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

by Choi Chun-Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 56-58). / Abstract also in Chinese. / Chapter 1 --- Radial symmetry for solutions of a semilinear el- liptic equation on a bounded domain --- p.6 / Chapter 2 --- Asymptotic symmetry of singular solutions to a semilinear elliptic equation --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Some preliminary analysis --- p.14 / Chapter 2.3 --- Proof of Theorem 2.1 --- p.20 / Chapter 3 --- Classification of non-negative solutions to Yamabe type equations --- p.32 / Chapter 3.1 --- Introduction --- p.32 / Chapter 3.2 --- The Proof of Theorem 3.1 for k > 0 --- p.38 / Chapter 3.3 --- Case k <0 --- p.48 / Bibliography

Wang Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Some Preparations --- p.10 / Chapter 3 --- Proof of Theorem 1.1 --- p.24 / Chapter 4 --- Location of Blow-up Points (for n=2) --- p.26 / Chapter 5 --- Location of Blow-up Points (for General n) --- p.35 / Chapter 6 --- Asymptotic behavior of solutions near blow-up point --- p.46 / Chapter 7 --- Appendix --- p.57 / Bibliography --- p.61

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Includes bibliographical references. / by Robert R. Reitano. / Ph.D.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 68-70. / Ph.D.

Photocopy of typescript. / Digitized by Kansas Correctional Industries

Thesis. 1977. Sc.D.--Massachusetts Institute of Technology. Dept. of Physics. / M̲i̲c̲ṟo̲f̲i̲c̲ẖe̲ c̲o̲p̲y̲ a̲v̲a̲i̲ḻa̲ḇḻe̲ i̲ṉ A̲ṟc̲ẖi̲v̲e̲s̲ a̲ṉḏ S̲c̲i̲e̲ṉc̲e̲. / Vita. / Bibliography : leaves 167-170. / by Steven R. Rogers. / Sc.D.

Lee, Ho Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 75-79). / Abstracts in English and Chinese. / Chapter 1 --- Upscaling methods for high contrast problems --- p.6 / Chapter 1.1 --- Review on upscaling methods --- p.7 / Chapter 1.2 --- Upscaling method with high contrast of the conductivity --- p.11 / Chapter 2 --- Multiscale finite element methods for high contrast problems --- p.19 / Chapter 2.1 --- Review on Multiscale finite element methods --- p.20 / Chapter 2.2 --- Local spectral basis functions --- p.23 / Chapter 2.3 --- Discussion for MsFEM with spectral multiscale basis functions . --- p.25 / Chapter 3 --- Elliptic equations in high-contrast heterogeneous media --- p.28 / Chapter 3.1 --- Preliminaries --- p.29 / Chapter 3.2 --- Integral representation --- p.32 / Chapter 3.3 --- The well-posedness of the integral equation --- p.37 / Chapter 4 --- A numerical approach for the Elliptic equations in high-contrast heterogeneous media --- p.45 / Chapter 4.1 --- Introduction --- p.46 / Chapter 4.2 --- A new approach --- p.47 / Chapter 4.3 --- Discussion of the results --- p.50 / Chapter 4.4 --- Numerical experiment --- p.51 / Chapter 5 --- A preconditioner for high contrast problems using reduced-contrast approximations --- p.62 / Chapter 5.1 --- Reduced-contrast approximations for the solution of elliptic equations --- p.63 / Chapter 5.2 --- Review on multigrid methods --- p.66 / Chapter 5.3 --- Preconditioning and numerical experiments --- p.70 / Bibliography --- p.75

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.

Firstly, we establish the existence of a solution u epsilon concentrating along a curve Gammaepsilon near the non-degenerate Gamma, exponentially small in epsilon at any positive distance from the curve, provided epsilon is small and away from certain critical numbers. The concentrating curve Gammaepsilon will collapse to Gamma as epsilon &rarr; 0. / In this thesis, we consider the following problem 32Du-u+up= 0 and u&gt;0 in W , 6u6n= 0 on 6W, where O is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the outward normal of O and p > 1. Let Gamma be a straight line intersecting orthogonally with &part;O at exactly two points. We use the infinite dimensional Lyapunov-Schmidt reduction method, introduced by M. del Pino, M. Kowalczyk and J. Wei in [14], to deal with the non-invertibility caused by the critical eigenvalues of the linearized operator in the perturbed problems and then construct interior concentration layers near Gamma, which interact with the boundary. Moreover, the method of successive improvements of the approximation helps us decompose the interaction between the boundary and the interior layers. / Secondly, for any given integer N with N &ge; 2 and for small epsilon away from certain critical numbers, we construct another solution uepsilon exhibiting N concentration layers at mutual distances O(epsilon&mid; ln epsilon&mid;), whose concentration set will approach the non-degenerate and non-minimal Gamma as epsilon &rarr; 0, provided that the exponent p &ge; 2. Asymptotic location of these layers is governed by a Toda type system. / Yang, Jun. / "July 2007." / Adviser: Juncheng Wei. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 129-136). / 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, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.