important problems of compressible Navier-Stokes equations, including the well-posedness of the Cauchy problem, the regularity of the weak solutions constructed by Lions and Feireisl, and the dynamics of vacuum states, etc.. / Luo, Zhen. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 152-161). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Wong Chak-fu. / "September 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 451-475). / 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.

Guo Zhenhua. / "June 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 79-84). / 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.

Orientador: Paulo Fernando de Arruda Mancera / Banca: Helenice de Oliveira Florentino Silva / Banca: Valdemir Garcia Ferreira / Resumo: Considera-se a construção de métodos compacto e não compacto de quarta ordem para resolver numericamente as equações de Navier-Stokes na formulação função corrente em uma malha uniforme. Aplica-se esses métodos de alta ordem em um canal-tipo estenose e o conjunto das equações não lineares resultantes da discretização é resolvido pelo método de Newton. Erros RMS e máximo, bem como linhas de corrente são apresentados / Abstract: This work considers the development of compact and wide fourth-order schemes for solving the Navier-Stokes equations in the streamfunction formulation on a uniform grid. These high order schemes are applied in a stenosis channel-type and the set of nonlinear equations resulting from the discretization is solved by Newton's method. The RMS and maximum errors, and also the streamlines are shown / Mestre

Neste trabalho desenvolvemos um algoritmo para a geração de malhas quadtree com o objetivo de utilizá-las na simulação de escoamento de fluidos, onde muitas vezes faz-se necessário o uso de malhas finas. A idéia central da geração das malhas quadtree está baseada na estrutura de árvore quaternária em que cada nodo possui quatro filhos. Assim uma estrutura de árvore está associada a uma malha que pode ter espaçamento uniforme ou refinamento em regiões específicas. Para evitar problemas de discretização, uma malha quadtree deve satisfazer um critério chamado de balanceamento e este é tratado de forma detalhada no desenvolvimento do trabalho. Podemos destacar também outro ponto importantíssimo na implementação de malhas quadtree, que é a busca de vizinhos dos nodos. Além disso apresentamos a discretização dos operadores em diferenças finitas, que é feita a partir do conhecimento dos vizinhos dos quadrantes da malha e de técnicas de interpolação. A ordem do método é verificada a partir de testes com a equação do calor, tanto para malhas uniformes quanto para malhas com níveis de refinamento diferenciados e concluímos, assim, que o método desenvolvido e apresentado é satisfatório. / In this work we developed an algorithm to generate meshes quadtree in order to use them to simulate uid ow, which often makes it necessary to use ne meshes. The central idea of the generation of quadtree meshes is based on the quaternary structure of the tree, where each node has four children, and thus a tree structure is associated with a mesh that can be evenly or locally re ned. To avoid problems of discretization, a quadtree mesh must satisfy a criterion called the balancing and this is addressed in detail in the development of this work. We also highlight another important point in the implementation of quadtree meshes, which is the search for neighboring nodes. Additionally, we present the nite di erences discretization of operators, which uses the knowledge of the mesh quadrant neighboors and interpolation techniques. The order of the method is checked by tests with the heat equation for both uniform meshes and for meshes with di erent levels of re nement and we conclude that method here presented is satisfactory.

Dans cette thèse, nous nous intéressons à deux approximations hyperboliques des équations de Navier-Stokes incompressibles en dimensions 2 et 3 d'espace. Dans un premier temps, on considère une perturbation hyperbolique de l'équation de la chaleur, introduite par Cattaneo en 1949, pour remédier au paradoxe de la propagation instantanée de cette équation. En 2004, Brenier, Natalini et Puel remarquent que la même perturbation, qui consiste à rajouter ε∂tt à l'équation, intervient en relaxant les équations d'Euler. En dimension 2, les auteurs montrent que, pour des sonnées régulières et sous certaines hypothèses de petitesse, la solution globale de la perturbation converge vers l'unique solution globale de (NS). En 2007, Paicu et Raugel améliorent les résultats de [BNP] en étendant la théorie à la dimension 3 et en prenant des données beaucoup moins régulières. Nous avons obtenu des résultats de convergence, avec données de régularité quasi-critique, qui complètent et prolongent ceux de [BNP] et [PR]. La seconde approximation que l'on considère est un nouveau modèle hyperbolique à vitesse de propagation finie. Ce modèle est obtenu en pénalisant la contrainte d'incompressibilité dans la perturbation de Cattaneo. Nous démontrons que les résultats d'existence globale et de convergence du précédent modèle sont encore vérifiés pour celui-ci. / In this work, we are interested in two hyperbolic approximations of the 2D and 3D Navier-Stokes equations. The first model we consider comes from Cattaneo's hyperbolic perturbation of the heat equation to obtain a finite speed of propagation equation. Brenier, Natalini and Puel studied the same perturbation as a relaxed version of the 2D Euler equations and proved that the solution to this relaxation converges towards the solution to (NS) with smooth data, provided some smallness assumptions. Later, Paicu and Raugel improved their results, extending the theory to the 3D setting and requiring significantly less regular data. Following [BNP] and [PR], we prove global existence and convergence results with quasi-critical regularity assumptions on the initial data. In the second part, we introduce a new hyperbolic model with finite speed of propagation, obtained by penalizing the incompressibility constraint in Cattaneo's perturbation. We prove that the same global existence and convergence results hold for this model as well as for the first one.