Next, we consider that the fluids are isentropic and the domain is also bounded, smooth, simply connected in R2 . We show that the estimates are uniform in all time if the smallness assumption on the initial data is prescribed. It follows that the solutions of compressible Navier-Stokes equations converge to the incompressible ones uniformly in both spatial and temporal variables as the Mach number vanishes. / This thesis deals with the low Mach number limit of the compressible Navier-Stokes equations. It is to verify that the compressible fluids become incompressible as Mach number tends to zero. In another words, the pressure due to compression can be neglected. This is a singular limit. / We will show that, as the Mach number tends to zero, the local smooth solutions of compressible Navier-Stokes equations with zero thermal conductivity coefficient converge strongly to the solutions of incompressible Navier-Stokes equations, provided that the initial data satisfy the "bounded derivative conditions". The key point, which is one of our main contributions, is the uniform high norm estimates in Mach number. We will study two cases. The first case is that, the domain is a finite interval and the boundary condition for the velocity is no-slip. In the second case, the domain is bounded, smooth, and simply connected in R2 . The boundary condition for the velocity is replaced by the slip-type's, thus the vorticity and the divergence of velocity can be estimated separately. / Ou, Yaobin. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3546. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 107-111). / 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.

Yu, Rongfeng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 57-63). / Abstracts in English and Chinese. / Introduction --- p.3 / Chapter 1 --- Preliminaries --- p.11 / Chapter 1.1 --- Notations and function spaces --- p.11 / Chapter 1.2 --- Some useful inequalities --- p.12 / Chapter 1.3 --- Fundamental lemmas --- p.15 / Chapter 2 --- Compressible Navier-Stokes Equations for Quantum Fluids --- p.16 / Chapter 2.1 --- Background --- p.17 / Chapter 2.2 --- Derivation of model --- p.17 / Chapter 3 --- Global Weak Solutions to Barotropic Navier-Stokes Equations for Quantum Fluids --- p.22 / Chapter 3.1 --- Reformulation and main results --- p.23 / Chapter 3.2 --- Construction of approximate solutions --- p.27 / Chapter 3.3 --- A priori estimates --- p.39 / Chapter 3.4 --- Proof of Theorem 3.1.6 --- p.40 / Chapter 4 --- Global Existence and Large Time Behavior of Weak Solutions to Quantum Navier-Stokes-Poisson Equa-tions --- p.46 / Chapter 4.1 --- Global existence of weak solutions --- p.47 / Chapter 4.2 --- Large time behavior --- p.50 / Chapter 5 --- Discussions and Future Work --- p.55 / Bibliography --- p.56

We present an immersed interface algorithm for the incompressible Navier Stokes equations. The interface is represented by cubic splines which are interpolated through a set of Lagrangian control points. The position of the control points is implicitly updated using the fluid velocity. The forces that the interface exerts on the fluid are computed from the constitutive relation of the interface and are applied to the fluid through jumps in the pressure and jumps in the derivatives of pressure and velocity. A projection method is used to time advance the Navier-Stokes equations on a uniform cartesian mesh. The Poisson-like equations required for the implicit solution of the diffusive and pressure terms are solved using a fast Fourier transform algorithm. The position of the interface is updated implicitly using a quasi-Newton method (BFGS) within each timestep. Several examples are presented to illustrate the flexibility of the presented approach. / Singapore-MIT Alliance (SMA)

A new construction of regular solutions to the three dimensional Navier{Stokes equa- tions is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderl??of type estimates are used to establish su??- cient conditions for the convergence of those expansions. The construction also de??nes a system of inhomogeneous di??erential equations, called the extended Navier{Stokes equations, which turns out to have global solutions in suitably constructed normed spaces. Moreover, in these spaces, the normal form of the Navier{Stokes equations associated with the terms of the asymptotic expansions is a well-behaved in??nite system of di??erential equations. An application of those asymptotic expansions of regular solutions is the analysis of the helicity for large times. The dichotomy of the helicity's asymptotic behavior is then established. Furthermore, the relations between the helicity and the energy in several cases are described.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 73-77).

Thesis (M.S. in Aeronautical Engineering)--Naval Postgraduate School, September 2003. / Thesis advisor(s): Kevin D. Jones, Max F. Platzer. Includes bibliographical references (p. 55-56). Also available online.

Thesis--University of Kentucky (M.S.), 2003. / Title from document title page. Document formatted into pages; contains ix, 110 p. : ill. Includes abstract and vita. Includes bibliographical references (p. 99-109).

Thesis (Aeronautical and Astronautical Engineer)--Naval Postgraduate School, December 2002. / Thesis advisor(s): Kevin James, Max Platzer. Includes bibliographical references (p. 139). Also available online.

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