In this thesis, we consider the non-uniqueness of weak solutions to the compressible isentropic Euler equations on a bounded domain. We establish a criterion for the existence of infinitely many solutions with the same initial data. Using this criterion, we prove that for general smooth initial density and velocity, there exists infinitely many weak solutions. Furthermore, given smooth initial density bounded away from zero, we can construct a initial velocity field such taht there exists infinitely many weak solutions with this initial data, and these solutions satisfy the entropy inequality for a positive finite time. The results also generalizes to the free boundary value problems. / 本論文考慮可壓等熵歐拉方程在有界區域上其弱解的非唯一性。我們建立了關於存在無窮多個有同一初始狀態的弱解的判別準則。利用此準則，我們證明了對於一般的光滑的初始密度和速度場，存在無窮多個弱解:進一步的，對於光滑有正下界的初始密度場，我們構造了一小初始速度場，證明存在滿足這樣初始資料的無窮多個弱解，並且，在一個有限時間內，這些弱解滿足熵不等式。這些結果可以推廣到自由邊界情形。 / Detailed summary in vernacular field only. / Luo, Tianwen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 55-58). / Abstracts also in Chinese. / Introduction --- p.6 / Chapter 1 --- Preliminaries --- p.18 / Chapter 1.1 --- Notational Conventions and Function Spaces --- p.18 / Chapter 1.2 --- Elementary Inequalities --- p.19 / Chapter 1.3 --- Fundamental Lemmas --- p.21 / Chapter 2 --- Formulation and Main Results --- p.22 / Chapter 2.1 --- Mathematical Formulation --- p.23 / Chapter 2.2 --- Main Theorems and the Criterion --- p.24 / Chapter 2.3 --- The Space of Subsolutions --- p.27 / Chapter 3 --- Proof of the Main Theorems --- p.30 / Chapter 3.1 --- Proof of the Subsolution Lemma --- p.31 / Chapter 3.2 --- The Perturbation Property --- p.34 / Chapter 3.3 --- Construction of Suitable Initial Data --- p.39 / Chapter 3.4 --- Convex Integration --- p.43 / Chapter 4 --- Free Boundary Problem --- p.47 / Chapter 4.1 --- Formulation and Results --- p.48 / Chapter 4.2 --- Remarks on the Proofs --- p.51 / Chapter 5 --- Discussions and Future Work --- p.54 / Bibliography --- p.55

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Environmental flows (e.g. river and atmospheric flows) governed by the shallow water equations (SWE) are usually dominated by the advective mechanism over multiple time-scales. The combination of time dependency and nonlinear advection creates difficulties in the numerical solution of the SWE. A fully-implicit scheme is desirable because a relatively large time step may be used in a simulation. However, nonlinearity in a fully implicit method results in a system of nonlinear equations to be solved at each time step. To address this difficulty, a new method for implicit solution of unsteady nonlinear advection equations is developed in this research. This Time-Centered Split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically-nonlinear coupled equations for any number of variables. To explore its characteristics, the TCS algorithm is first applied to onedimensional problems and compared to the conventional nonlinear solution methods. The temporal accuracy and practical stability of the method is confirmed using these 1D examples. It is shown that TCS can computationally linearize unsteady nonlinear advection problems without either 1) outer iteration or 2) calculation of the Jacobian. A family of the TCS method is created in one general form by introducing weighting factors to different terms. We prove both analytically and by examples that the value of the weighting factors does not affect the order of accuracy of the scheme. In addition, the TCS method can not only computationally linearize but also decouple an equation system of coupled variables using special combinations of weighting factors. Hence, the TCS method provides flexibilities and efficiency in applications. / text

Bibliography: leaves 70-72 / vi, 73 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1987

Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2007. / Typescript. Includes bibliographical references (leaves 143-151).